import OAI.Dynamics.ThreeTorus.ColorBlocks import OAI.Dynamics.ThreeTorus.Prediction import OAI.Dynamics.ThreeTorus.SymmetricPackets namespace OAI /-! Symmetric anchored blocks, unconditional block production, physical mass, and estimator regularity. -/ section /-! Concrete finite colored iteration. The cofinal-frequency tree is built from actual neutral--signal pairs. The error budget is fixed before the initial anchor state and before any future packet index type is known. -/ noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators namespace ThreeTorus def ColoredSignalReflectedRealizes (b D : ℝ) (d : ℝ → ℂ) (γ ell : ℝ) : Prop := ∀ (R : Type) [DecidableEq R], ∀ (C : ColoredState R), ∀ τR : R → R, C.Reflection τR → ∀ margin : ℝ, 0 < margin → C.InBand b D margin → ∀ (a : R → UnitAddCircle → ℂ), (∀ r, Continuous (a r)) → ∀ ω : ℝ, 0 ≤ ω → C.Matches d a ω → ∀ ε η κ : ℝ, 0 < ε → 0 < η → 0 < κ → ∀ᶠ N : ℕ in atTop, ∃ s : ColoredPair C N, s.Tracks η ∧ s.output.Reflection τR ∧ (∃ δ : ℝ, 0 < δ ∧ δ < 1/8 ∧ δ < κ ∧ s.final.GoodArc δ) ∧ ∀ r, physicalError (C.vector r) (s.output.vector r ∘ s.chart) < ε ∧ (∫ z, |s.output.density r z-C.density r z*colorFactor a ell (N:ℤ) r z| ∂phaseHaar) < 2*γ*ω*(∫ z, C.density r z ∂phaseHaar)+ε /-- The interface is discharged by the actual geometric construction. -/ theorem exists_concrete_colored_reflected_signal {b D : ℝ} (hb : 0 < b) (hbD : b < D) {d : ℝ → ℂ} (hd : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) d) (hdc : HasCompactSupport d) (hdr : ∀ v, d (-v) = starRingEnd ℂ (d v)) (hd0 : d 0 = 0) (hds : support d ⊆ Icc (-D) D) : ∃ L : ℕ, ∃ γ₀ : ℝ, 2 ≤ L ∧ D/b < L ∧ 0 < γ₀ ∧ ∀ γ : ℝ, 0 < γ → γ ≤ γ₀ → (∀ v, 2*γ*‖d v‖ ≤ 1/2) ∧ ColoredSignalReflectedRealizes b D d γ (refreshedStrength (signalPrimeProduct L) γ) := exists_colored_reflected_signal hb hbD hd hdc hdr hd0 hds /-- Fixed data and a small coefficient tolerance first; the threshold tree is chosen on the actual initial state; ALL diagonals above that tree are realized with small product-model and physical errors. No independence of the diagonal, modeled next state, or unproved packet producer is used. -/ theorem ColoredSignalReflectedRealizes.iterate {b D γ ell : ℝ} {d : ℝ → ℂ} (hsignal : ColoredSignalReflectedRealizes b D d γ ell) (hd : Continuous d) (hdc : HasCompactSupport d) (hγ : 0 < γ) (hell : 0 ≤ ell) (hells : 6*ell ≤ 1) (m n : ℕ) (τR : Fin m → Fin m) (a : Fin m → UnitAddCircle → ℂ) (ha : ∀ r, Continuous (a r)) (ha3 : ∀ r z, ‖a r z‖ ≤ 3) {M ε κ : ℝ} (hM : 0 ≤ M) (hε : 0 < ε) (hκ : 0 < κ) : ∃ ω : ℝ, 0 < ω ∧ ∀ C : ColoredState (Fin m), C.Reflection τR → colorMass C.density ≤ M → ∀ margin : ℝ, 0 < margin → C.InBand b D margin → C.Matches d a ω → ∃ B : (j : ℕ) → (Fin j → ℤ) → ℤ, ∀ L : Fin n → ℤ, (∀ j : Fin n, B j (fun i => L ⟨i.val,i.isLt.trans j.isLt⟩) < L j) → ∃ E : ColoredState (Fin m), ∃ α : Equiv.Perm TripleTorus, ColoredPath C L E α ∧ colorL1Distance (colorModel C.density a ell L) E.density < ε ∧ (∀ r, physicalError (C.vector r) (E.vector r ∘ α) < ε) ∧ ∃ μ : ℝ, 0 < μ ∧ E.InBand b D μ ∧ E.Matches d a (((n:ℝ)+1)*ω) ∧ E.Reflection τR ∧ (0 < n → ∃ δ : ℝ, 0 < δ ∧ δ < 1/8 ∧ δ < κ ∧ E.packets.GoodArc δ) := by classical obtain ⟨t,ω,U,ht,hω,hU,hfinal,hmassBound,hsubstitution⟩ := finite_iteration_budget n hM hγ hε obtain ⟨e,he,het,hem⟩ := finite_color_tolerance m ht refine ⟨ω,hω,?_⟩ intro C hrC hmass margin hmargin hband hmatch let Reach (k : ℕ) (L : Fin k → ℤ) : Prop := ∃ E : ColoredState (Fin m), ∃ α : Equiv.Perm TripleTorus, ∃ μ : ℝ, 0 < μ ∧ ColoredPath C L E α ∧ E.InBand b D μ ∧ E.Matches d a (((k:ℝ)+1)*ω) ∧ colorL1Distance (colorModel C.density a ell L) E.density ≤ iterationError t k ∧ (∀ r, physicalError (C.vector r) (E.vector r ∘ α) ≤ iterationError t k) ∧ E.Reflection τR ∧ (0 < k → ∃ δ : ℝ, 0 < δ ∧ δ < 1/8 ∧ δ < κ ∧ E.packets.GoodArc δ) have hzero : Reach 0 (fun i => Fin.elim0 i) := by refine ⟨C,Equiv.refl _,margin,hmargin,ColoredPath.nil,hband,?_,?_,?