import OAI.Dynamics.ThreeTorus.PassageFamilies namespace OAI /-! Closed finite packet states, canonical densities, residue reindexing, anchors, reflection, clipping, and state tracking. -/ section /-! Concrete finite-packet input data and the support bookkeeping used by iteration. Field regularity is inherited from literal amplitudes. -/ noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators namespace ThreeTorus namespace PassagePackets variable {ι : Type*} (F : PassagePackets ι) def core (i : ι) : Set ℝ := tsupport (F.A₀ i 0) lemma core_subset (i : ι) : F.core i ⊆ Icc F.l F.u := closure_minimal ((F.support_lift i 0).trans Ioc_subset_Icc_self) isClosed_Icc lemma core_compact (i : ι) : IsCompact (F.core i) := isCompact_Icc.of_isClosed_subset (isClosed_tsupport _) (F.core_subset i) lemma support_subset_core (hmod : F.SpareIndependent) (i : ι) (a : UnitAddCircle) : support (F.A₀ i a) ⊆ F.core i := by intro x hx apply subset_tsupport _ intro hz apply hx apply norm_eq_zero.mp rw [hmod.1 i a x,hz,norm_zero] lemma amplitude_bound [Finite ι] : ∃ B : ℝ, 0 ≤ B ∧ ∀ i p, ‖F.A i p‖^2 ≤ B := by classical have hb i := (isCompact_range ((F.continuous_amplitude i).norm.pow 2)).isBounded.exists_norm_le choose B hB using hb obtain ⟨C,hC⟩ := (Set.finite_range B).isCompact.isBounded.exists_norm_le refine ⟨max C 0,le_max_right _ _,fun i p => ?_⟩ have h₁ := hB i _ (mem_range_self p) have h₂ := hC _ (mem_range_self i) exact (le_abs_self _).trans (h₁.trans ((le_abs_self _).trans (h₂.trans (le_max_left _ _)))) /-- Adding an integral carrier changes no temporal eigenfrequency modulo one. -/ def carrier (c : ℤ) : PassagePackets ι := { F with V := fun x => (c:ℝ)+F.V x periodic := fun x => by dsimp; rw [F.periodic x] smooth_speed := contDiff_const.add F.smooth_speed positive_deriv := fun x hx => by simpa only [deriv_const_add] using F.positive_deriv x hx } lemma carrier_spare (c : ℤ) (hmod : F.SpareIndependent) : (F.carrier c).SpareIndependent := hmod lemma carrier_circle (c : ℤ) (x : ℝ) : ((F.carrier c).V x:UnitAddCircle) = (F.V x:UnitAddCircle) := by change (((c:ℝ)+F.V x:ℝ):UnitAddCircle) = _ rw [AddCircle.coe_add] have hc : ((c:ℝ):UnitAddCircle) = 0 := (AddCircle.coe_eq_zero_iff 1).mpr ⟨c,by simp⟩ rw [hc,zero_add] /-- Scalar cutoffs with arbitrarily small support tracking error exist for EVERY sufficiently large carrier. Choosing the carrier is therefore still compatible with the signal-law and normalized-index error bounds. -/ theorem tracked_cutoffs [Finite ι] (hi : Injective F.m) (hmod : F.SpareIndependent) {η : ℝ} (hη : 0 < η) : ∀ᶠ c : ℕ in atTop, 0 < c ∧ ∃ χ : ι → ℝ → ℝ, (∀ i, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (χ i) ∧ HasCompactSupport (χ i) ∧ χ i 0 = 0 ∧ (∀ s, 0 ≤ χ i s ∧ χ i s ≤ 1) ∧ (∀ s ∈ tsupport (χ i), ∃ x ∈ F.core i, |s-(F.m i:ℝ)*((c:ℝ)+F.V x)| < η) ∧ ∀ (C : Set ℝ) a s, stationaryWeight C (fun x => (F.m i:ℝ)*((c:ℝ)+F.V x)) (F.A₀ i a) s ≠ 0 → χ i s = 1) ∧ Pairwise (fun i j => Disjoint (tsupport (χ i)) (tsupport (χ j))) := by have hv : Continuous F.periodic.lift := by apply