import Mathlib import OAI.RepresentationTheory.SignedTensor.Basic import OAI.RepresentationTheory.Tensor.Postselection import OAI.Analysis.Matrix.TensorMoment namespace OAI noncomputable section open scoped BigOperators Matrix.Norms.L2Operator ComplexOrder attribute [local instance] Classical.propDecidable noncomputable section open scoped BigOperators ComplexConjugate Matrix.Norms.L2Operator attribute [local instance] Classical.propDecidable noncomputable section open scoped BigOperators ComplexOrder MatrixOrder attribute [local instance] Classical.propDecidable namespace CoordinateSweeps.SignedTensor open PositiveTensor /- Partial-trace invariance of the actual regrouping signs on each fixed exception sector. No assumption of complex conditional probabilities occurs. -/ theorem partialTrace_regroup {p l : ℕ} (E : Finset (Fin l)) (v : Word p l × Word p l → ℂ) (hv : ∀ u w, v (u,w) ≠ 0 → ∀ i, (odd (u i) ↔ ¬ odd (w i)) ↔ i ∈ E) : partialTrace (PositiveTensor.outer (fun uw => regroupPhase uw.1 uw.2 * v uw)) = partialTrace (PositiveTensor.outer v) := by ext u u' apply Finset.sum_congr rfl intro w _ change (regroupPhase u w * v (u,w)) * star (regroupPhase u' w * v (u',w)) = v (u,w)*star (v (u',w)) by_cases hu : v (u,w) = 0 · simp [hu] by_cases hu' : v (u',w) = 0 · simp [hu'] have he : regroupPhase u w = regroupPhase u' w := by apply regroupPhase_of_same_parity intro i have hi := hv u w hu i have hi' := hv u' w hu' i tauto rw [star_mul,regroupPhase_star,he] calc _ = (regroupPhase u' w * regroupPhase u' w) * (v (u,w)*star (v (u',w))) := by ring _ = _ := by rw [regroupPhase_square,one_mul] abbrev Pair (p : ℕ) := Letter p × Letter p abbrev MatchedPair (p : ℕ) := {a : Pair p // odd a.1 ↔ odd a.2} /- Valid pinned exceptional data. Injectivity is essential: it is supplied by the proved repeated-mismatch cancellation, not imposed on the kernel. -/ structure ExceptionPattern (p l : ℕ) where sites : Finset (Fin l) pair : sites → Pair p mismatch : ∀ i, odd (pair i).1 ↔ ¬ odd (pair i).2 injective : Function.Injective pair deriving Fintype /- Every nonzero commuting matrix entry has literal valid exceptional data. -/ def entryPattern {p l : ℕ} (A : commutant p l) (u w : Word p l) (h : A.val u w ≠ 0) : ExceptionPattern p l where sites := mismatchSlots u w pair i := (u i.val,w i.val) mismatch i := (Finset.mem_filter.mp i.property).2 injective := by intro i j he apply Subtype.ext by_contra hij exact h (commutant_entry_zero_repeated_mismatch A u w i.val j.val hij (congrArg Prod.fst he) (congrArg Prod.snd he) (Finset.mem_filter.mp i.property).2) namespace ExceptionPattern variable {p l : ℕ} def encode (F : ExceptionPattern p l) (a : Pair p) : Option (Fin l) := if h : ∃ i : F.sites, F.pair i = a then some (Classical.choose h).val else none lemma encode_eq_some (F : ExceptionPattern p l) (a : Pair p) (i : Fin l) : F.encode a = some i ↔ ∃ hi : i ∈ F.sites, F.pair ⟨i,hi⟩ = a := by unfold encode split_ifs with h · constructor · intro he have hi := Option.some.inj he subst i exact ⟨(Classical.choose h).property, Classical.choose_spec h⟩ · rintro ⟨hi,he⟩ have hh : Classical.choose h = ⟨i,hi⟩ := F.injective ((Classical.choose_spec h).trans he.symm) rw [hh] · constructor · intro he; cases he · rintro ⟨hi,he⟩ exact (h ⟨⟨i,hi⟩,he⟩).elim lemma mem_sites_iff (F : ExceptionPattern p l) (i : Fin l) : i ∈ F.sites ↔ ∃ a, F.encode a = some i := by constructor · intro hi exact ⟨F.pair ⟨i,hi⟩,(F.encode_eq_some _ i).mpr ⟨hi,rfl⟩⟩ · rintro ⟨a,ha⟩ exact ((F.encode_eq_some a i).mp ha).choose lemma encode_injective : Function.Injective (encode (p := p) (l := l)) := by intro F G he have hs : F.sites = G.sites := by ext i rw [F.mem_sites_iff,G.mem_sites_iff] simp only [he] cases F with | mk E f hf hfi => cases G with | mk E' g hg hgi => dsimp at hs subst E' have hp : f = g := by funext i have hh : (ExceptionPattern.mk E f hf hfi).encode (f i) = some i.val := ((ExceptionPattern.mk E f hf hfi).encode_eq_some _ _).mpr ⟨i.property,rfl⟩ rw [congrFun he (f i)] at hh obtain ⟨hi,hv⟩ := ((ExceptionPattern.mk E g hg hgi).encode_eq_some _ _).mp hh exact hv.symm subst g rfl lemma card_le : Fintype.card (ExceptionPattern p l) ≤ (l+1)^((2*p)*(2*p)) := by have hc := Fintype.card_le_of_injective (encode (p := p) (l := l)) encode_injective simpa only [Fintype.card_fun,Fintype.card_option,Fintype.card_fin, Fintype.card_prod] using hc lemma sites_card_le (F : ExceptionPattern p l) : F.sites.card ≤ (2*p)*(2*p) := by have hc := Fintype.card_le_of_injective F.pair F.injective simpa only [Fintype.card_coe,Fintype.card_prod,Fintype.card_fin] using hc /- Slots not occupied by a pinned exceptional symbol. -/ abbrev Free (F : ExceptionPattern p l) := {i : Fin l // i ∉ F.sites} abbrev FreeWord (F : ExceptionPattern p l) := F.Free → MatchedPair p def fillPair (F : ExceptionPattern p l) (x : F.FreeWord) (i : Fin l) : Pair p := if hi : i ∈ F.sites then F.pair ⟨i,hi⟩ else (x ⟨i,hi⟩).val def fill (F : ExceptionPattern p l) (x : F.FreeWord) : Word p l × Word p l := (fun i => (F.fillPair x i).1, fun i => (F.fillPair x i).2) lemma fillPair_at_pin (F : ExceptionPattern p l) (x : F.FreeWord) (i : F.sites) : F.fillPair x i.val = F.pair i := by simp [fillPair,i.property] lemma fillPair_at_free (F : ExceptionPattern p l) (x : F.FreeWord) (i : F.Free) : F.fillPair x i.val = (x i).val := by simp [fillPair,i.property] lemma fill_mismatch_iff (F : ExceptionPattern p l) (x : F.FreeWord) (i : Fin l) : (odd ((F.fill x).1 i) ↔ ¬ odd ((F.fill x).2 i)) ↔ i ∈ F.sites := by change (odd (F.fillPair x i).1 ↔ ¬ odd (F.fillPair x i).2) ↔ _ by_cases hi : i ∈ F.sites · rw [show F.fillPair x i = F.pair ⟨i,hi⟩ from F.fillPair_at_pin x ⟨i,hi⟩] exact iff_of_true (F.mismatch ⟨i,hi⟩) hi · rw [show F.fillPair x i = (x ⟨i,hi⟩).val from F.fillPair_at_free x ⟨i,hi⟩] have hm := (x ⟨i,hi⟩).property tauto lemma fill_injective (F : ExceptionPattern p l) : Function.Injective F.fill := by intro x y he funext i apply Subtype.ext have hh : F.fillPair x i.val = F.fillPair y i.val := Prod.ext (congrFun (congrArg Prod.fst he) i.val) (congrFun (congrArg Prod.snd he) i.val) simpa only [F.fillPair_at_free] using hh lemma fill_disjoint {F G : ExceptionPattern p l} (x : F.FreeWord) (y : G.FreeWord) (he : F.fill x = G.fill y) : F = G := by have hs : F.sites = G.sites := by ext i rw [← F.fill_mismatch_iff x i,← G.fill_mismatch_iff y i,he] cases F with | mk E f hf hfi => cases G with | mk E' g hg hgi => dsimp at hs subst E' have hp : f = g := by funext i have hh : (ExceptionPattern.mk E f hf hfi).fillPair x i.val = (ExceptionPattern.mk E g hg hgi).fillPair y i.val := Prod.ext (congrFun (congrArg Prod.fst he) i.val) (congrFun (congrArg Prod.snd he) i.val) simpa only [fillPair_at_pin] using hh subst g rfl lemma entry_in_range (A : commutant p l) (u w : Word p l) (h : A.val u w ≠ 0) : ∃ x : (entryPattern A u w h).FreeWord, (entryPattern A u w h).fill x = (u,w) := by let F := entryPattern A u w h let x : F.FreeWord := fun i => ⟨(u i.val,w i.val),by have hi : i.val ∉ mismatchSlots u w := i.property have hm : ¬ (odd (u i.val) ↔ ¬ odd (w i.val)) := by simpa only [mismatchSlots,Finset.mem_filter,Finset.mem_univ,true_and] using hi tauto⟩ refine ⟨x,?