_,hrC,(by intro h; omega)⟩ · simpa using hmatch · rw [colorModel_zero] simpa [colorL1Distance,iterationError] using ht.le · intro r simpa [physicalError,iterationError] using ht.le have hstep : ∀ k (L : Fin k → ℤ), k < n → Reach k L → ∀ᶠ N : ℤ in atTop, Reach (k+1) (Fin.snoc L N) := by intro k L hk hreach obtain ⟨E,α,μ,hμ,hpath,hbandE,hmatchE,herror,hphysical,hrE,_hgoodE⟩ := hreach have hmassE : colorMass E.density ≤ U := by have hmodel := colorModel_mass_bound C.density_continuous C.density_nonneg ha ha3 hell hells L have hbase := mul_le_mul_of_nonneg_right hmass (show (0:ℝ) ≤ 2^k by positivity) exact (colorMass_le_add_distance (continuous_colorModel C.density_continuous ha ell L) E.density_continuous).trans ((add_le_add (hmodel.trans hbase) herror).trans (hmassBound k hk.le)) obtain ⟨η₀,hη₀,hcoeff⟩ := core_coefficient_drift_budget E.packets hd hdc (fun i => a (E.color i)) (fun i => ha (E.color i)) hω hmatchE let η := min η₀ (μ/2) have hη : 0 < η := lt_min hη₀ (half_pos hμ) have hωk : 0 ≤ ((k:ℝ)+1)*ω := by positivity have hready := hsignal (Fin m) E τR hrE μ hμ hbandE a ha (((k:ℝ)+1)*ω) hωk hmatchE e η κ he hη hκ apply eventually_int_of_eventually_nat filter_upwards [hready] with N hN obtain ⟨s,htrack,hrNext,hgoodNext,hcolor⟩ := hN have hbandNext : s.output.InBand b D (μ/2) := htrack.band_margin hbandE (min_le_right _ _) have hcoeffNext : s.output.Matches d a ((((k+1:ℕ):ℝ)+1)*ω) := by intro i y _hy have hh := hcoeff _ s.final (Sigma.fst ∘ Sigma.fst) s.output.scale (htrack.mono (min_le_left _ _)) i y change ‖d ((s.final.m i:ℝ)/s.output.scale)- a (E.color ((Sigma.fst ∘ Sigma.fst) i)) ((s.final.m i:ℝ)*s.final.V y:UnitAddCircle)‖ ≤ _ have heq : ((((k+1:ℕ):ℝ)+1)*ω) = ((k:ℝ)+1)*ω+ω := by push_cast; ring rw [heq] exact hh have hdensityStep : colorL1Distance (fun r z => E.density r z*colorFactor a ell (N:ℤ) r z) s.output.density ≤ t := by rw [colorL1Distance_symm] have hh := Finset.sum_le_sum (s := Finset.univ) (fun r _ => (hcolor r).2.le) simp only [Finset.sum_add_distrib,← Finset.mul_sum,Finset.sum_const, Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] at hh change colorL1Distance s.output.density (fun r z => E.density r z*colorFactor a ell (N:ℤ) r z) ≤ 2*γ*(((k:ℝ)+1)*ω)*colorMass E.density+(m:ℝ)*e at hh have hb := mul_le_mul_of_nonneg_left hmassE (show 0 ≤ 2*γ*(((k:ℝ)+1)*ω) by positivity) have hs := hsubstitution k hk.le linarith refine ⟨s.output,α.trans s.chart,μ/2,half_pos hμ,hpath.snoc s,hbandNext,hcoeffNext,?_,?_,hrNext,(fun _ => hgoodNext)⟩ · have hh := colorModel_step_error C.density_continuous E.density_continuous s.output.density_continuous ha ha3 hell hells L (N:ℤ) have hs := iterationError_step ht.le k linarith · intro r have htwo := hpath.physicalError_eq (E.vector r) (s.output.vector r ∘ s.chart) have hh := physicalError_triangle (C.vector_continuous r) ((E.vector_continuous r).comp hpath.continuous) ((s.output.vector_continuous r).comp (s.chart_continuous.comp hpath.continuous)) change physicalError (C.vector r) (s.output.vector r ∘ (α.trans s.chart)) ≤ 2*physicalError (C.vector r) (E.vector r ∘ α)+ 2*physicalError (E.vector r ∘ α) ((s.output.vector r ∘ s.chart) ∘ α) at hh rw [htwo] at hh have hs := iterationError_step ht.le k have hp := hphysical r have hp' := (hcolor r).1 linarith obtain ⟨B,hB⟩ := finite_frequency_tree Reach hzero hstep refine ⟨B,?_⟩ intro L hL obtain ⟨E,α,μ,hμ,hpath,hbandE,hmatchE,herr,hphys,hrE,hgoodE⟩ := hB L hL exact ⟨E,α,hpath,herr.trans_lt hfinal,fun r => (hphys r).trans_lt hfinal, μ,hμ,hbandE,hmatchE,hrE,hgoodE⟩ end ThreeTorus end end section noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators namespace ThreeTorus /-- Accuracy requirements for the actual anchor are fixed BEFORE that anchor is made. Given any such actual anchor, the two block tests are achieved on one actual endpoint of a sequence of neutral/signal pairs. -/ theorem ColoredSignalReflectedRealizes.block {b D γ ell : ℝ} {d : ℝ → ℂ} (hsignal : ColoredSignalReflectedRealizes b D d γ ell) (hdc : Continuous d) (hds : HasCompactSupport d) (hγ : 0 < γ) {m n : ℕ} {p : Fin m → UnitAddCircle → ℝ} (hp : ∀ r z, 0 ≤ p r z) (hc : ∀ r, Continuous (p r)) (tags : Fin m → ℂ) (htags : ∀ r, ‖tags r‖ ≤ 1) {q A : UnitAddCircle → ℂ} (hq : Continuous q) (hqB : ∀ z, ‖q z‖ ≤ 2) (hA : Continuous A) (k : ℕ) {delta nu ε : ℝ} (hell : 0 < ell) (hells : 6*ell ≤ 1) (hd : 0 < delta) (hd1 : delta ≤ 1/100) (hn0 : delta/2 ≤ (n:ℝ)*ell^2) (hn1 : (n:ℝ)*ell^2 ≤ delta) (hmass : (∫ z, blockTotal p z ∂phaseHaar) ≤ 