Continuous.quotient_lift exact F.smooth_speed.continuous filter_upwards [eventually_integral_speed_separation F.m F.periodic.lift F.nonzero hv] with c hc have hzero i : 0 ∉ (fun x => (F.m i:ℝ)*((c:ℝ)+F.V x)) '' F.core i := by rintro ⟨x,_hx,he⟩ exact hc.2.1 i (x:UnitAddCircle) he have hsep : Pairwise (fun i j => Disjoint ((fun x => (F.m i:ℝ)*((c:ℝ)+F.V x)) '' F.core i) ((fun x => (F.m j:ℝ)*((c:ℝ)+F.V x)) '' F.core j)) := by intro i j hij rw [Set.disjoint_left] rintro s ⟨x,_hx,rfl⟩ ⟨y,_hy,he⟩ exact hij (hi (hc.2.2 i j (x:UnitAddCircle) (y:UnitAddCircle) he.symm)) obtain ⟨χ,hχ,hd⟩ := tracked_frequency_cutoffs F.m (fun x => (c:ℝ)+F.V x) F.core (continuous_const.add F.smooth_speed.continuous) F.core_compact hzero hsep hη exact ⟨hc.1,χ,fun i => ⟨(hχ i).1,(hχ i).2.1,(hχ i).2.2.1, (hχ i).2.2.2.1,(hχ i).2.2.2.2.2.1,fun C a s => (hχ i).2.2.2.2.2.2 C (F.A₀ i a) (F.support_subset_core hmod i a) s⟩,hd⟩ end PassagePackets end ThreeTorus end end section /-! Canonical smooth density of a literal passage-state packet. These are computed from the actual amplitude and speed; no spectral property is a field. -/ noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators namespace ThreeTorus lemma continuous_density_unique {p q : UnitAddCircle → ℝ} (hp : Continuous p) (hq : Continuous q) (hp0 : ∀ z, 0 ≤ p z) (hq0 : ∀ z, 0 ≤ q z) (he : AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal (p z)) = AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal (q z))) : p = q := by apply (Continuous.ae_eq_iff_eq AddCircle.haarAddCircle hp hq).mp have hh := (withDensity_eq_iff_of_sigmaFinite hp.measurable.ennreal_ofReal.aemeasurable hq.measurable.ennreal_ofReal.aemeasurable).mp he filter_upwards [hh] with z hz have hh := congrArg ENNReal.toReal hz simpa only [ENNReal.toReal_ofReal (hp0 z), ENNReal.toReal_ofReal (hq0 z)] using hh lemma fold_density_nonzero {q : ℝ → ℝ} {z : UnitAddCircle} (hz : foldCircleDensity q z ≠ 0) : ∃ s : ℝ, (s:UnitAddCircle) = z ∧ q s ≠ 0 := by obtain ⟨x,rfl⟩ := Quotient.mk_surjective z rw [foldCircleDensity_coe_all] at hz have hk : ∃ k : ℤ, q (x+k) ≠ 0 := by by_contra hh push Not at hh exact hz (by simp only [hh,tsum_zero]) obtain ⟨k,hk⟩ := hk refine ⟨x+k,?_,hk⟩ rw [AddCircle.coe_add] have he : ((k:ℝ):UnitAddCircle) = 0 := (AddCircle.coe_eq_zero_iff 1).mpr ⟨k,by simp⟩ rw [he,add_zero] lemma foldCircleDensity_finsetSum_eq {ι : Type*} (S : Finset ι) (q : ι → ℝ → ℝ) (hq : ∀ i ∈ S, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (q i)) (hc : ∀ i ∈ S, HasCompactSupport (q i)) : foldCircleDensity (fun x => ∑ i ∈ S, q i x) = fun z => ∑ i ∈ S, foldCircleDensity (q i) z := by have hqS : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x => ∑ i ∈ S, q i x) := ContDiff.sum hq have hcS : HasCompactSupport (fun x => ∑ i ∈ S, q i x) := by convert HasCompactSupport.finset_sum hc using 1 funext point simp only [Finset.sum_apply] apply (Continuous.ae_eq_iff_eq AddCircle.haarAddCircle (continuous_of_continuous_lift (foldCircleDensity_smooth_lift hqS hcS).continuous) (continuous_finsetSum _ (fun i hi => continuous_of_continuous_lift (foldCircleDensity_smooth_lift (hq i hi) (hc i hi)).continuous))).mp exact foldCircleDensity_finsetSum_ae S q (fun i hi => (hq i hi).continuous.measurable) (fun i hi => (hq i hi).continuous.integrable_of_hasCompactSupport (hc i hi)) namespace PassagePackets variable {ι : Type*} (F : PassagePackets ι) def weight (i : ι) : ℝ → ℝ := stationaryWeight (Icc (F.l-F.R) (F.u+F.R)) (fun x => (F.m i:ℝ)*F.V x) (F.A₀ i 0) def density (i : ι) : UnitAddCircle → ℝ := foldCircleDensity (F.weight i) lemma weight_nonneg (i : ι) (s : ℝ) : 0 ≤ F.weight i s := branchDensity_nonneg (fun _ => sq_nonneg _) s lemma density_nonneg (i : ι) (z : UnitAddCircle) : 0 ≤ F.density i z := foldCircleDensity_nonneg (F.weight_nonneg i) z lemma smooth_zero_lift (i : ι) : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (F.A₀ i 0) := by simpa only [Function.comp_def, AddCircle.coe_zero] using (F.smooth_lift i).comp (show ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => (x,(0:ℝ))) from contDiff_id.prodMk contDiff_const) lemma weight_regular (i : ι) : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (F.weight i) ∧ HasCompactSupport (F.weight i) := by let q := fun x => ‖F.A₀ i 0 x‖^2 have hqs : tsupport q ⊆ F.core i := by apply closure_mono intro x hx hz exact hx (by simp [q,hz]) have hqc : HasCompactSupport q := (F.core_compact i).of_isClosed_subset isClosed_closure hqs have hs : tsupport q ⊆ Ioo (F.l-F.R) (F.u+F.R) := by intro x hx have hh := F.core_subset i (hqs hx) constructor <;> linarith [F.Rpos,hh.1,hh.2] have has : support (F.A₀ i 0) ⊆ Ioo (F.l-F.R) (F.u+F.R) := by intro x hx have hh := F.support_lift i 0 hx constructor <;> linarith [F.Rpos,hh.1,hh.2] have hb := scaled_speed_good_branch (F.nonzero i) F.Rpos F.smooth_speed F.positive_deriv have he := stationaryWeight_change_branch Ioo_subset_Icc_self hb.1 has change ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (stationaryWeight _ _ _) ∧ HasCompactSupport (stationaryWeight _ _ _) rw [he] exact ⟨branchDensity_smooth (contDiff_const.mul F.smooth_speed) isOpen_Ioo (hb.1.mono Ioo_subset_Icc_self) hb.2 ((F.smooth_zero_lift i).norm_sq (𝕜 := ℂ)) hqc hs, branchDensity_compact (continuous_const.mul F.smooth_speed.continuous) hqc⟩ lemma density_smooth (i : ι) : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => F.density i (x:UnitAddCircle)) := foldCircleDensity_smooth_lift (F.weight_regular i).1 (F.weight_regular i).2 lemma density_continuous (i : ι) : Continuous (F.density i) := continuous_of_continuous_lift (F.density_smooth i).continuous lemma speed_continuous : Continuous F.periodic.lift := by apply Continuous.quotient_lift exact F.smooth_speed.continuous /-- The canonical density is the spectral measure of the actual angular packet. -/ theorem density_spectral (hmod : F.SpareIndependent) (i : ι) : packetSpectralMeasure (fun u => (F.periodic.lift u:UnitAddCircle)) (F.m i) (F.A i) = AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal (F.density i z)) := by have hb := scaled_speed_good_branch (F.nonzero i) F.Rpos F.smooth_speed F.positive_deriv