_⟩ have he (i : Fin l) : F.fillPair x i = (u i,w i) := by unfold fillPair split_ifs <;> rfl exact Prod.ext (funext fun i => congrArg Prod.fst (he i)) (funext fun i => congrArg Prod.snd (he i)) /- Extend a permutation of the free slots by the identity on the exceptions. -/ def extendPerm (F : ExceptionPattern p l) (σ : Equiv.Perm F.Free) : Equiv.Perm (Fin l) := Equiv.Perm.ofSubtype σ lemma extendPerm_pin (F : ExceptionPattern p l) (σ : Equiv.Perm F.Free) (i : F.sites) : F.extendPerm σ i.val = i.val := by simp [extendPerm,Equiv.Perm.ofSubtype_apply_of_not_mem,i.property] lemma extendPerm_free (F : ExceptionPattern p l) (σ : Equiv.Perm F.Free) (i : F.Free) : F.extendPerm σ i.val = (σ i).val := by exact Equiv.Perm.ofSubtype_apply_of_mem σ i.property lemma fill_transport (F : ExceptionPattern p l) (σ : Equiv.Perm F.Free) (x : F.FreeWord) : F.fill (fun i => x (σ⁻¹ i)) = (wordPerm (F.extendPerm σ) (F.fill x).1, wordPerm (F.extendPerm σ) (F.fill x).2) := by have hinv : (F.extendPerm σ)⁻¹ = F.extendPerm σ⁻¹ := by simp only [extendPerm,map_inv] have he (i : Fin l) : F.fillPair (fun i => x (σ⁻¹ i)) i = F.fillPair x ((F.extendPerm σ)⁻¹ i) := by rw [hinv] by_cases hi : i ∈ F.sites · rw [show F.extendPerm σ⁻¹ i = i from F.extendPerm_pin σ⁻¹ ⟨i,hi⟩] simp [fillPair,hi] · rw [show F.extendPerm σ⁻¹ i = (σ⁻¹ ⟨i,hi⟩).val from F.extendPerm_free σ⁻¹ ⟨i,hi⟩] rw [show F.fillPair (fun i => x (σ⁻¹ i)) i = (x (σ⁻¹ ⟨i,hi⟩)).val from F.fillPair_at_free (fun i => x (σ⁻¹ i)) ⟨i,hi⟩,F.fillPair_at_free] exact Prod.ext (funext fun i => congrArg Prod.fst (he i)) (funext fun i => congrArg Prod.snd (he i)) def purified (F : ExceptionPattern p l) (A : commutant p l) (x : F.FreeWord) : ℂ := regroupPhase (F.fill x).1 (F.fill x).2 * A.val (F.fill x).1 (F.fill x).2 lemma purified_invariant (F : ExceptionPattern p l) (A : commutant p l) (σ : Equiv.Perm F.Free) (x : F.FreeWord) : F.purified A (fun i => x (σ⁻¹ i)) = F.purified A x := by unfold purified rw [F.fill_transport] apply regrouped_commutant_invariant intro i hi exact F.extendPerm_pin σ ⟨i,(F.fill_mismatch_iff x i).mp hi⟩ end ExceptionPattern end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor.ExceptionPattern open PositiveTensor variable {p l : ℕ} def embedding (F : ExceptionPattern p l) : F.FreeWord ↪ (Word p l × Word p l) := ⟨F.fill,F.fill_injective⟩ def component (F : ExceptionPattern p l) (A : commutant p l) (uw : Word p l × Word p l) : ℂ := if uw ∈ Set.range F.fill then A.val uw.1 uw.2 else 0 lemma component_image (F : ExceptionPattern p l) (A : commutant p l) (x : F.FreeWord) : F.component A (F.fill x) = A.val (F.fill x).1 (F.fill x).2 := by simp [component] lemma component_inclusion (F : ExceptionPattern p l) (A : commutant p l) : F.component A = (inclusion F.embedding).mulVec (fun x => A.val (F.fill x).1 (F.fill x).2) := by funext q by_cases hq : q ∈ Set.range F.fill · obtain ⟨x,rfl⟩ := hq have hh := inclusion_mulVec_image F.embedding (fun x => A.val (F.fill x).1 (F.fill x).2) x change (inclusion F.embedding).mulVec (fun x => A.val (F.fill x).1 (F.fill x).2) (F.fill x) = _ at hh exact (F.component_image A x).trans hh.symm · rw [show F.component A q = 0 from ite_eq_right hq] exact (inclusion_mulVec_notin F.embedding _ q hq).symm lemma purified_inclusion (F : ExceptionPattern p l) (A : commutant p l) : (inclusion F.embedding).mulVec (F.purified A) = fun q => regroupPhase q.1 q.2 * F.component A q := by funext q by_cases hq : q ∈ Set.range F.fill · obtain ⟨x,rfl⟩ := hq have hh := inclusion_mulVec_image F.embedding (F.purified A) x change (inclusion F.embedding).mulVec (F.purified A) (F.fill x) = _ at hh rw [hh,F.component_image] rfl · rw [inclusion_mulVec_notin F.embedding _ q hq] simp [component,hq] lemma component_partialTrace (F : ExceptionPattern p l) (A : commutant p l) : partialTrace (PositiveTensor.outer ((inclusion F.embedding).mulVec (F.purified A))) = partialTrace (PositiveTensor.outer (F.component A)) := by rw [F.purified_inclusion] apply partialTrace_regroup F.sites intro u w h i have hq : (u,w) ∈ Set.range F.fill := by by_contra hn exact h (ite_eq_right hn) obtain ⟨x,hx⟩ := hq have he := F.fill_mismatch_iff x i simpa only [hx] using he lemma component_sum (A : commutant p l) : ∑ F : ExceptionPattern p l, F.component A = fun q => A.val q.1 q.2 := by funext q rw [Finset.sum_apply] by_cases hq : A.val q.1 q.2 = 0 · simp [component,hq] · let F := entryPattern A q.1 q.2 hq obtain ⟨x,hx⟩ := entry_in_range A q.1 q.2 hq have hmem : q ∈ Set.range F.fill := ⟨x,hx⟩ rw [Finset.sum_eq_single F] · exact ite_eq_left hmem · intro G _ hG apply ite_eq_right rintro ⟨y,hy⟩ exact hG (fill_disjoint y x (hy.trans hx.symm)) · simp lemma component_square_sum (A : commutant p l) (q : Word p l × Word p l) : ∑ F : ExceptionPattern p l, F.component A q * star (F.component A q) = A.val q.1 q.2 * star (A.val q.1 q.2) := by by_cases hq : A.val q.1 q.2 = 0 · simp [component,hq] · let F := entryPattern A q.1 q.2 hq obtain ⟨x,hx⟩ := entry_in_range A q.1 q.2 hq have hmem : q ∈ Set.range F.fill := ⟨x,hx⟩ rw [Finset.sum_eq_single F] · simp [component,hmem] · intro G _ hG have hn : q ∉ Set.range G.fill := by rintro ⟨y,hy⟩ exact hG (fill_disjoint y x (hy.trans hx.symm)) simp [component,hn] · simp lemma purified_trace (F : ExceptionPattern p l) (A : commutant p l) : Matrix.trace (OrdinaryTensor.outer (F.purified A)) = Matrix.trace (PositiveTensor.outer (F.component A)) := by rw [show OrdinaryTensor.outer (F.purified A) = Matrix.vecMulVec (F.purified A) (star (F.purified A)) from rfl, ← inclusion_trace_outer F.embedding,F.purified_inclusion] apply Finset.sum_congr rfl intro q _ change (regroupPhase q.1 q.2 * F.component A q) * star (regroupPhase q.1 q.2 * F.component A q) = F.component A q * star (F.component A q) rw [star_mul,regroupPhase_star] calc _ = (regroupPhase q.1 q.2 * regroupPhase q.1 q.2) * (F.component A q * star (F.component A q)) := by ring _ = _ := by rw [regroupPhase_square,one_mul] lemma sum_purified_trace (A : commutant p l) : ∑ F : ExceptionPattern p l, Matrix.trace (OrdinaryTensor.outer (F.purified A)) = Matrix.trace (A.val*A.val.conjTranspose) := by simp_rw [purified_trace] simp only [PositiveTensor.outer,Matrix.trace,Matrix.diag_apply,Matrix.vecMulVec_apply,Pi.star_apply] rw [Finset.sum_comm] simp_rw [component_square_sum] simp only [Fintype.sum_prod_type,Matrix.mul_apply,Matrix.conjTranspose_apply] end CoordinateSweeps.SignedTensor.ExceptionPattern namespace CoordinateSweeps.SignedTensor.ExceptionPattern