2) (hcenter : (∫ z, Real.sqrt (blockTotal p z)*‖blockTagMean p tags z-q z‖^2 ∂phaseHaar) ≤ delta) (hnu : 0 < nu) (hε : 0 < ε) (τR : Fin m → Fin m) {κ : ℝ} (hκ : 0 < κ) : ∃ e ω : ℝ, 0 < e ∧ 0 < ω ∧ ∀ C : ColoredState (Fin m), C.Reflection τR → colorL1Distance p C.density ≤ e → ∀ margin : ℝ, 0 < margin → C.InBand b D margin → C.Matches d (fun r z => tags r-q z) ω → ∃ L : Fin n → ℤ, (∀ j, 0 < L j) ∧ ∃ E : ColoredState (Fin m), ∃ α : Equiv.Perm TripleTorus, ColoredPath C L E α ∧ (∀ r, physicalError (C.vector r) (E.vector r ∘ α) < ε) ∧ ((∫ z, Real.sqrt (blockTotal p z) ∂phaseHaar)-300*delta^2-nu ≤ ∫ z, Real.sqrt (blockTotal E.density z) ∂phaseHaar) ∧ ((∑ r, ∫ z, E.density r z* ‖A z+blockDiagonalEstimator q ell L z-((k:ℂ)+1)*tags r‖^2 ∂phaseHaar) ≤ (∫ z, blockPreviousError p tags A k z ∂phaseHaar)+4/delta+nu) ∧ E.Reflection τR ∧ ∃ δ : ℝ, 0 < δ ∧ δ < 1/8 ∧ δ < κ ∧ E.packets.GoodArc δ := by obtain ⟨e0,he0,htests⟩ := block_diagonal_transfer hp hc tags htags hq hqB hA k hell hells hd hd1 hn0 hn1 hmass hcenter hnu let a : Fin m → UnitAddCircle → ℂ := fun r z => tags r-q z have ha : ∀ r, Continuous (a r) := fun r => continuous_const.sub hq have ha3 : ∀ r z, ‖a r z‖ ≤ 3 := fun r z => (norm_sub_le _ _).trans (by linarith [htags r,hqB z]) obtain ⟨ω,hω,hiteration⟩ := hsignal.iterate hdc hds hγ hell.le hells m n τR a ha ha3 (M := 3) (by norm_num) (lt_min (half_pos he0) hε) hκ let e : ℝ := min 1 (e0/(2*2^n)) have he : 0 < e := lt_min zero_lt_one (div_pos he0 (by positivity)) refine ⟨e,ω,he,hω,?_⟩ intro C hrC hclose margin hmargin hband hmatch have hmassC : colorMass C.density ≤ 3 := by have h0 := colorMass_le_add_distance hc C.density_continuous rw [colorMass_eq_integral_total hc] at h0 have he1 : e ≤ 1 := min_le_left _ _ linarith obtain ⟨B,hB⟩ := hiteration C hrC hmassC margin hmargin hband hmatch obtain ⟨L,hL,hLB,htest⟩ := htests B obtain ⟨E,α,hpath,hdensity,hphysical,_μ,_hμ,_hbandE,_hmatchE,hrE,hgoodE⟩ := hB L hLB have hinitial := colorModel_initial_error hc C.density_continuous ha ha3 hell.le hells L have hsmall : 2^n*colorL1Distance p C.density ≤ e0/2 := by have he2 : e ≤ e0/(2*2^n) := min_le_right _ _ have hprod := (le_div_iff₀ (show (0:ℝ) < 2*2^n by positivity)).1 (hclose.trans he2) nlinarith have hmodels := colorL1Distance_triangle (continuous_colorModel hc ha ell L) (continuous_colorModel C.density_continuous ha ell L) E.density_continuous have herr : colorL1Distance (blockModelColors p tags q ell L) E.density ≤ e0 := by rw [← colorModel_eq_blockModelColors] change colorL1Distance (colorModel p a ell L) E.density ≤ e0 have hi := hinitial.trans hsmall have hd' := lt_of_lt_of_le hdensity (min_le_left (e0/2) ε) linarith obtain ⟨hsqrt,hpred⟩ := htest E.density E.density_continuous E.density_nonneg herr have hnpos : 0 < n := by by_contra hh have hn : n = 0 := by omega rw [hn] at hn0 norm_num at hn0 linarith exact ⟨L,hL,E,α,hpath,(fun r => (hphysical r).trans_le (min_le_right _ _)),hsqrt,hpred,hrE,hgoodE hnpos⟩ end ThreeTorus end end section /-! The correctly ordered anchor constructs the literal finite output state, with coefficient matching valid on every next packet core. -/ noncomputable section open Set Function Filter MeasureTheory Complex open scoped Topology BigOperators ComplexConjugate namespace ThreeTorus /-- Carrier first; signal, band and all block tolerances afterwards; anchor integer last and still arbitrarily large. No modeled next densities occur. -/ theorem eventually_reflected_anchor_state {ι : Type*} [Fintype ι] (F : PassagePackets ι) (hi : Injective F.m) (hmod : F.SpareIndependent) {τref : ι → ι} (hr : F.Reflection τref) : ∀ᶠ c : ℕ in atTop, ∀ (d : ℝ → ℂ) (q : UnitAddCircle → ℂ) (a : ι → ℂ), Continuous q → ∀ b D margin τ : ℝ, 0 < τ → (∀ i x, x ∈ F.core i → ∀ s, |s-(F.m i:ℝ)*((c:ℝ)+F.V x)| < τ → d s = a i-q (s:UnitAddCircle) ∧ b+margin ≤ |s| ∧ |s| ≤ D-margin) → ∀ ε ω κ : ℝ, 0 < ε → 0 < ω → 0 < κ → ∃ g : PassageGeometryFamily F.V, g.c = c ∧ ∃ δ : ℝ, ∃ cut : LawCutoffs δ, ∃ L : ℝ, 0 < δ ∧ δ < κ ∧ 0 < L ∧ g.J₀ = neutralJitterReal cut L 0 ∧ ∃ S : ℕ → ι → Finset ℤ, ∀ᶠ n : ℕ in atTop, 0 < n ∧ ∃ G : PassagePackets (PassageOutputIndex (S n)), Injective G.m ∧ G.SpareIndependent ∧ G.V = (fun x => (1+g.J₀ x/n)/n) ∧ (∀ i, G.m