have hs : support (F.A₀ i 0) ⊆ Ioo (F.l-F.R) (F.u+F.R) := by intro x hx have hh := F.support_lift i 0 hx constructor <;> linarith [F.Rpos,hh.1,hh.2] have he := oldPacket_spectralDensity_of_modulus (m := F.m i) (c := 0) (v := F.periodic.lift) (F.continuous_amplitude i) (F.smooth_zero_lift i).continuous (hmod.2 i) (F.lift_matches i 0) ((F.support_lift i 0).trans (by intro x hx; exact ⟨lt_of_le_of_lt F.lo hx.1, hx.2.trans F.uo⟩)) hs F.speed_continuous.measurable F.smooth_speed (fun _ _ => rfl) isOpen_Ioo Ioo_subset_Icc_self hb.1 hb.2 simpa only [Int.cast_zero,sub_zero,density,weight] using he lemma density_marginal (hmod : F.SpareIndependent) (i : ι) : foldCircleDensity (fun s => ∫ a, stationaryWeight (Icc (F.l-F.R) (F.u+F.R)) (fun x => (F.m i:ℝ)*F.V x) (F.A₀ i a) s ∂AddCircle.haarAddCircle) = F.density i := by congr 1 funext s simp [stationaryWeight_modulus_independent (hmod.1 i),weight] lemma density_carrier (hmod : F.SpareIndependent) (c : ℤ) (i : ι) : (F.carrier c).density i = F.density i := by apply continuous_density_unique ((F.carrier c).density_continuous i) (F.density_continuous i) ((F.carrier c).density_nonneg i) (F.density_nonneg i) rw [← (F.carrier c).density_spectral (F.carrier_spare c hmod) i,← F.density_spectral hmod i] congr 1 funext z obtain ⟨x,rfl⟩ := Quotient.mk_surjective z exact F.carrier_circle c x /-- A nonzero density value is realized on the true old amplitude support, not merely in the convex hull of its frequency range. -/ lemma density_support (i : ι) {z : UnitAddCircle} (hz : F.density i z ≠ 0) : ∃ x ∈ support (F.A₀ i 0), ((F.m i:ℝ)*F.V x:UnitAddCircle) = z := by obtain ⟨s,hs,hq⟩ := fold_density_nonzero hz have hy : s ∈ (fun x => (F.m i:ℝ)*F.V x) '' Icc (F.l-F.R) (F.u+F.R) := by by_contra hh exact hq (by simp [weight,stationaryWeight,branchDensity,hh]) let x := invFunOn (fun x => (F.m i:ℝ)*F.V x) (Icc (F.l-F.R) (F.u+F.R)) s have hx := invFunOn_pos hy refine ⟨x,?_,?_⟩ · intro hz change F.A₀ i 0 (invFunOn (fun x => (F.m i:ℝ)*F.V x) (Icc (F.l-F.R) (F.u+F.R)) s) = 0 at hz exact hq (by simp [weight,stationaryWeight,branchDensity,hy,hz]) · exact (congrArg (fun t : ℝ => (t:UnitAddCircle)) hx.2).trans hs end PassagePackets end ThreeTorus end end section /-! Exact identification of the canonical density of an iterated packet state with the literal finite density already controlled by the passage estimates. -/ noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators namespace ThreeTorus lemma exactRetainedPacketDensity_nonneg (I : Set ℝ) (J : ℝ → ℝ) (ψ : UnitAddCircle → ℝ) (χ : ℝ → UnitAddCircle → ℝ) (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) (A : UnitAddCircle × UnitAddCircle → ℂ) (S : Finset ℤ) (n : ℕ) (z : UnitAddCircle) : 0 ≤ exactRetainedPacketDensity I J ψ χ m H A S n z := by apply foldCircleDensity_nonneg intro t apply Finset.sum_nonneg intro j _hj exact unfoldedModeDensity_nonneg (circleDensityLift_nonneg (retainedCircleWeight_nonneg ψ χ _ n j) 0) (Nat.cast_nonneg n) j t lemma exactRetainedPacketDensity_singleton (I : Set ℝ) (J : ℝ → ℝ) (ψ : UnitAddCircle → ℝ) (χ : ℝ → UnitAddCircle → ℝ) (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) (A : UnitAddCircle × UnitAddCircle → ℂ) (j : ℤ) (n : ℕ) : exactRetainedPacketDensity I J ψ χ m H A {j} n = foldCircleDensity (unfoldedModeDensity I J (invFunOn J I) (deriv J) (circleDensityLift 0 (retainedCircleWeight ψ χ (fun n j => exactPassageCoeff m (H n) A j) n j)) n j) := by simp only [exactRetainedPacketDensity,Finset.sum_singleton] lemma exactRetainedPacketDensity_sum_singletons {I : Set ℝ} {J : ℝ → ℝ} {ψ : UnitAddCircle → ℝ} {ψ₀ : ℝ → ℝ} {χ : ℝ → UnitAddCircle → ℝ} {m : ℤ} {H : ℕ → UnitAddCircle → UnitAddCircle} {Hl : ℕ → ℝ → ℝ} {A : UnitAddCircle × UnitAddCircle → ℂ} {S : Finset ℤ} {n : ℕ} (hJ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) J) (hIo : IsOpen I) (hinj : InjOn J I) (hd : ∀ x ∈ I, deriv J x ≠ 0) (hψ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) ψ₀) (hc : HasCompactSupport ψ₀) (he : ∀ x ∈ Ioc (0:ℝ) (0+1), ψ (x:UnitAddCircle) = ψ₀ x) (hs : tsupport ψ₀ ⊆ Ioc (0:ℝ) (0+1) ∩ I) (hχ : CylinderSmooth χ.uncurry) (hHl : ∀ n, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (Hl n)) (hH : ∀ n (t : ℝ), H n (t:UnitAddCircle) = (Hl n t:UnitAddCircle)) (hA : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => A (p.1,p.2))) (hn : 0 < n) (hS : ∀ j ∈ S, j ≠ 0) : exactRetainedPacketDensity I J ψ χ m H A S n = fun z => ∑ j ∈ S, exactRetainedPacketDensity I J ψ χ m H A {j} n z := by unfold exactRetainedPacketDensity simp only [Finset.sum_singleton] simp_rw [retainedWeight_lift_eq χ m H A n _ he (hs.trans inter_subset_left)] apply foldCircleDensity_finsetSum_eq · intro j _hj exact unfoldedModeDensity_smooth hJ hIo hinj hd (realRetainedWeight_smooth hψ hχ hHl hH hA n j) (realRetainedWeight_compact hc χ m H A n j) ((realRetainedWeight_support ψ₀ χ m H A n j).trans (hs.trans inter_subset_right)) n j · intro j hj exact unfoldedModeDensity_compact hJ.continuous (realRetainedWeight_compact hc χ m H A n j) (by exact_mod_cast hn.ne') (by exact_mod_cast hS j hj) lemma exactRetainedPacketDensity_filtered_singletons {I : Set ℝ} {J : ℝ → ℝ} {ψ : UnitAddCircle → ℝ} {ψ₀ : ℝ → ℝ} {χ : ℝ → UnitAddCircle → ℝ} {m : ℤ} {H : ℕ → UnitAddCircle → UnitAddCircle} {Hl : ℕ → ℝ → ℝ} {A : UnitAddCircle × UnitAddCircle → ℂ} {S : Finset ℤ} {n : ℕ} (hJ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) J) (hIo : IsOpen I) (hinj : InjOn J I) (hd : ∀ x ∈ I, deriv J x ≠ 0) (hψ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) ψ₀) (hc : HasCompactSupport ψ₀) (he : ∀ x ∈ Ioc (0:ℝ) (0+1), ψ (x:UnitAddCircle) = ψ₀ x) (hs : tsupport ψ₀ ⊆ Ioc (0:ℝ) (0+1) ∩ I) (hχ : CylinderSmooth χ.uncurry) (hHl : ∀ n, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (Hl n)) (hH : ∀ n (t : ℝ), H n (t:UnitAddCircle) = (Hl n t:UnitAddCircle)) (hA : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => A (p.1,p.2))) (hn : 0 < n) (hS : ∀ j ∈ S, j ≠ 0) (E : ℤ → Prop) [DecidablePred E] : exactRetainedPacketDensity I J ψ χ m H A (S.filter E) n = fun z => ∑ j : ↥S, if E j.val then exactRetainedPacketDensity I J ψ χ m H A {j.val} n