open PositiveTensor variable {p l : ℕ} lemma free_card (F : ExceptionPattern p l) : Fintype.card F.Free = l-F.sites.card := by simp only [Free,Fintype.card_subtype_compl,Fintype.card_coe,Fintype.card_fin] lemma free_card_pos (F : ExceptionPattern p l) (hl : (2*p)*(2*p) < l) : 0 < Fintype.card F.Free := by rw [F.free_card] exact Nat.sub_pos_of_lt (lt_of_le_of_lt F.sites_card_le hl) lemma free_card_le (F : ExceptionPattern p l) : Fintype.card F.Free ≤ l := by rw [F.free_card] exact Nat.sub_le _ _ lemma matchedPair_nonempty (hp : 0 < p) : Nonempty (MatchedPair p) := ⟨⟨(⟨0,by omega⟩,⟨0,by omega⟩),Iff.rfl⟩⟩ /- Exact domination of a pinned component by a finite product mixture on its free slots, after the proved sign cancellation in the genuine partial trace. -/ theorem component_postselection (F : ExceptionPattern p l) (hp : 0 < p) (hl : (2*p)*(2*p) < l) (A : commutant p l) : ((Matrix.trace (OrdinaryTensor.outer (F.purified A)) * ((Fintype.card F.Free+1 : ℕ) : ℂ)^(2*Fintype.card (MatchedPair p))) • partialTrace (inclusion F.embedding * (OrdinaryTensor.finiteMixture (I := F.Free) (A := MatchedPair p)) * (inclusion F.embedding).conjTranspose) - partialTrace (PositiveTensor.outer (F.component A))).PosSemidef := by let : Nonempty (MatchedPair p) := matchedPair_nonempty hp have h := OrdinaryTensor.invariant_vector_postselection (I := F.Free) (A := MatchedPair p) (F.free_card_pos hl) (F.purified A) (F.purified_invariant A) have hc := h.mul_mul_conjTranspose_same (inclusion F.embedding) rw [Matrix.mul_sub,Matrix.sub_mul,Matrix.mul_smul,Matrix.smul_mul] at hc have ho : inclusion F.embedding * OrdinaryTensor.outer (F.purified A) * (inclusion F.embedding).conjTranspose = PositiveTensor.outer ((inclusion F.embedding).mulVec (F.purified A)) := mul_outer_mul_star _ _ rw [ho] at hc have hp := partialTrace_posSemidef _ hc simpa only [partialTrace_sub,partialTrace_smul,F.component_partialTrace A] using hp end CoordinateSweeps.SignedTensor.ExceptionPattern namespace CoordinateSweeps.SignedTensor def pairAmplitude {p : ℕ} (z : MatchedPair p → ℂ) : Matrix (Letter p) (Letter p) ℂ := fun a b => if h : odd a ↔ odd b then z ⟨(a,b),h⟩ else 0 lemma pairAmplitude_unit_trace {p : ℕ} (z : MatchedPair p → ℂ) (hz : ∑ a, z a * star (z a) = 1) : Matrix.trace (pairAmplitude z * (pairAmplitude z).conjTranspose) = 1 := by have hs := Fintype.sum_of_injective (Subtype.val : MatchedPair p → Pair p) Subtype.val_injective (fun a => z a * star (z a)) (fun a : Pair p => pairAmplitude z a.1 a.2 * star (pairAmplitude z a.1 a.2)) (by intro a ha have hn : ¬ (odd a.1 ↔ odd a.2) := by intro h exact ha ⟨⟨a,h⟩,rfl⟩ simp [pairAmplitude,hn]) (by intro a; simp [pairAmplitude,a.property]) rw [hz] at hs simpa only [Matrix.trace,Matrix.diag_apply,Matrix.mul_apply,Matrix.conjTranspose_apply, Fintype.sum_prod_type] using hs.symm lemma pairAmplitude_even {p : ℕ} (z : MatchedPair p → ℂ) (a b : Letter p) (hab : odd a ↔ ¬ odd b) : (pairAmplitude z * (pairAmplitude z).conjTranspose) a b = 0 := by rw [Matrix.mul_apply] apply Finset.sum_eq_zero intro c _ by_cases hac : odd a ↔ odd c · have hbc : ¬ (odd b ↔ odd c) := by tauto simp [Matrix.conjTranspose_apply,pairAmplitude,hbc] · simp [Matrix.conjTranspose_apply,pairAmplitude,hac] /- The real even one-site density obtained by the literal auxiliary partial trace of a unit matched-pair vector. -/ def pairDensity {p : ℕ} (z : MatchedPair p → ℂ) (hz : ∑ a, z a * star (z a) = 1) : EvenDensity p := ⟨pairAmplitude z * (pairAmplitude z).conjTranspose, Matrix.posSemidef_self_mul_conjTranspose _, pairAmplitude_unit_trace z hz, pairAmplitude_even z⟩ end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor variable {p : ℕ} def basisDensity (a : Letter p) : Matrix (Letter p) (Letter p) ℂ := Matrix.diagonal (fun b => if b=a then 1 else 0) lemma basisDensity_posSemidef (a : Letter p) : (basisDensity a).PosSemidef := by apply Matrix.PosSemidef.diagonal intro b change (0 : ℂ) ≤ if b=a then _ else _ split_ifs <;> norm_num lemma basisDensity_le_one (a : Letter p) : (1-basisDensity a).PosSemidef := by have he : (1-basisDensity a) = Matrix.diagonal (fun b => if b=a then (0 : ℂ) else 1) := by ext i j by_cases hij : i=j · subst j by_cases hi : i=a <;> simp [basisDensity,Matrix.diagonal,hi] · simp [basisDensity,Matrix.diagonal,hij] rw [he] apply Matrix.PosSemidef.diagonal intro b change (0 : ℂ) ≤ if b=a then _ else _ split_ifs <;> norm_num lemma basisDensity_trace (a : Letter p) : Matrix.trace (basisDensity a) = 1 := by simp [basisDensity,Matrix.trace_diagonal] def pinAmplitude (a : Pair p) : Matrix (Letter p) (Letter p) ℂ := fun i j => if (i,j)=a then 1 else 0 lemma pinAmplitude_density (a : Pair p) : pinAmplitude a * (pinAmplitude a).conjTranspose = basisDensity a.1 := by ext i j simp only [Matrix.mul_apply,Matrix.conjTranspose_apply,pinAmplitude] rw [Finset.sum_eq_single a.2] · by_cases hi : i=a.1 <;> by_cases hj : j=a.1 <;> simp [Prod.ext_iff,hi,hj,basisDensity,Matrix.diagonal,eq_comm] · intro k _ hk simp [Prod.ext_iff,hk] · simp def keep (s : ℕ) : ℝ := s/(s+1) def noise (s p : ℕ) : ℝ := 1/((s+1)*(2*p)) lemma keep_nonneg (s : ℕ) : 0 ≤ keep s := div_nonneg (Nat.cast_nonneg _) (by positivity) lemma noise_nonneg (s p : ℕ) : 0 ≤ noise s p := by unfold noise; positivity lemma keep_add_noise (s p : ℕ) (hp : 0 < p) : keep s + noise s p*(2*p) = 1 := by have hp' : (p : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hp.ne' have hs : (s : ℝ)+1 ≠ 0 := by positivity unfold keep noise field_simp def depolarized (s : ℕ) (r : EvenDensity p) : Matrix (Letter p) (Letter p) ℂ := (keep s : ℂ) • r.val + (noise s p : ℂ) • 1 lemma depolarized_posSemidef (s : ℕ) (r : EvenDensity p) : (depolarized s r).PosSemidef := by exact (r.property.1.smul (by exact_mod_cast keep_nonneg s)).add (Matrix.PosSemidef.one.smul (by exact_mod_cast noise_nonneg s p)) lemma depolarized_trace (s : ℕ) (hp : 0 < p) (r : EvenDensity p) : Matrix.trace (depolarized s r) = 1 := by rw [depolarized,Matrix.trace_add,Matrix.trace_smul,Matrix.trace_smul,r.property.2.1] simp only [Matrix.trace_one,Fintype.card_fin,smul_eq_mul,mul_one] exact_mod_cast keep_add_noise s p hp lemma depolarized_even (s : ℕ) (r : EvenDensity p) (i j : Letter p) (hij : odd i ↔ ¬ odd j) : depolarized s r i j = 0 := by have hne : i ≠ j := by intro h; subst j; tauto simp [depolarized,Matrix.smul_apply,r.property.2.2 i j hij,hne] def depolarizedDensity (s : ℕ) (hp : 0 < p) (r : EvenDensity p) : EvenDensity p := ⟨depolarized s r,depolarized_posSemidef s r,depolarized_trace s hp r,depolarized_even s r⟩ lemma depolarized_free (s : ℕ) (r : EvenDensity p) : (depolarized s r - (keep s : ℂ) • r.val).PosSemidef := by simpa only [depolarized,add_sub_cancel_left] using (Matrix.PosSemidef.one (n := Letter p) (R := ℂ)).smul (by exact_mod_cast noise_nonneg s p) lemma depolarized_pin (s : ℕ) (r : EvenDensity p) (a : Letter p) : (depolarized s r - (noise s p : ℂ) • basisDensity a).PosSemidef := by have ht : (0 : ℂ) ≤ (keep s : ℂ) := by exact_mod_cast keep_nonneg s have hu : (0 : ℂ) ≤ (noise s p : ℂ) := by exact_mod_cast noise_nonneg s p have h := (r.property.1.smul ht).add ((basisDensity_le_one a).smul hu) simpa only [depolarized,smul_sub,add_sub_assoc] using h end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor open PositiveTensor variable {p l : ℕ} def pinnedDensity (E : Finset (Fin l)) (a : E → Letter p) (r : EvenDensity p) : Matrix (Word p l) (Word p l) ℂ := productMatrix (fun i => if hi : i ∈ E then basisDensity (a ⟨i,hi⟩) else r.val) lemma pinnedDensity_posSemidef (E : Finset (Fin l)) (a : E → Letter p) (r : EvenDensity p) : (pinnedDensity E a r).PosSemidef := by apply productMatrix_posSemidef intro i split_ifs with hi · exact basisDensity_posSemidef _ · exact r.property.1 lemma keep_pow_lower (s n : ℕ) (hn : n ≤ s) : 1 ≤ (s+1 : ℝ) * (keep s)^n := by have hs : (0 : ℝ) < s+1 := by positivity have hi : (s+1 : ℝ)⁻¹ ≤ 1 := (inv_le_one₀ hs).mpr (by linarith [Nat.cast_nonneg (α := ℝ) s]) have hb := one_add_mul_le_pow (a := -((s+1 : ℝ)⁻¹)) (by linarith : -2 ≤ -((s+1 : ℝ)⁻¹)) n have he : (1 - (s+1 : ℝ)⁻¹) = keep s := by unfold keep; field_simp; ring calc 1 ≤ (s+1 : ℝ)-n := by have hn' : (n : ℝ) ≤ s := by exact_mod_cast hn linarith _ = (s+1 : ℝ)*(1+n * (-((s+1 : ℝ)⁻¹))) := by field_simp; ring _ ≤ (s+1 : ℝ)*(1+(-((s+1 : ℝ)⁻¹)))^n := mul_le_mul_of_nonneg_left hb hs.le _ = _ := by rw [show 1+(-((s+1 : ℝ)⁻¹)) = keep s from he] lemma pin_coefficient_le (s p k : ℕ) (hp : 0 < p) (hps : p ≤ s) (hk : k ≤ (2*p)*(2*p)) : (s+1 : ℝ) * ((s+1 : ℝ)*(2*p))^k ≤ (s+1 : ℝ)^(13*p^2) := by have hs : (1 : ℝ) ≤ s+1 := by linarith [Nat.cast_nonneg (α := ℝ) s] have hp' : (p : ℝ) ≤ s := by exact_mod_cast hps have ha : (2*p : ℝ) ≤ (s+1 : ℝ)^2 := by nlinarith [sq_nonneg (s : ℝ)] have hbase : (s+1 : ℝ)*(2*p) ≤ (s+1 : ℝ)^3 := by nlinarith [mul_le_mul_of_nonneg_left ha (by positivity : (0 : ℝ) ≤ s+1)] have hexp : 3*k+1 ≤ 13*p^2 := by have hp1 : 1 ≤ p*p := by nlinarith nlinarith [hk] calc _ ≤ (s+1 : ℝ) * ((s+1 : ℝ)^3)^k := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (by positivity) hbase k) (by positivity) _ = (s+1 : ℝ)^(3*k+1) := by rw [← pow_mul,pow_succ]; ring _ ≤ _ := pow_le_pow_right₀ hs hexp /- Each pinned ordinary product density is dominated by a tensor power of a single even density, with only polynomial cost per actual exception. -/ theorem pinned_absorption (s : ℕ) (hp : 0 < p) (hls : l ≤ s) (E : Finset (Fin l)) (a : E → Letter p) (r : EvenDensity p) : (((s+1 : ℝ) * ((s+1 : ℝ)*(2*p))^E.card : ℝ) : ℂ) • densityPower l (depolarized s r) - pinnedDensity E a r |>.PosSemidef := by let c : Fin l → ℂ := fun i => if i ∈ E then (noise s p : ℂ) else (keep s : ℂ) let f : Fin l → Matrix (Letter p) (Letter p) ℂ := fun i => if hi : i ∈ E then basisDensity (a ⟨i,hi⟩) else r.val have hlocal (i : Fin l) : (c i • f i).PosSemidef := by by_cases hi : i ∈ E · simpa only [c,f,ite_eq_left hi,dite_eq_left hi] using (basisDensity_posSemidef (a ⟨i,hi⟩)).smul (show (0 : ℂ) ≤ (noise s p : ℂ) by exact_mod_cast noise_nonneg s p) · simpa only [c,f,ite_eq_right hi,dite_eq_right hi] using r.property.1.smul (show (0 : ℂ) ≤ (keep s : ℂ) by exact_mod_cast keep_nonneg s) have hdiff (i : Fin l) : (depolarized s r - c i • f i).PosSemidef := by by_cases hi : i ∈ E · simpa only [c,f,ite_eq_left hi,dite_eq_left hi] using depolarized_pin s r (a ⟨i,hi⟩) · simpa only [c,f,ite_eq_right hi,dite_eq_right hi] using depolarized_free s r have h := productMatrix_mono (fun i => c i • f i) (fun _ => depolarized s r) hlocal hdiff rw [productMatrix_smul] at h have hc : (∏ i, c i) = ((keep s)^(l-E.card) * (noise s p)^E.card : ℝ) := by have hE : Finset.univ.filter (fun i : Fin l => i ∈ E) = E := by ext i; simp have hC : Finset.univ.filter (fun i : Fin l => i ∉ E) = Eᶜ := by ext i; simp simp only [c,Finset.prod_ite,hE,hC,Finset.prod_const,Finset.card_compl,Fintype.card_fin] push_cast ring rw [hc] at h let C : ℝ := (s+1 : ℝ) * ((s+1 : ℝ)*(2*p))^E.card have hC : (0 : ℝ) ≤ C := by dsimp [C]; positivity have hu : ((s+1 : ℝ)*(2*p))*noise s p = 1 := by have hp' : (p : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hp.ne' unfold noise field_simp have hCW : 1 ≤ C*((keep s)^(l-E.card)*(noise s p)^E.card) := by calc _ ≤ (s+1 : ℝ)*(keep s)^(l-E.card) := keep_pow_lower s _ (le_trans (Nat.sub_le _ _) hls) _ = _ := by dsimp [C] calc _ = ((s+1 : ℝ)*(keep s)^(l-E.card))*((((s+1 : ℝ)*(2*p))*noise s p)^E.card) := by rw [hu,one_pow,mul_one] _ = _ := by rw [mul_pow]; ring have hh := h.smul (show (0 : ℂ) ≤ (C : ℂ) by exact_mod_cast hC) have hz := (pinnedDensity_posSemidef E a r).smul (show (0 : ℂ) ≤ ((C*((keep s)^(l-E.card)*(noise s p)^E.card)-1 : ℝ) : ℂ) by exact_mod_cast sub_nonneg.mpr hCW) have ht := hh.add hz change ((C : ℂ) • (densityPower l (depolarized s r) - (((keep s)^(l-E.card)*(noise s p)^E.card : ℝ) : ℂ) • pinnedDensity E a r) + ((C*((keep s)^(l-E.card)*(noise s p)^E.card)-1 : ℝ) : ℂ) • pinnedDensity E a r).PosSemidef at ht change ((C : ℂ) • densityPower l (depolarized s r) - pinnedDensity E a r).PosSemidef convert ht using 1 simp only [smul_sub,smul_smul,Complex.ofReal_sub,Complex.ofReal_mul,Complex.ofReal_one, sub_smul,one_smul] module end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor open PositiveTensor variable {p l : ℕ} theorem pinned_absorption_uniform (s : ℕ) (hp : 0 < p) (hps : p ≤ s) (hls : l ≤ s) (E : Finset (Fin l)) (hE : E.card ≤ (2*p)*(2*p)) (a : E → Letter p) (r : EvenDensity p) : ((((s+1 : ℝ)^(13*p^2)) : ℂ) • densityPower l (depolarized s r) - pinnedDensity E a r).PosSemidef := by have h := pinned_absorption s hp hls E a r have hle := pin_coefficient_le s p E.card hp hps hE have hr : (densityPower l (depolarized s r)).PosSemidef := productMatrix_posSemidef (fun _ : Fin l => depolarized s r) (fun _ => depolarized_posSemidef s r) have hh := (hr.smul (show (0 : ℂ) ≤ (((s+1 : ℝ)^(13*p^2) - (s+1 : ℝ)*((s+1 : ℝ)*(2*p))^E.card : ℝ) : ℂ) by exact_mod_cast sub_nonneg.mpr hle)).add h convert hh using 1 push_cast module end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor.ExceptionPattern open PositiveTensor variable {p l : ℕ} def siteAmplitude (F : ExceptionPattern p l) (z : MatchedPair p → ℂ) (i : Fin l) : Matrix (Letter p) (Letter p) ℂ := if hi : i ∈ F.sites then pinAmplitude (F.pair ⟨i,hi⟩) else pairAmplitude z def pairProduct (F : ExceptionPattern p l) (z : MatchedPair p → ℂ) (q : Word p l × Word p l) : ℂ := ∏ i, F.siteAmplitude z i (q.1 i) (q.2 i) lemma siteAmplitude_pin (F : ExceptionPattern p l) (z : MatchedPair p → ℂ) (x : F.FreeWord) (i : F.sites) : F.siteAmplitude z i.val ((F.fill x).1 i.val) ((F.fill x).2 i.val) = 1 := by change F.siteAmplitude z i.val (F.fillPair