i = i.2.val) ∧ (∀ i, (∫ x : TripleTorus, ‖packet (F.m i) (F.A i) x- outputVector G i (passageChart (g.H n) x)‖^2 ∂tripleVolume) < ε ∧ (∫ z, |outputDensity G i z-F.density i z| ∂phaseHaar) < ε) ∧ (∀ i, b+margin ≤ |(G.m i:ℝ)/n| ∧ |(G.m i:ℝ)/n| ≤ D-margin) ∧ (∀ i y, ‖d ((G.m i:ℝ)/n)- (a i.1-q ((G.m i:ℝ)*G.V y:UnitAddCircle))‖ < ω) ∧ ∃ τG : PassageOutputIndex (S n) → PassageOutputIndex (S n), (∀ i, (τG i).1 = τref i.1) ∧ G.Reflection τG ∧ G.GoodArc δ := by filter_upwards [eventually_neutral_anchor_reflected_family F hi hmod hr] with c hc intro d q a hq b D margin τ hτ hroom ε ω κ hε hω hκ obtain ⟨hcp,δ,cut,L,χ,Δ,H,hδ,_hδ',hδκ,hL,hχ,_hχd,hΔ,hΔt,hHc,hHe, hgeo,hgraph,hlimit,S,_hSd,hS,hgood⟩ := hc 1 (by norm_num) ε τ κ hε hτ hκ let g : PassageGeometryFamily F.V := ⟨c,hcp,Δ,H,neutralJitter cut L 0,neutralJitterReal cut L 0,(fun _ => rfl), continuous_neutralJitter cut L 0,hΔ,hΔt,hHc,hHe,hgeo,hgraph,hlimit⟩ obtain ⟨R,_hR,hRJ⟩ := periodic_real_bound (neutralJitterReal_smooth cut L 0).continuous (neutralJitterReal_periodic cut L 0) have hbudget := anchor_coefficient_budget (d := d) (D := D-margin) hq a hω R refine ⟨g,rfl,δ,cut,L,hδ,hδκ,hL,rfl,S,?_⟩ filter_upwards [hgood,hbudget] with n hn hnω obtain ⟨hnp,haul,G,hGi,hGmod,hGV,hGm,hGA,_hGden,hgroup,hreflect⟩ := hn have hloc (i : PassageOutputIndex (S n)) : d ((G.m i:ℝ)/n) = a i.1-q (((G.m i:ℝ)/(n:ℝ)):UnitAddCircle) ∧ b+margin ≤ |(G.m i:ℝ)/n| ∧ |(G.m i:ℝ)/n| ≤ D-margin := by obtain ⟨x,hx,hx'⟩ := (hχ i.1).2.2.2.2 _ (subset_tsupport _ ((hS n hnp i.1 _).mp i.2.property)) rw [hGm] exact hroom i.1 x hx _ hx' refine ⟨hnp,G,hGi,hGmod,hGV,hGm,?_,fun i => (hloc i).2,?_,hreflect⟩ · intro i have hv := outputVector_eq_retained F G (neutralCircleCutoff cut) (fun i s _ => χ i s) H n hGm hGA i have hp : exactRetainedPacketDensity (Ioo δ (1-δ)) (neutralJitterReal cut L 0) (neutralCircleCutoff cut) (fun s _ => χ i s) (F.m i) H (F.A i) (S n i) n = outputDensity G i := by have hh := hgroup i (fun _ => True) change _ = fun z => ∑ j : ↥(S n i), G.density ⟨i,j⟩ z simpa only [Finset.filter_true,ite_true] using hh have hh := haul i dsimp only at hh rw [← hv,hp] at hh exact ⟨hh.1,hh.2.1⟩ · intro i y rw [hGV] exact hnω.2 (G.m i) i.1 (neutralJitterReal cut L 0 y) (hloc i).2.2 (hRJ y) (hloc i).1 end ThreeTorus end end section noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators namespace ThreeTorus /-- The carrier precedes the block signal and every block accuracy requirement. The output retains the literal actual geometry and finite amplitudes. -/ theorem eventually_colored_reflected_anchor {m : ℕ} (C : ColoredState (Fin m)) {τR : Fin m → Fin m} (hrC : C.Reflection τR) : ∀ᶠ c : ℕ in atTop, ∀ (d : ℝ → ℂ) (q : UnitAddCircle → ℂ) (tags : Fin m → ℂ), Continuous q → ∀ b D margin τ : ℝ, 0 < τ → (∀ i x, x ∈ C.packets.core i → ∀ s, |s-(C.packets.m i:ℝ)*((c:ℝ)+C.packets.V x)| < τ → d s = tags (C.color i)-q (s:UnitAddCircle) ∧ b+margin ≤ |s| ∧ |s| ≤ D-margin) → ∀ ε e ω κ : ℝ, 0 < ε → 0 < e → 0 < ω → 0 < κ → ∀ᶠ n : ℕ in atTop, ∃ s : ColoredAnchor C n, s.geometry.c = c ∧ (∃ δ : ℝ, ∃ cut : LawCutoffs δ, ∃ K : ℝ, 0 < δ ∧ δ < κ ∧ 0 < K ∧ s.geometry.J₀ = neutralJitterReal cut K 0) ∧ (∀ r, physicalError (C.vector r) (s.output.vector r ∘ s.chart) < ε) ∧ colorL1Distance C.density s.output.density < e ∧ s.output.InBand b D margin ∧ s.output.Matches d (fun r z => tags r-q z) ω ∧ s.output.Reflection τR ∧ ∃ δ : ℝ, 0 < δ ∧ δ < κ ∧ s.final.GoodArc δ := by classical obtain ⟨τref,hr,hτref⟩ := hrC filter_upwards [eventually_reflected_anchor_state C.packets C.injective C.spare hr] with c hc intro d q tags hq b D margin τ hτ hroom ε e ω κ hε he hω hκ let K : ℝ := Fintype.card C.Index have hK : 0 ≤ K := Nat.cast_nonneg _ let e' : ℝ := min (ε/(K^2+1)) (e/((m:ℝ)*K+1)) have he' : 0 < e' := lt_min (div_pos hε (by positivity)) (div_pos he (by positivity)) have hphysicalBudget : K^2*e' < ε := by have he1 : e' ≤ ε/(K^2+1) := min_le_left _ _ have hp := (le_div_iff₀ (show 0 < K^2+1 by positivity)).1 he1 nlinarith have hdensityBudget : (m:ℝ)*K*e' < e := by have he1 : e' ≤ e/((m:ℝ)*K+1) := min_le_right _ _ have hp := (le_div_iff₀ (show 0 < (m:ℝ)*K+1 by positivity)).1 he1 nlinarith obtain ⟨g,hgc,δ,cut,L,hδ,hδκ,hL,hgJ,S,hready⟩ := hc d q (tags ∘ C.color) hq b D margin τ hτ hroom e' ω κ he' hω hκ filter_upwards [hready] with n hn obtain ⟨hnp,G,hGi,hGmod,hGV,hGm,herror,hband,hmatch,τG,hτG,hrG,hgoodG⟩ := hn let s : ColoredAnchor C n := ⟨g,S n,hnp,G,hGi,hGmod,hGV,hGm⟩ refine ⟨s,hgc,⟨δ,cut,L,hδ,hδκ,hL,hgJ⟩,?_,?_,hband,?_,⟨τG,hrG,?_⟩,⟨δ,hδ,hδκ,hgoodG⟩⟩ · intro r have hh := colored_physical_bound (fun i => continuous_packet (C.packets.continuous_amplitude i)) (fun i => (continuous_outputVector G i).comp (continuous_passageChart (g.Hcontinuous n))) C.color r he'.le (fun i => (herror i).1.le) hphysicalBudget change physicalError (C.vector r) _ < ε at hh have heq : (s.output.vector r ∘ s.chart) = (fun x => ∑ i, if C.color i = r then outputVector G i (passageChart (g.H n) x) else 0) := by funext x exact colorVector_output G C.color r _ rw [heq] exact hh · have hcolor r : (∫ z, |C.density r z-s.output.density r z| ∂phaseHaar) ≤ K*e' := by have hh := colored_density_bound (fun i => continuous_outputDensity G i) C.packets.density_continuous C.color r (fun _ => 0) (γ := 0) (ω := 0) he'.le (fun i => by simpa using (herror i).2.le) simp only [zero_mul,zero_add] at hh have heq : (fun z => ∑ i, if C.color i = r then outputDensity G i z else 0) = s.output.density r := by funext z exact (colorDensity_output G C.color r z).symm change (∫ z, |(fun z => ∑ i, if C.color i = r then outputDensity G i z else 0) z- C.density r z| ∂phaseHaar) ≤ K*e' at hh rw [heq] at hh simpa only [abs_sub_comm] using hh apply (Finset.sum_le_sum (s := Finset.univ) (fun r _ => hcolor r)).trans_lt simpa only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_assoc] using hdensityBudget · intro i y _hy exact (hmatch i y).le · intro i change C.color ((τG i).1) = τR (C.color i.1) exact (congrArg C.color (hτG i)).trans (hτref i.1) end ThreeTorus end end section noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators ComplexConjugate namespace ThreeTorus /-- All signal parameters and tolerances are fixed before the anchor. The endpoint is a genuine actual state linked by its anchor and finite passage path, and satisfies both prescribed source block inequalities. The input reflection hypotheses are the ordinary packet reality conditions needed to construct the real-symmetric band signal. -/ theorem actual_reflected_block_of_center {m : ℕ} (C : ColoredState (Fin m)) (τR : Fin m → Fin m) (hrC : C.Reflection τR) (tags : Fin m → ℂ) (htags : ∀ r, ‖tags r‖ ≤ 1) (htrR : ∀ r, tags (τR r) = conj (tags r)) {q A : UnitAddCircle → ℂ} (hq : Continuous q) (hqs : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => q (x:UnitAddCircle))) (hqr : ∀ z, q (-z) = conj (q z)) (hqB : ∀ z, ‖q z‖ ≤ 2) (hA : Continuous A) (k : ℕ) {delta nu ε κ : ℝ} (hd : 0 < delta) (hd1 : delta ≤ 1/100) (hnu : 0 < nu) (hε : 0 < ε) (hκ : 0 < κ) (hmass : (∫ z, blockTotal C.density z ∂phaseHaar) ≤ 2) (hcenter : (∫ z, Real.sqrt (blockTotal C.density z)* ‖blockTagMean C.density tags z-q z‖^2 ∂phaseHaar) ≤ delta) : ∃ n : ℕ, ∃ ell : ℝ, 0 < n ∧ 0 < ell ∧ 6*ell ≤ 1 ∧ delta/2 ≤ (n:ℝ)*ell^2 ∧ (n:ℝ)*ell^2 ≤ delta ∧ ∃ N : ℕ, ∃ s : ColoredAnchor C N, (∃ δ : ℝ, ∃ cut : LawCutoffs δ, ∃ K : ℝ, 0 < δ ∧ δ < κ ∧ 0 < K ∧ s.geometry.J₀ = neutralJitterReal cut K 0) ∧ ∃ L : Fin n → ℤ, (∀ j, 0 < L j) ∧ ∃ E : ColoredState (Fin m), ∃ α : Equiv.Perm TripleTorus, ColoredPath s.output L E α ∧ (∀ r, physicalError (C.vector r) (E.vector r ∘ s.chart.trans α) < ε) ∧ ((∫ z, Real.sqrt (blockTotal C.density z) ∂phaseHaar)-300*delta^2-nu ≤ ∫ z, Real.sqrt (blockTotal E.density z) ∂phaseHaar) ∧ ((∑ r, ∫ z, E.density r z* ‖A z+blockDiagonalEstimator q ell L z-((k:ℂ)+1)*tags r‖^2 ∂phaseHaar) ≤ (∫ z, blockPreviousError C.density tags A k z ∂phaseHaar)+4/delta+nu) ∧ E.Reflection τR ∧ ∃ δ : ℝ, 0 < δ ∧ δ < 1/8 ∧ δ < κ ∧ E.packets.GoodArc δ := by classical obtain ⟨τ,hr,hτ⟩ := hrC have hmr := hr.index have hcore i : C.packets.core i ⊆ C.packets.core (τ i) := by rw [hr.core] have htr i : tags (C.color (τ i)) = conj (tags (C.color i)) := by rw [hτ,htrR] obtain ⟨c,hsep,hanchor⟩ := ((eventually_integral_speed_separation C.packets.m C.packets.periodic.lift C.packets.nonzero C.packets.speed_continuous).and (eventually_colored_reflected_anchor C ⟨τ,hr,hτ⟩)).exists let K : C.Index → Set ℝ := fun i => (fun x => (C.packets.m i:ℝ)*((c:ℝ)+C.packets.V x)) '' C.packets.core i have hK i : IsCompact (K i) := (C.packets.core_compact i).image (continuous_const.mul (continuous_const.add C.packets.smooth_speed.continuous)) have hK0 i : 0 ∉ K i := by rintro ⟨x,_hx,he⟩ exact hsep.2.1 i (x:UnitAddCircle) he have hKd : Pairwise (Disjoint on K) := by intro i j hij change Disjoint (K i) (K j) rw [Set.disjoint_left] rintro z ⟨x,_hx,rfl⟩ ⟨y,_hy,he⟩ exact hij (C.injective (hsep.2.2 i j (x:UnitAddCircle) (y:UnitAddCircle) he.symm)) have hKr i x (hx : x ∈ K i) : -x ∈ K (τ i) := by obtain ⟨y,hy,rfl⟩ := hx refine ⟨y,hcore i hy,?