x i.val).1 (F.fillPair x i.val).2 = 1 rw [F.fillPair_at_pin] simp [siteAmplitude,i.property,pinAmplitude] lemma siteAmplitude_free (F : ExceptionPattern p l) (z : MatchedPair p → ℂ) (x : F.FreeWord) (i : F.Free) : F.siteAmplitude z i.val ((F.fill x).1 i.val) ((F.fill x).2 i.val) = z (x i) := by change F.siteAmplitude z i.val (F.fillPair x i.val).1 (F.fillPair x i.val).2 = z (x i) rw [F.fillPair_at_free] simp [siteAmplitude,i.property,pairAmplitude,(x i).property] lemma pairProduct_fill (F : ExceptionPattern p l) (z : MatchedPair p → ℂ) (x : F.FreeWord) : F.pairProduct z (F.fill x) = OrdinaryTensor.tensorVector z x := by unfold pairProduct rw [← Fintype.prod_subtype_mul_prod_subtype (fun i => i ∈ F.sites)] simp only [F.siteAmplitude_pin,F.siteAmplitude_free,Finset.prod_const_one,one_mul] rfl lemma pairProduct_nonzero (F : ExceptionPattern p l) (z : MatchedPair p → ℂ) (q : Word p l × Word p l) (hq : F.pairProduct z q ≠ 0) : q ∈ Set.range F.fill := by have hi (i : Fin l) : F.siteAmplitude z i (q.1 i) (q.2 i) ≠ 0 := (Finset.prod_ne_zero_iff.mp hq) i (Finset.mem_univ i) have hp (i : F.sites) : (q.1 i.val,q.2 i.val) = F.pair i := by by_contra hn exact hi i.val (by simp [siteAmplitude,i.property,pinAmplitude,hn]) have hf (i : F.Free) : odd (q.1 i.val) ↔ odd (q.2 i.val) := by by_contra hn exact hi i.val (by simp [siteAmplitude,i.property,pairAmplitude,hn]) let x : F.FreeWord := fun i => ⟨(q.1 i.val,q.2 i.val),hf i⟩ have he (i : Fin l) : F.fillPair x i = (q.1 i,q.2 i) := by by_cases hi : i ∈ F.sites · exact (F.fillPair_at_pin x ⟨i,hi⟩).trans (hp ⟨i,hi⟩).symm · exact F.fillPair_at_free x ⟨i,hi⟩ exact ⟨x,Prod.ext (funext fun i => congrArg Prod.fst (he i)) (funext fun i => congrArg Prod.snd (he i))⟩ lemma pairProduct_inclusion (F : ExceptionPattern p l) (z : MatchedPair p → ℂ) : (inclusion F.embedding).mulVec (OrdinaryTensor.tensorVector z) = F.pairProduct z := by funext q by_cases hq : q ∈ Set.range F.fill · obtain ⟨x,rfl⟩ := hq have hh := inclusion_mulVec_image F.embedding (OrdinaryTensor.tensorVector z) x change (inclusion F.embedding).mulVec (OrdinaryTensor.tensorVector z) (F.fill x) = _ at hh exact hh.trans (F.pairProduct_fill z x).symm · rw [inclusion_mulVec_notin F.embedding _ q hq] exact (not_ne_iff.mp (fun hh => hq (F.pairProduct_nonzero z q hh))).symm /- Partial tracing a filled product vector yields pinned basis densities at the true exceptions and the same even density at every remaining slot. -/ lemma filled_product_partialTrace (F : ExceptionPattern p l) (z : MatchedPair p → ℂ) (hz : ∑ a, z a * star (z a) = 1) : partialTrace (PositiveTensor.outer ((inclusion F.embedding).mulVec (OrdinaryTensor.tensorVector z))) = pinnedDensity F.sites (fun i => (F.pair i).1) (pairDensity z hz) := by rw [F.pairProduct_inclusion] have he : partialTrace (PositiveTensor.outer (F.pairProduct z)) = productMatrix (fun i => F.siteAmplitude z i * (F.siteAmplitude z i).conjTranspose) := by ext u v simpa only [partialTrace,PositiveTensor.outer,Matrix.vecMulVec_apply,Pi.star_apply,pairProduct] using pair_product_partialTrace (F.siteAmplitude z) u v rw [he] congr 1 funext i by_cases hi : i ∈ F.sites · simp only [siteAmplitude,hi,dite_eq_left,pinAmplitude_density] · simp [siteAmplitude,hi,pairDensity] end CoordinateSweeps.SignedTensor.ExceptionPattern namespace CoordinateSweeps.SignedTensor.ExceptionPattern open PositiveTensor variable {p l : ℕ} lemma purified_trace_le (F : ExceptionPattern p l) (A : commutant p l) : Matrix.trace (OrdinaryTensor.outer (F.purified A)) ≤ Matrix.trace (A.val*A.val.conjTranspose) := by rw [← sum_purified_trace A] exact Finset.single_le_sum (fun G _ => OrdinaryTensor.outer_trace_nonneg (G.purified A)) (Finset.mem_univ F) abbrev Data (F : ExceptionPattern p l) := OrdinaryTensor.Histogram F.Free (MatchedPair p) × OrdinaryTensor.PhaseSpace F.Free (MatchedPair p) def dataDensity (F : ExceptionPattern p l) (hn : 0 < Fintype.card F.Free) (d : F.Data) : EvenDensity p := pairDensity (OrdinaryTensor.empiricalVector d.1 d.2) (OrdinaryTensor.empiricalVector_unit hn d.1 d.2) def dataPoint (F : ExceptionPattern p l) (s : ℕ) (hp : 0 < p) (hn : 0 < Fintype.card F.Free) (d : F.Data) : EvenDensity p := depolarizedDensity s hp (F.dataDensity hn d) def rawPinned (F : ExceptionPattern p l) (hn : 0 < Fintype.card F.Free) : Matrix (Word p l) (Word p l) ℂ := ∑ d : F.Data, pinnedDensity F.sites (fun i => (F.pair i).1) (F.dataDensity hn d) def rawDensity (F : ExceptionPattern p l) (s : ℕ) (hp : 0 < p) (hn : 0 < Fintype.card F.Free) : Matrix (Word p l) (Word p l) ℂ := ∑ d : F.Data, densityPower l (F.dataPoint s hp hn d).val lemma rawPinned_posSemidef (F : ExceptionPattern p l) (hn : 0 < Fintype.card F.Free) : (F.rawPinned hn).PosSemidef := by exact Matrix.posSemidef_sum _ (fun d _ => pinnedDensity_posSemidef _ _ _) lemma rawDensity_posSemidef (F : ExceptionPattern p l) (s : ℕ) (hp : 0 < p) (hn : 0 < Fintype.card F.Free) : (F.rawDensity s hp hn).PosSemidef := by exact Matrix.posSemidef_sum _ (fun d _ => densityPower_posSemidef _ _ (F.dataPoint s hp hn d).property.1) lemma rawMixture_partialTrace (F : ExceptionPattern p l) (hn : 0 < Fintype.card F.Free) : partialTrace (inclusion F.embedding * (OrdinaryTensor.rawMixture (I := F.Free) (A := MatchedPair p)) * (inclusion F.embedding).conjTranspose) = F.rawPinned hn := by unfold OrdinaryTensor.rawMixture rawPinned rw [Fintype.sum_prod_type] simp only [Matrix.mul_sum,Matrix.sum_mul,partialTrace_sum] apply Finset.sum_congr rfl intro h _ apply Finset.sum_congr rfl intro t _ have he := congrArg (partialTrace (J := Word p l) (K := Word p l)) (mul_outer_mul_star (inclusion F.embedding) (OrdinaryTensor.tensorVector (OrdinaryTensor.empiricalVector h t))) exact he.trans (F.filled_product_partialTrace _ (OrdinaryTensor.empiricalVector_unit hn h t)) lemma rawPinned_domination (F : ExceptionPattern p l) (hp : 0 < p) (hn : 0 < Fintype.card F.Free) : (F.rawPinned hn - partialTrace (inclusion F.embedding * (OrdinaryTensor.finiteMixture (I := F.Free) (A := MatchedPair p)) * (inclusion F.embedding).conjTranspose)).PosSemidef := by let : Nonempty (MatchedPair p) := matchedPair_nonempty hp have h := (OrdinaryTensor.rawMixture_domination (I := F.Free) (A := MatchedPair p)).mul_mul_conjTranspose_same (inclusion F.embedding) rw [Matrix.mul_sub,Matrix.sub_mul] at h have hh := partialTrace_posSemidef _ h simpa only [partialTrace_sub,F.rawMixture_partialTrace hn] using hh lemma rawPinned_absorption (F : ExceptionPattern p l) (s : ℕ) (hp : 0 < p) (hps : p ≤ s) (hls : l ≤ s) (hn : 0 < Fintype.card F.Free) : ((((s+1 : ℝ)^(13*p^2)) : ℂ) • F.rawDensity s hp hn - F.rawPinned hn).PosSemidef := by have hh := Matrix.posSemidef_sum Finset.univ (fun d _ => pinned_absorption_uniform s hp hps hls F.sites F.sites_card_le (fun i => (F.pair i).1) (F.dataDensity hn d)) simpa only [Finset.sum_sub_distrib,← Finset.smul_sum,rawPinned,rawDensity,dataPoint,depolarizedDensity] using hh end CoordinateSweeps.SignedTensor.ExceptionPattern namespace CoordinateSweeps.SignedTensor open PositiveTensor variable {p l : ℕ} lemma matchedPair_card_le (p : ℕ) : Fintype.card (MatchedPair p) ≤ (2*p)*(2*p) := by simpa only [Pair,Fintype.card_prod,Letter,Fintype.card_fin] using (Fintype.card_subtype_le (fun a : Pair p => odd a.1 ↔ odd a.2)) namespace ExceptionPattern def empty (p l : ℕ) : ExceptionPattern p l where sites := ∅ pair i := False.elim (Finset.notMem_empty i.val i.property) mismatch i := False.elim (Finset.notMem_empty i.val i.property) injective i := False.elim (Finset.notMem_empty i.val i.property) instance : Nonempty (ExceptionPattern p l) := ⟨empty p l⟩ lemma data_card_le (F : ExceptionPattern p l) : Fintype.card F.Data ≤ (l+1)^(8*p^2) := by have hh := OrdinaryTensor.histogram_card_le (I := F.Free) (A := MatchedPair p) have ht : Fintype.card (OrdinaryTensor.PhaseSpace F.Free (MatchedPair p)) = (Fintype.card F.Free+1)^Fintype.card (MatchedPair p) := by simp [OrdinaryTensor.PhaseSpace] rw [Fintype.card_prod,ht] calc _ ≤ (Fintype.card F.Free+1)^(2*Fintype.card (MatchedPair p)) := by rw [two_mul (Fintype.card (MatchedPair p)),pow_add] exact Nat.mul_le_mul_right _ hh _ ≤ (l+1)^(2*Fintype.card (MatchedPair p)) := Nat.pow_le_pow_left (Nat.add_le_add_right F.free_card_le 1) _ _ ≤ _ := Nat.pow_le_pow_right (by omega) (by nlinarith [matchedPair_card_le p]) lemma free_power_le (F : ExceptionPattern p l) (s : ℕ) (hls : l ≤ s) : (Fintype.card F.Free+1 : ℕ)^(2*Fintype.card (MatchedPair p)) ≤ (s+1)^(8*p^2) := by calc _ ≤ (s+1)^(2*Fintype.card (MatchedPair p)) := Nat.pow_le_pow_left (Nat.add_le_add_right (F.free_card_le.trans hls) 1) _ _ ≤ _ := Nat.pow_le_pow_right (by omega) (by nlinarith [matchedPair_card_le p]) lemma component_uniform_domination (F : ExceptionPattern p l) (s : ℕ) (hp : 0 < p) (hps : p ≤ s) (hls : l ≤ s) (hl : (2*p)*(2*p) < l) (A : commutant p l) : ((Matrix.trace (A.val*A.val.conjTranspose) * ((s+1 : ℕ) : ℂ)^(21*p^2)) • F.rawDensity s hp (F.free_card_pos hl) - partialTrace (PositiveTensor.outer (F.component A))).PosSemidef := by let hn := F.free_card_pos hl let M := partialTrace (inclusion F.embedding * (OrdinaryTensor.finiteMixture (I := F.Free) (A := MatchedPair p)) * (inclusion F.embedding).conjTranspose) have hM : M.PosSemidef := partialTrace_posSemidef _ (OrdinaryTensor.finiteMixture_posSemidef.mul_mul_conjTranspose_same _) have ht : (0 : ℂ) ≤ Matrix.trace (A.val*A.val.conjTranspose) := (Matrix.posSemidef_self_mul_conjTranspose A.val).trace_nonneg have ha : (0 : ℂ) ≤ Matrix.trace (A.val*A.val.conjTranspose) * ((s+1 : ℕ) : ℂ)^(8*p^2) := mul_nonneg ht (by positivity) have hc : Matrix.trace (OrdinaryTensor.outer (F.purified A)) * ((Fintype.card F.Free+1 : ℕ) : ℂ)^(2*Fintype.card (MatchedPair p)) ≤ Matrix.trace (A.val*A.val.conjTranspose) * ((s+1 : ℕ) : ℂ)^(8*p^2) := by exact mul_le_mul (F.purified_trace_le A) (by exact_mod_cast F.free_power_le s hls) (by positivity) ht have h₁ := domination_pad _ M _ _ hM hc (F.component_postselection hp hl A) have h₂ := domination_trans _ M (F.rawPinned hn) _ 1 ha h₁ (by simpa only [one_smul] using F.rawPinned_domination hp hn) rw [mul_one] at h₂ have h₃ := domination_trans _ (F.rawPinned hn) (F.rawDensity s hp hn) _ (((s+1 : ℝ)^(13*p^2)) : ℂ) ha h₂ (F.rawPinned_absorption s hp hps hls hn) convert h₃ using 1 push_cast rw [mul_assoc,← pow_add] congr 2; ring end ExceptionPattern abbrev DensityProfile (p l : ℕ) := Σ F : ExceptionPattern p l, F.Data lemma densityProfile_card_le (p l : ℕ) : Fintype.card (DensityProfile p l) ≤ (l+1)^(12*p^2) := by rw [Fintype.card_sigma] calc _ ≤ ∑ _F : ExceptionPattern p l, (l+1)^(8*p^2) := Finset.sum_le_sum (fun F _ => F.data_card_le) _ = Fintype.card (ExceptionPattern p l) * (l+1)^(8*p^2) := by simp _ ≤ (l+1)^((2*p)*(2*p)) * (l+1)^(8*p^2) := Nat.mul_le_mul_right _ (ExceptionPattern.card_le (p := p) (l := l)) _ = _ := by rw [← pow_add]; congr 1; ring lemma densityProfile_nonempty (hp : 0 < p) : Nonempty (DensityProfile p l) := by let : Nonempty (MatchedPair p) := ExceptionPattern.matchedPair_nonempty hp let F := ExceptionPattern.empty p l exact ⟨⟨F,Classical.choice inferInstance⟩⟩ def profilePoint (s p l : ℕ) (hp : 0 < p) (hl : (2*p)*(2*p) < l) (d : DensityProfile p l) : EvenDensity p := d.1.dataPoint s hp (d.1.free_card_pos hl) d.2 def rawProfileDensity (s p l : ℕ) (hp : 0 < p) (hl : (2*p)*(2*p) < l) : Matrix (Word p l) (Word p l) ℂ := ∑ d : DensityProfile p l, densityPower l (profilePoint s p l hp hl d).val lemma rawProfileDensity_eq (s p l : ℕ) (hp : 0 < p) (hl : (2*p)*(2*p) < l) : rawProfileDensity s p l hp hl = ∑ F : ExceptionPattern p l, F.rawDensity s hp (F.free_card_pos hl) := by exact Fintype.sum_sigma _ lemma rawProfileDensity_posSemidef (s p l : ℕ) (hp : 0 < p) (hl : (2*p)*(2*p) < l) : (rawProfileDensity s p l hp hl).PosSemidef := by exact Matrix.posSemidef_sum _ (fun d _ => densityPower_posSemidef _ _ (profilePoint s p l hp hl d).property.1) /- Finite postselection for the literal signed action commutant. -/ theorem commutant_covariance_domination (s p l : ℕ) (hp : 0 < p) (hps : p ≤ s) (hls : l ≤ s) (hl : (2*p)*(2*p) < l) (A : commutant p l) : ((Matrix.trace (A.val*A.val.conjTranspose) * ((s+1 : ℕ) : ℂ)^(25*p^2)) • rawProfileDensity s p l hp hl - A.val*A.val.conjTranspose).PosSemidef := by have hc := partialTrace_posSemidef _ (outer_sum_domination (fun F : ExceptionPattern p l => F.component A)) rw [ExceptionPattern.component_sum,partialTrace_sub,partialTrace_smul,partialTrace_sum, partialTrace_outer_matrix] at hc have hsum := Matrix.posSemidef_sum Finset.univ (fun (F : ExceptionPattern p l) _ => F.component_uniform_domination s hp hps hls hl A) rw [Finset.sum_sub_distrib,← Finset.smul_sum,← rawProfileDensity_eq s p l hp hl] at hsum have h := domination_trans _ _ _ (Fintype.card (ExceptionPattern p l) : ℂ) _ (by positivity) hc hsum have hcard : Fintype.card (ExceptionPattern p l) ≤ (s+1)^(4*p^2) := by calc _ ≤ (l+1)^((2*p)*(2*p)) := ExceptionPattern.card_le (p := p) (l := l) _ ≤ (s+1)^((2*p)*(2*p)) := Nat.pow_le_pow_left (Nat.add_le_add_right hls 1) _ _ = _ := by congr 1; ring apply domination_pad _ _ _ _ (rawProfileDensity_posSemidef s p l hp hl) _ h calc _ ≤ ((s+1 : ℕ) : ℂ)^(4*p^2) * (Matrix.trace (A.val*A.val.conjTranspose) * ((s+1 : ℕ) : ℂ)^(21*p^2)) := mul_le_mul_of_nonneg_right (by exact_mod_cast hcard) (mul_nonneg (Matrix.posSemidef_self_mul_conjTranspose _).trace_nonneg (by positivity)) _ = _ := by rw [mul_left_comm,← pow_add]; congr 2; ring end CoordinateSweeps.SignedTensor noncomputable section open scoped BigOperators ENNReal open MeasureTheory namespace CoordinateSweeps.FiniteAtomic variable {I X V : Type*} [Fintype I] [Nonempty I] [MeasurableSpace