_⟩ simp only [hmr,Int.cast_neg,neg_mul] obtain ⟨d,hds,hdcompact,hd0,hdr,hdlocal⟩ := exists_symmetric_anchor_band K hK hKd hK0 τ hKr (tags ∘ C.color) htr hqs hqr obtain ⟨b,D,μ,t,hb,hbD,hμ,ht,hdsupport,hroom⟩ := anchor_uniform_room K hK hK0 hdcompact (fun i y => tags (C.color i)-q (y:UnitAddCircle)) hdlocal obtain ⟨P,γ₀,_hP,_hPD,hγ₀,hsignal⟩ := exists_concrete_colored_reflected_signal hb hbD hds hdcompact hdr hd0 hdsupport let γ : ℝ := min γ₀ (delta/2)/2 have hγ : 0 < γ := half_pos (lt_min hγ₀ (half_pos hd)) have hγ₀' : γ ≤ γ₀ := by have hh := min_le_left γ₀ (delta/2) dsimp [γ] linarith [lt_min hγ₀ (half_pos hd)] have hγd : γ ≤ delta/2 := by have hh := min_le_right γ₀ (delta/2) dsimp [γ] linarith [lt_min hγ₀ (half_pos hd)] let ell := refreshedStrength (signalPrimeProduct P) γ have hell : 0 < ell := refreshedStrength_pos _ hγ have helg : ell ≤ γ := refreshedStrength_le _ hγ.le have hells : 6*ell ≤ 1 := by linarith have helld : ell^2 ≤ delta/2 := by have hel1 : ell ≤ 1 := by linarith nlinarith obtain ⟨n,hn,hn0,hn1⟩ := exists_block_length hell hd helld obtain ⟨_hbound,hsig⟩ := hsignal γ hγ hγ₀' obtain ⟨e,ω,he,hω,hblock⟩ := hsig.block hds.continuous hdcompact hγ C.density_nonneg C.density_continuous tags htags hq hqB hA k hell hells hd hd1 hn0 hn1 hmass hcenter hnu (show 0 < ε/4 by positivity) τR hκ have hanchorReady := hanchor d q tags hq b D μ t ht (fun i x hx s hs => hroom i _ ⟨x,hx,rfl⟩ s hs) (ε/4) e ω κ (by positivity) he hω hκ obtain ⟨N,s,_hsc,hjitter,hphysical,hclose,hband,hmatch,hrAnchor,_hgoodAnchor⟩ := hanchorReady.exists obtain ⟨L,hL,E,α,hpath,hmovement,hsqrt,hpred,hrE,hgoodE⟩ := hblock s.output hrAnchor hclose.le μ hμ hband hmatch refine ⟨n,ell,hn,hell,hells,hn0,hn1,N,s,hjitter,L,hL,E,α,hpath,?_,hsqrt,hpred,hrE,hgoodE⟩ intro r have hmove : physicalError (s.output.vector r ∘ s.chart) ((E.vector r ∘ α) ∘ s.chart) < ε/4 := by change physicalError (s.output.vector r ∘ passageChart (s.geometry.H N)) ((E.vector r ∘ α) ∘ passageChart (s.geometry.H N)) < ε/4 rw [physicalError_chart (s.geometry.Hcontinuous N).measurable] exact hmovement r have htriangle := physicalError_triangle (C.vector_continuous r) ((s.output.vector_continuous r).comp s.chart_continuous) (((E.vector_continuous r).comp hpath.continuous).comp s.chart_continuous) change physicalError (C.vector r) ((E.vector r ∘ α) ∘ s.chart) < ε have hfirst := hphysical r linarith end ThreeTorus end end section noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators ComplexConjugate namespace ThreeTorus /-- An actual finite block, centered using its ACTUAL input packet densities. The finite chain contains the anchor and every signal/refresh pair. -/ structure ActualColoredBlock {m : ℕ} (C E : ColoredState (Fin m)) (β : Equiv.Perm TripleTorus) (D : UnitAddCircle → ℂ) : Prop where witness : ∃ n : ℕ, ∃ ell : ℝ, 0 < n ∧ 0 < ell ∧ ∃ q : UnitAddCircle → ℂ, Continuous q ∧ ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => q (x:UnitAddCircle)) ∧ (∀ z, q (-z) = conj (q z)) ∧ ∃ N : ℕ, ∃ s : ColoredAnchor C N, ∃ L : Fin n → ℤ, ∃ α : Equiv.Perm TripleTorus, ColoredPath s.output L E α ∧ β = s.chart.trans α ∧ D = blockDiagonalEstimator q ell L namespace ActualColoredBlock variable {m : ℕ} {C E : ColoredState (Fin m)} {β : Equiv.Perm TripleTorus} {D : UnitAddCircle → ℂ} (h : ActualColoredBlock C E β D) include h lemma continuous : Continuous β := by obtain ⟨_,_,_,_,_,_,_,_,_,s,_,_,hp,rfl,_⟩ := h.witness exact hp.continuous.comp s.chart_continuous lemma preserves : MeasurePreserving β tripleVolume tripleVolume := by obtain ⟨_,_,_,_,_,_,_,_,_,s,_,_,hp,rfl,_⟩ := h.witness exact hp.preserves.comp s.chart_preserves lemma physicalError_eq (f g : TripleTorus → ℂ) : physicalError (f ∘ β) (g ∘ β) = physicalError f g := by obtain ⟨_,_,_,_,_,_,_,_,_,s,_,α,hp,rfl,_⟩ := h.witness change physicalError ((f ∘ α) ∘ passageChart (s.geometry.H _)) ((g ∘ α) ∘ passageChart (s.geometry.H _)) = _ rw [physicalError_chart (s.geometry.Hcontinuous _).measurable,hp.physicalError_eq] lemma estimator_continuous : Continuous D := by obtain ⟨_,_,_,_,_,hq,_,_,_,_,_,_,_,_,rfl⟩ := h.witness exact continuous_blockPhaseEstimator_diagonal hq _ _ end ActualColoredBlock theorem actual_reflected_block {m : ℕ} (C : ColoredState (Fin m)) (τR : Fin m → Fin m) (hτR : Involutive τR) (hrC : C.Reflection τR) (tags : Fin m → ℂ) (htags : ∀ r, ‖tags r‖ ≤ 1) (htrR : ∀ r, tags (τR r) = conj (tags r)) {A : UnitAddCircle → ℂ} (hA : Continuous A) (k : ℕ) {delta nu ε κ : ℝ} (hd : 0 < delta) (hd1 : delta ≤ 1/100) (hnu : 0 < nu) (hε : 0 < ε) (hκ : 0 < κ) (hmass : (∫ z, blockTotal C.density z ∂phaseHaar) ≤ 2) : ∃ E : ColoredState (Fin m), ∃ β : Equiv.Perm TripleTorus, ∃ D : UnitAddCircle → ℂ, ActualColoredBlock C E β D ∧ (∀ r, physicalError (C.vector r) (E.vector r ∘ β) < ε) ∧ ((∫ z, Real.sqrt (blockTotal C.density z) ∂phaseHaar)-300*delta^2-nu ≤ ∫ z, Real.sqrt (blockTotal E.density z) ∂phaseHaar) ∧ ((∑ r, ∫ z, E.density r z*‖A z+D z-((k:ℂ)+1)*tags r‖^2 ∂phaseHaar) ≤ (∫ z, blockPreviousError C.density tags A k z ∂phaseHaar)+4/delta+nu) ∧ E.Reflection τR ∧ ∃ δ : ℝ, 0 < δ ∧ δ < 1/8 ∧ δ < κ ∧ E.packets.GoodArc δ := by obtain ⟨q,hq,hqs,hqB,hqr,hcenter⟩ := C.exists_center hrC hτR htags htrR hd obtain ⟨n,ell,hn,hell,_hells,_hn0,_hn1,N,s,_hjitter,L,_hL,E,α,hpath,hphys,hsqrt,hpred,hr,hgood⟩ := actual_reflected_block_of_center C τR hrC tags htags htrR hq hqs hqr hqB hA k hd hd1 hnu hε hκ hmass hcenter.le exact ⟨E,s.chart.trans α,blockDiagonalEstimator q ell L, ⟨n,ell,hn,hell,q,hq,hqs,hqr,N,s,L,α,hpath,rfl,rfl⟩, hphys,hsqrt,hpred,hr,hgood⟩ end ThreeTorus end end section noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators namespace ThreeTorus def physicalEnergy (f : TripleTorus → ℂ) : ℝ := ∫ x, ‖f x‖^2 ∂tripleVolume lemma physicalEnergy_nonneg (f : TripleTorus → ℂ) : 0 ≤ physicalEnergy f := integral_nonneg (fun _ => sq_nonneg _) lemma physicalEnergy_integrable {f : TripleTorus → ℂ} (hf : Continuous f) : Integrable (fun x => ‖f x‖^2) tripleVolume := (hf.norm.pow 2).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) lemma scalar_energy_comparison {a b t : ℝ} (ht : 0 < t) : (a+b)^2 ≤ (1+t)*a^2+(1+1/t)*b^2 := by have he : ((1+t)*a^2+(1+1/t)*b^2-(a+b)^2)*t = (t*a-b)^2 := by field_simp ring have hh := sq_nonneg (t*a-b) nlinarith lemma physicalEnergy_comparison {f g : TripleTorus → ℂ} (hf : Continuous f) (hg : Continuous g) {t : ℝ} (ht : 0 < t) : physicalEnergy g ≤ (1+t)*physicalEnergy f+(1+1/t)*physicalError f g := by have hp x : ‖g x‖^2 ≤ (1+t)*‖f x‖^2+(1+1/t)*‖f x-g x‖^2 := by calc _ ≤ (‖f x‖+‖f x-g x‖)^2 := by gcongr simpa only [dist_eq_norm,sub_zero,norm_sub_rev,add_comm] using dist_triangle (g x) (f x) 0 _ ≤ _ := scalar_energy_comparison ht unfold physicalEnergy physicalError calc _ ≤ ∫ x, (1+t)*‖f x‖^2+(1+1/t)*‖f x-g x‖^2 ∂tripleVolume := integral_mono (physicalEnergy_integrable hg) (((physicalEnergy_integrable hf).const_mul _).add ((physicalError_integrable hf hg).const_mul _)) hp _ = _ := by rw [integral_add ((physicalEnergy_integrable hf).const_mul _) ((physicalError_integrable hf hg).const_mul _),integral_const_mul,integral_const_mul] lemma exists_physical_mass_tolerance {f : TripleTorus → ℂ} (hf : Continuous f) (hm : physicalEnergy f < 2) : ∃ ε : ℝ, 0 < ε ∧ ∀ g : TripleTorus → ℂ, Continuous g → physicalError f g < ε → physicalEnergy g < 2 := by let M := physicalEnergy f have hM : 0 ≤ M := physicalEnergy_nonneg f let t := (2-M)/(2*(M+1)) have ht : 0 < t := div_pos (sub_pos.mpr hm) (by positivity) have htM : t*(2*(M+1)) = 2-M := div_mul_cancel₀ _ (by positivity) have hgap : 0 < 2-(1+t)*M := by nlinarith let ε := (2-(1+t)*M)/(2*(1+1/t)) have hε : 0 < ε := div_pos hgap (by positivity) have heq : ε*(2*(1+1/t)) = 2-(1+t)*M := div_mul_cancel₀ _ (by positivity) refine ⟨ε,hε,?