X] [MeasurableSingletonClass X] [NormedAddCommGroup V] [NormedSpace ℝ V] [CompleteSpace V] def uniform (x : I → X) : Measure X := (Fintype.card I : ℝ≥0∞)⁻¹ • ∑ i, Measure.dirac (x i) instance uniform_probability (x : I → X) : IsProbabilityMeasure (uniform x) := by constructor simp [uniform, Measure.finsetSum_apply, ENNReal.inv_mul_cancel, Fintype.card_ne_zero] omit [Nonempty I] in lemma uniform_integral (x : I → X) (f : X → V) : (∫ z, f z ∂uniform x) = (Fintype.card I : ℝ)⁻¹ • ∑ i, f (x i) := by unfold uniform rw [integral_smul_measure,integral_finsetSum_measure] · simp [integral_dirac] · intro i _ exact integrable_dirac enorm_lt_top end CoordinateSweeps.FiniteAtomic namespace CoordinateSweeps.SignedTensor open PositiveTensor MeasureTheory instance (p : ℕ) : MeasurableSingletonClass (EvenDensity p) := by let : MeasurableSpace (Matrix (Letter p) (Letter p) ℂ) := borel _ let : BorelSpace (Matrix (Letter p) (Letter p) ℂ) := ⟨rfl⟩ change MeasurableSingletonClass {r : Matrix (Letter p) (Letter p) ℂ // _} infer_instance lemma isotypic_mem_commutant {p l : ℕ} (ρ : UnitaryIrrep (Equiv.Perm (Fin l))) : isotypic (p := p) ρ ∈ commutant p l := by intro σ rw [isotypic_eq_projector] exact (ρ.projector_commute (action p l) σ).symm lemma isotypic_self_mul {p l : ℕ} (ρ : UnitaryIrrep (Equiv.Perm (Fin l))) : isotypic (p := p) ρ * (isotypic (p := p) ρ).conjTranspose = isotypic (p := p) ρ := by rw [(isotypic_posSemidef ρ).isHermitian.eq,isotypic_eq_projector,ρ.projector_idempotent] lemma isotypic_trace_le (s p l : ℕ) (hls : l ≤ s) (ρ : UnitaryIrrep (Equiv.Perm (Fin l))) : Matrix.trace (isotypic (p := p) ρ) ≤ (ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(4*p^2) := by rw [isotypic_trace] apply mul_le_mul_of_nonneg_left _ (by positivity) have hh := (multiplicity_le p l ρ).trans (Nat.pow_le_pow_left (Nat.add_le_add_right hls 1) _) have he : (2*p)*(2*p) = 4*p^2 := by ring rw [he] at hh exact_mod_cast hh /- The large-slot case, as an actual probability mixture of even densities. -/ theorem density_domination_large (s p l : ℕ) (hp : 0 < p) (hps : p ≤ s) (hls : l ≤ s) (hl : (2*p)*(2*p) < l) (ρ : UnitaryIrrep (Equiv.Perm (Fin l))) : ∃ μ : Measure (EvenDensity p), IsProbabilityMeasure μ ∧ (((ρ.dimension : ℂ) * ((s+1 : ℕ) : ℂ)^(41*p^2)) • (∫ r : EvenDensity p, densityPower l r.val ∂μ) - isotypic (p := p) ρ).PosSemidef := by let : Nonempty (DensityProfile p l) := densityProfile_nonempty hp let x := profilePoint s p l hp hl let μ := FiniteAtomic.uniform x refine ⟨μ,inferInstance,?_⟩ have hint : (∫ r : EvenDensity p, densityPower l r.val ∂μ) = (Fintype.card (DensityProfile p l) : ℂ)⁻¹ • rawProfileDensity s p l hp hl := by rw [FiniteAtomic.uniform_integral] change (Fintype.card (DensityProfile p l) : ℝ)⁻¹ • rawProfileDensity s p l hp hl = _ ext u v simp [Complex.real_smul] rw [hint] let A : commutant p l := ⟨isotypic ρ,isotypic_mem_commutant ρ⟩ have h := commutant_covariance_domination s p l hp hps hls hl A change ((_ * _) • _ - isotypic ρ * (isotypic ρ).conjTranspose).PosSemidef at h simp only [A,isotypic_self_mul] at h have hcard : Fintype.card (DensityProfile p l) ≤ (s+1)^(12*p^2) := (densityProfile_card_le p l).trans (Nat.pow_le_pow_left (Nat.add_le_add_right hls 1) _) have hc0 : (Fintype.card (DensityProfile p l) : ℂ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero have hbig : ((s+1 : ℕ) : ℂ)^(12*p^2) * (Fintype.card (DensityProfile p l) : ℂ)⁻¹ ≥ 1 := by have hi : (0 : ℂ) ≤ (Fintype.card (DensityProfile p l) : ℂ)⁻¹ := by positivity have hh := mul_le_mul_of_nonneg_right (show (Fintype.card (DensityProfile p l) : ℂ) ≤ ((s+1 : ℕ) : ℂ)^(12*p^2) by exact_mod_cast hcard) hi simpa only [mul_inv_cancel₀ hc0] using hh rw [smul_smul] apply domination_pad _ _ _ _ (rawProfileDensity_posSemidef s p l hp hl) _ h calc _ ≤ ((ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(4*p^2)) * ((s+1 : ℕ) : ℂ)^(25*p^2) := mul_le_mul_of_nonneg_right (isotypic_trace_le s p l hls ρ) (by positivity) _ = (ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(29*p^2) := by rw [mul_assoc,← pow_add]; congr 2; ring _ ≤ ((ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(29*p^2)) * (((s+1 : ℕ) : ℂ)^(12*p^2) * (Fintype.card (DensityProfile p l) : ℂ)⁻¹) := by have hh := mul_le_mul_of_nonneg_left hbig (show (0 : ℂ) ≤ (ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(29*p^2) by positivity) simpa only [mul_one] using hh _ = _ := by rw [← mul_assoc,mul_assoc (ρ.dimension : ℂ),← pow_add] congr 3; ring def uniformDensity (p : ℕ) (hp : 0 < p) : EvenDensity p := ⟨((2*p : ℕ) : ℂ)⁻¹ • 1, Matrix.PosSemidef.one.smul (by positivity), by rw [Matrix.trace_smul,Matrix.trace_one] have hc : ((2*p : ℕ) : ℂ) ≠ 0 := Nat.cast_ne_zero.mpr (by omega) simpa only [Letter,Fintype.card_fin,smul_eq_mul] using inv_mul_cancel₀ hc, by intro i j hij have hne : i ≠ j := by intro he; subst j; tauto simp [hne]⟩ lemma densityPower_one (p l : ℕ) : densityPower l (1 : Matrix (Letter p) (Letter p) ℂ) = 1 := by ext u v by_cases huv : u=v · subst v simp [densityPower] · have hi : ∃ i, u i ≠ v i := Function.ne_iff.mp huv obtain ⟨i,hi⟩ := hi simp only [densityPower,Matrix.one_apply,ite_eq_right huv] exact Finset.prod_eq_zero (Finset.mem_univ i) (ite_eq_right hi) lemma densityPower_smul {p : ℕ} (l : ℕ) (c : ℂ) (r : Matrix (Letter p) (Letter p) ℂ) : densityPower l (c • r) = c^l • densityPower l r := by ext u v simp [densityPower,Finset.prod_mul_distrib] lemma uniformDensity_power (p l : ℕ) (hp : 0 < p) : densityPower l (uniformDensity p hp).val = (((2*p : ℕ) : ℂ)^l)⁻¹ • 1 := by change densityPower l ((((2*p : ℕ) : ℂ)⁻¹) • 1) = _ rw [densityPower_smul,densityPower_one,inv_pow] lemma small_power_le (s p l : ℕ) (hps : p ≤ s) (hl : l ≤ (2*p)*(2*p)) : (2*p)^l ≤ (s+1)^(8*p^2) := by have hb : 2*p ≤ (s+1)^2 := by nlinarith calc _ ≤ ((s+1)^2)^l := Nat.pow_le_pow_left hb l _ = (s+1)^(2*l) := by rw [pow_mul] _ ≤ _ := Nat.pow_le_pow_right (by omega) (by nlinarith) /- Small slot counts, including zero, require no exceptional-sector chart. -/ theorem density_domination_small (s p l : ℕ) (hp : 0 < p) (hps : p ≤ s) (hls : l ≤ s) (hl : l ≤ (2*p)*(2*p)) (ρ : UnitaryIrrep (Equiv.Perm (Fin l))) : ∃ μ : Measure (EvenDensity p), IsProbabilityMeasure μ ∧ (((ρ.dimension : ℂ) * ((s+1 : ℕ) : ℂ)^(12*p^2)) • (∫ r : EvenDensity p, densityPower l r.val ∂μ) - isotypic (p := p) ρ).PosSemidef := by refine ⟨Measure.dirac (uniformDensity p hp),inferInstance,?_⟩ rw [integral_dirac,uniformDensity_power,smul_smul] have hP := trace_domination (isotypic (p := p) ρ) (isotypic_posSemidef ρ) apply domination_pad _ 1 _ _ Matrix.PosSemidef.one _ hP have hc0 : (((2*p : ℕ) : ℂ)^l) ≠ 0 := pow_ne_zero _ (Nat.cast_ne_zero.mpr (by omega)) have hi : (0 : ℂ) ≤ (((2*p : ℕ) : ℂ)^l)⁻¹ := by positivity have hh := mul_le_mul_of_nonneg_right (show (((2*p : ℕ) : ℂ)^l) ≤ ((s+1 : ℕ) : ℂ)^(8*p^2) by exact_mod_cast small_power_le s p l hps hl) hi rw [mul_inv_cancel₀ hc0] at hh calc _ ≤ (ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(4*p^2) := isotypic_trace_le s p l hls ρ _ ≤ ((ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(4*p^2)) * (((s+1 : ℕ) : ℂ)^(8*p^2)*(((2*p : ℕ) : ℂ)^l)⁻¹) := by simpa only [mul_one] using mul_le_mul_of_nonneg_left hh (show (0 : ℂ) ≤ (ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(4*p^2) by positivity) _ = _ := by rw [← mul_assoc,mul_assoc (ρ.dimension : ℂ),← pow_add]; congr 3; ring end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor open PositiveTensor MeasureTheory lemma domination_coefficient (D s p : ℕ) (hD : 0 < D) : (Complex.ofReal (Real.exp (Real.log D+41*p^2*Real.log (s+1)))) = (D : ℂ)*((s+1 : ℕ) : ℂ)^(41*p^2) := by have hD' : (0 : ℝ) < D := by exact_mod_cast hD have hs : (0 : ℝ) < s+1 := by positivity rw [Real.exp_add,Real.exp_log hD'] have he : (41 : ℝ)*p^2*Real.log (s+1) = ((41*p^2 : ℕ) : ℝ)*Real.log (s+1) := by push_cast; ring rw [he,Real.exp_nat_mul,Real.exp_log hs] push_cast rfl /- Source05 equation13, with a concrete absolute constant. The measure is constructed from finite empirical phase mixtures, not an added hypothesis. -/ theorem signed_density_domination : Domination := by refine ⟨41,by norm_num,?_⟩ intro s p hp hps l hls ρ have hp0 : 0 < p := by omega rw [domination_coefficient ρ.dimension s p ρ.positive] by_cases hl : (2*p)*(2*p) < l · exact density_domination_large s p l hp0 hps hls hl ρ · obtain ⟨μ,hμ,h⟩ := density_domination_small s p l hp0 hps hls (by omega) ρ refine ⟨μ,hμ,?_⟩ have hM : (∫ r : EvenDensity p, densityPower l r.val ∂μ).PosSemidef := by -- Positivity of the integral follows from the explicitly proved lower -- bound by a PSD projection; no measurability of arbitrary kernels is needed. have ha : 0 < (ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(12*p^2) := by exact_mod_cast mul_pos (show (0 : ℝ) < ρ.dimension by exact_mod_cast ρ.positive) (pow_pos (show (0 : ℝ) < s+1 by positivity) _) have hh := h.add (isotypic_posSemidef ρ) rw [sub_add_cancel] at hh have hc : (ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(12*p^2) ≠ 0 := ne_of_gt ha have hh' := hh.smul (le_of_lt (inv_pos.mpr ha)) rwa [smul_smul,inv_mul_cancel₀ hc,one_smul] at hh' apply domination_pad _ _ _ _ hM _ h apply mul_le_mul_of_nonneg_left _ (by positivity) have he : 12*p^2 ≤ 41*p^2 := by omega exact_mod_cast Nat.pow_le_pow_right (show 0 < s+1 by omega) he end CoordinateSweeps.SignedTensor namespace CoordinateSweeps.SignedTensor open PositiveTensor lemma finite_profile_domination (s p l : ℕ) (hp : 0 < p) (hps : p ≤ s) (hls : l ≤ s) (hl : (2*p)*(2*p) < l) (ρ : UnitaryIrrep (Equiv.Perm (Fin l))) : (((ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(41*p^2)) • ((Fintype.card (DensityProfile p l) : ℂ)⁻¹ • rawProfileDensity s p l hp hl) - isotypic (p := p) ρ).PosSemidef := by let : Nonempty (DensityProfile p l) := densityProfile_nonempty hp let A : commutant p l := ⟨isotypic ρ,isotypic_mem_commutant ρ⟩ have h := commutant_covariance_domination s p l hp hps hls hl A change ((_ * _) • _ - isotypic ρ * (isotypic ρ).conjTranspose).PosSemidef at h simp only [A,isotypic_self_mul] at h have hcard : Fintype.card (DensityProfile p l) ≤ (s+1)^(12*p^2) := (densityProfile_card_le p l).trans (Nat.pow_le_pow_left (Nat.add_le_add_right hls 1) _) have hc0 : (Fintype.card (DensityProfile p l) : ℂ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero have hbig : ((s+1 : ℕ) : ℂ)^(12*p^2) * (Fintype.card (DensityProfile p l) : ℂ)⁻¹ ≥ 1 := by have hi : (0 : ℂ) ≤ (Fintype.card (DensityProfile p l) : ℂ)⁻¹ := by positivity have hh := mul_le_mul_of_nonneg_right (show (Fintype.card (DensityProfile p l) : ℂ) ≤ ((s+1 : ℕ) : ℂ)^(12*p^2) by exact_mod_cast hcard) hi simpa only [mul_inv_cancel₀ hc0] using hh rw [smul_smul] apply domination_pad _ _ _ _ (rawProfileDensity_posSemidef s p l hp hl) _ h calc _ ≤ ((ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(4*p^2)) * ((s+1 : ℕ) : ℂ)^(25*p^2) := mul_le_mul_of_nonneg_right (isotypic_trace_le s p l hls ρ) (by positivity) _ = (ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(29*p^2) := by rw [mul_assoc,← pow_add]; congr 2; ring _ ≤ ((ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(29*p^2)) * (((s+1 : ℕ) : ℂ)^(12*p^2) * (Fintype.card (DensityProfile p l) : ℂ)⁻¹) := by have hh := mul_le_mul_of_nonneg_left hbig (show (0 : ℂ) ≤ (ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(29*p^2) by positivity) simpa only [mul_one] using hh _ = _ := by rw [← mul_assoc,mul_assoc (ρ.dimension : ℂ),← pow_add] congr 3; ring lemma finite_uniform_domination (s p l : ℕ) (hp : 0 < p) (hps : p ≤ s) (hls : l ≤ s) (hl : l ≤ (2*p)*(2*p)) (ρ : UnitaryIrrep (Equiv.Perm (Fin l))) : (((ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(12*p^2)) • densityPower l (uniformDensity p hp).val-isotypic (p := p) ρ).PosSemidef := by rw [uniformDensity_power,smul_smul] have hP := trace_domination (isotypic (p := p) ρ) (isotypic_posSemidef ρ) apply domination_pad _ 1 _ _ Matrix.PosSemidef.one _ hP have hc0 : (((2*p : ℕ) : ℂ)^l) ≠ 0 := pow_ne_zero _ (Nat.cast_ne_zero.mpr (by omega)) have hi : (0 : ℂ) ≤ (((2*p : ℕ) : ℂ)^l)⁻¹ := by positivity have hh := mul_le_mul_of_nonneg_right (show (((2*p : ℕ) : ℂ)^l) ≤ ((s+1 : ℕ) : ℂ)^(8*p^2) by exact_mod_cast small_power_le s p l hps hl) hi rw [mul_inv_cancel₀ hc0] at hh calc _ ≤ (ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(4*p^2) := isotypic_trace_le s p l hls ρ _ ≤ ((ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(4*p^2)) * (((s+1 : ℕ) : ℂ)^(8*p^2)*(((2*p : ℕ) : ℂ)^l)⁻¹) := by simpa only [mul_one] using mul_le_mul_of_nonneg_left hh (show (0 : ℂ) ≤ (ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(4*p^2) by positivity) _ = _ := by rw [← mul_assoc,mul_assoc (ρ.dimension : ℂ),← pow_add]; congr 3; ring /- Finite (not a hypothetical measure) version of source05:eq13. -/ theorem finite_density_domination (s p l : ℕ) (hp : 0 < p) (hps : p ≤ s) (hls : l ≤ s) (ρ : UnitaryIrrep (Equiv.Perm (Fin l))) : ∃ (I : Type) (_ : Fintype I) (_ : Nonempty I) (r : I → EvenDensity p), (((ρ.dimension : ℂ)*((s+1 : ℕ) : ℂ)^(41*p^2)) • ((Fintype.card I : ℂ)⁻¹ • ∑ i, densityPower l (r i).val)-isotypic (p := p) ρ).PosSemidef := by by_cases hl : (2*p)*(2*p) uniformDensity p hp,?_⟩ simp only [Fintype.card_punit,Nat.cast_one,inv_one,Finset.univ_unique,Finset.sum_singleton,one_smul] apply domination_pad _ _ _ _ (densityPower_posSemidef _ _ (uniformDensity p hp).property.1) _ (finite_uniform_domination s p l hp hps hls (by omega) ρ) apply mul_le_mul_of_nonneg_left _ (by positivity) exact_mod_cast Nat.pow_le_pow_right (show 0 < s+1 by omega) (show 12*p^2≤41*p^2 by omega) end CoordinateSweeps.SignedTensor end end end end open scoped Matrix.Norms.L2Operator end OAI