_⟩ intro g hg he have hc := physicalEnergy_comparison hf hg ht have hmul : (1+1/t)*physicalError f g < (1+1/t)*ε := mul_lt_mul_of_pos_left he (by positivity) dsimp only [M] at * nlinarith end ThreeTorus end end section noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators ComplexConjugate namespace ThreeTorus lemma ActualColoredBlock.physicalEnergy_eq {m : ℕ} {C E : ColoredState (Fin m)} {β : Equiv.Perm TripleTorus} {D : UnitAddCircle → ℂ} (h : ActualColoredBlock C E β D) (f : TripleTorus → ℂ) : physicalEnergy (f ∘ β) = physicalEnergy f := by have he := h.physicalError_eq (fun _ => 0) f simpa [physicalError,physicalEnergy,Function.comp_def] using he lemma ColoredState.mass_transport_tolerance {m : ℕ} (C : ColoredState (Fin m)) (hm : (∫ z, blockTotal C.density z ∂phaseHaar) < 2) : ∃ ε : ℝ, 0 < ε ∧ ∀ (E : ColoredState (Fin m)) (β : Equiv.Perm TripleTorus) (D : UnitAddCircle → ℂ), ActualColoredBlock C E β D → (∀ r, physicalError (C.vector r) (E.vector r ∘ β) < ε) → (∫ z, blockTotal E.density z ∂phaseHaar) < 2 := by rw [C.total_mass_energy] at hm obtain ⟨η,hη,hclose⟩ := exists_physical_mass_tolerance C.totalVector_continuous hm let ε : ℝ := η/((m:ℝ)+1)^2 have hm1 : 0 < (m:ℝ)+1 := by positivity have hε : 0 < ε := div_pos hη (sq_pos_of_pos hm1) have heq : ε*((m:ℝ)+1)^2 = η := div_mul_cancel₀ _ (ne_of_gt (sq_pos_of_pos hm1)) refine ⟨ε,hε,?_⟩ intro E β D hblock herrors have hs := physicalError_sum C.vector_continuous (fun r => (E.vector_continuous r).comp hblock.continuous) simp only [Fintype.card_fin] at hs have hsum : (∑ r, physicalError (C.vector r) (E.vector r ∘ β)) ≤ (m:ℝ)*ε := by simpa only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] using Finset.sum_le_sum (s := Finset.univ) (fun r _ => (herrors r).le) have hmul := mul_le_mul_of_nonneg_left hsum (Nat.cast_nonneg m : (0:ℝ) ≤ m) have hsmall : (m:ℝ)*((m:ℝ)*ε) < η := by nlinarith [Nat.cast_nonneg (α := ℝ) m] have herr : physicalError C.totalVector (E.totalVector ∘ β) < η := hs.trans_lt (hmul.trans_lt hsmall) have henergy := hclose _ (E.totalVector_continuous.comp hblock.continuous) herr rw [hblock.physicalEnergy_eq] at henergy rw [E.total_mass_energy] exact henergy end ThreeTorus end end section noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators ComplexConjugate namespace ThreeTorus lemma fourier_diagonal_smooth_lift (n : ℤ) : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => fourier 1 (n • (x:UnitAddCircle))) := by have he (x : ℝ) : fourier 1 (n • (x:UnitAddCircle)) = fourier n (x:UnitAddCircle) := by simp only [fourier_apply,one_zsmul] simp_rw [he,fourier_coe_apply] exact Complex.contDiff_exp.comp ((contDiff_const.mul Complex.ofRealCLM.contDiff).div_const _) lemma blockDiagonalEstimator_smooth {n : ℕ} {q : UnitAddCircle → ℂ} (hq : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => q (x:UnitAddCircle))) (ell : ℝ) (L : Fin n → ℤ) : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => blockDiagonalEstimator q ell L (x:UnitAddCircle)) := by unfold blockDiagonalEstimator blockPhaseEstimator exact hq.add (contDiff_const.mul (ContDiff.sum (fun i _ => fourier_diagonal_smooth_lift (L i)))) lemma blockDiagonalEstimator_reflection {n : ℕ} {q : UnitAddCircle → ℂ} (hq : ∀ z, q (-z) = conj (q z)) (ell : ℝ) (L : Fin n → ℤ) (z : UnitAddCircle) : blockDiagonalEstimator q ell L (-z) = conj (blockDiagonalEstimator q ell L z) := by have hf (x : UnitAddCircle) : fourier 1 (-x) = conj (fourier 1 x) := by simpa only [fourier_apply,zsmul_neg] using (fourier_neg' (n := 1) (x := x)) unfold blockDiagonalEstimator blockPhaseEstimator simp only [hq,zsmul_neg,hf,map_add,map_mul,map_sum,map_inv₀, map_natCast,Complex.conj_ofReal] namespace ActualColoredBlock variable {m : ℕ} {C E : ColoredState (Fin m)} {β : Equiv.Perm TripleTorus} {D : UnitAddCircle → ℂ} (h : ActualColoredBlock C E β D) include h lemma estimator_smooth : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => D (x:UnitAddCircle)) := by obtain ⟨_,_,_,_,_,_,hqs,_,_,_,_,_,_,_,rfl⟩ := h.witness exact blockDiagonalEstimator_smooth hqs _ _ lemma estimator_reflection (z : UnitAddCircle) : D (-z) = conj (D z) := by obtain ⟨_,_,_,_,_,_,_,hqr,_,_,_,_,_,_,rfl⟩ := h.witness exact blockDiagonalEstimator_reflection hqr _ _ _ end ActualColoredBlock end ThreeTorus end end end OAI