z else 0 := by rw [exactRetainedPacketDensity_sum_singletons hJ hIo hinj hd hψ hc he hs hχ hHl hH hA hn (fun j hj => hS j (Finset.mem_filter.mp hj).1)] funext z rw [Finset.sum_filter] exact (Finset.sum_coe_sort S (fun j => if E j then exactRetainedPacketDensity I J ψ χ m H A {j} n z else 0)).symm /-- Pointwise, not just a.e., identity of a literal output packet's canonical density. The hypotheses below are only smooth lifts and the displayed literal coefficient/speed identities; no new spectral law is assumed. -/ theorem literal_packet_canonical_density {ι : Type*} (F : PassagePackets ι) (hmod : F.SpareIndependent) (i : ι) {I : Set ℝ} {J : UnitAddCircle → ℝ} {J₀ : ℝ → ℝ} {ψ : UnitAddCircle → ℝ} {ψ₀ : ℝ → ℝ} {χ : ℝ → UnitAddCircle → ℝ} {m j : ℤ} {n : ℕ} {H : ℕ → UnitAddCircle → UnitAddCircle} {Hl : ℕ → ℝ → ℝ} {A : UnitAddCircle × UnitAddCircle → ℂ} (hm : F.m i = j) (hamp : F.A i = retainedAmplitude ψ χ m H A n j) (hspeed : ∀ a, (F.periodic.lift a:UnitAddCircle) = (((1+J a/n)/n:ℝ):UnitAddCircle)) (hJc : Continuous J) (hJ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) J₀) (hJe : ∀ x ∈ I, J (x:UnitAddCircle) = J₀ x) (hIo : IsOpen I) (hinj : InjOn J₀ I) (hd : ∀ x ∈ I, deriv J₀ x ≠ 0) (hψc : Continuous ψ) (hψ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) ψ₀) (hc : HasCompactSupport ψ₀) (he : ∀ x ∈ Ioc (0:ℝ) (0+1), ψ (x:UnitAddCircle) = ψ₀ x) (hs : tsupport ψ₀ ⊆ Ioc (0:ℝ) (0+1) ∩ I) (hχc : Continuous χ.uncurry) (hχ : CylinderSmooth χ.uncurry) (hHc : ∀ n, Continuous (H n)) (hHl : ∀ n, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (Hl n)) (hH : ∀ n (t : ℝ), H n (t:UnitAddCircle) = (Hl n t:UnitAddCircle)) (hAc : Continuous A) (hA : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => A (p.1,p.2))) (hn : 0 < n) : F.density i = exactRetainedPacketDensity I J₀ ψ χ m H A {j} n := by have hj : j ≠ 0 := hm ▸ F.nonzero i have hps := exactRetainedPacketDensity_smooth_lift (m := m) hJ hIo hinj hd hψ hc he hs hχ hHl hH hA hn (S := {j}) (fun k hk => by simpa only [Finset.mem_singleton.mp hk] using hj) apply continuous_density_unique (F.density_continuous i) (continuous_of_continuous_lift hps.continuous) (F.density_nonneg i) (exactRetainedPacketDensity_nonneg I J₀ ψ χ m H A {j} n) rw [← F.density_spectral hmod i,hm,hamp,show (fun a => (F.periodic.lift a:UnitAddCircle)) = (fun a => (((1+J a/n)/n:ℝ):UnitAddCircle)) from funext hspeed, exactRetainedPacketDensity_singleton] exact retainedAmplitude_spectralDensity hψc hχc hHc hAc (fun x hx hp => (hs (subset_tsupport ψ₀ (by rwa [he x hx] at hp))).2) hJc.measurable hJ hJe hIo hinj hd hn hj end ThreeTorus end end section /-! The residue slices are slices of the actual retained angular list, not independently fabricated Fourier arrays. -/ noncomputable section open MeasureTheory Set Function Filter open scoped Topology BigOperators namespace ThreeTorus lemma sum_residue_rename {M : Type*} [AddCommMonoid M] (f : ℤ → M) (S T : Finset ℤ) (Q h : ℤ) (hQ : Q ≠ 0) (hS : ∀ k, k ∉ S → f (h+k*Q) = 0) (hT : ∀ j, j ∉ T → f j = 0) : ∑ k ∈ S, f (h+k*Q) = ∑ j ∈ T.filter (fun j => j % Q = h % Q), f j := by classical let a : ℤ → ℤ := fun k => h+k*Q have hai : Injective a := fun x y he => mul_right_cancel₀ hQ (add_left_cancel he) let F : ℤ → M := fun j => if j % Q = h % Q then f j else 0 have hFa k : F (a k) = f (a k) := by simp [F,a,Int.add_mul_emod_self_right] have hSa j (hj : j ∉ S.image a) : F j = 0 := by dsimp only [F] split_ifs with he · let k := j/Q-h/Q have hk : a k = j := by dsimp [a,k] have hh := Int.emod_add_ediv_mul h Q have hj' := Int.emod_add_ediv_mul j Q rw [he] at hj' nlinarith have hkS : k ∉ S := fun hks => hj (hk ▸ Finset.mem_image.mpr ⟨k,hks,rfl⟩) rw [← hk] exact hS k hkS · rfl have hTF j (hj : j ∉ T) : F j = 0 := by simp [F,hT j hj] calc ∑ k ∈ S, f (a k) = ∑ k ∈ S, F (a k) := by simp_rw [hFa] _ = ∑ j ∈ S.image a, F j := (Finset.sum_image hai.injOn).symm _ = ∑ j ∈ T, F j := sum_eq_of_zero_off F (S.image a) T hSa hTF _ = _ := (Finset.sum_filter _ f).symm lemma exactRetainedPacketDensity_residue_rename (I : Set ℝ) (J₀ : ℝ → ℝ) (ψ : UnitAddCircle → ℝ) (χ : ℝ → UnitAddCircle → ℝ) (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) (A : UnitAddCircle × UnitAddCircle → ℂ) (S T : Finset ℤ) (n : ℕ) (Q h : ℤ) (hQ : Q ≠ 0) (hS : ∀ k, k ∉ S → ∀ a, χ (((h+k*Q:ℤ):ℝ)/n) a = 0) (hT : ∀ j, j ∉ T → ∀ a, χ ((j:ℝ)/n) a = 0) : foldCircleDensity (fun t => ∑ k ∈ S, unfoldedModeDensity I J₀ (invFunOn J₀ I) (deriv J₀) (circleDensityLift 0 (retainedCircleWeight ψ χ (fun n j => exactPassageCoeff m (H n) A j) n (h+k*Q))) n ((h+k*Q:ℤ):ℝ) t) = exactRetainedPacketDensity I J₀ ψ χ m H A (T.filter (fun j => j % Q = h % Q)) n := by unfold exactRetainedPacketDensity congr 1 funext t apply sum_residue_rename (fun j : ℤ => unfoldedModeDensity I J₀ (invFunOn J₀ I) (deriv J₀) (circleDensityLift 0 (retainedCircleWeight ψ χ (fun n j => exactPassageCoeff m (H n) A j) n j)) n (j:ℝ) t) S T Q h hQ · intro k hk have hz := hS k hk simp only [Int.cast_add,Int.cast_mul] at hz simp [unfoldedModeDensity,scaledBranchKernel,branchDensity, circleDensityLift,retainedCircleWeight,hz] · intro j hj simp [unfoldedModeDensity,scaledBranchKernel,branchDensity, circleDensityLift,retainedCircleWeight,hT j hj] end ThreeTorus end end section /-! Neutral refresh splits the literal next angular list, simultaneously for all residues, with total error independent of the number of residue tests. -/ noncomputable section open MeasureTheory Set Function Filter open scoped Topology BigOperators namespace ThreeTorus lemma foldCircleDensity_const_mul (w : ℝ → ℝ) (c : ℝ) : foldCircleDensity (fun s => c*w s) = fun z => c*foldCircleDensity w z := by funext z induction z using Quotient.inductionOn with | h x => simp only [foldCircleDensity_coe_all,tsum_mul_left] theorem exact_neutral_residue_refresh {A : UnitAddCircle × UnitAddCircle → ℂ} {A₀ : UnitAddCircle → ℝ → ℂ} {Pr V : ℝ → ℝ} {m : ℤ} {Δ : ℕ → ℝ} {H : ℕ → UnitAddCircle → UnitAddCircle} {o l u R : ℝ} (hm : m ≠ 0) (hH : ∀ n (x : ℝ), H n (x : UnitAddCircle) = ((n:ℝ)*(Pr x+Δ n*x) : UnitAddCircle)) (hA : ∀ a x, x ∈ Ioc o (o+1) → A ((x : UnitAddCircle),a) = A₀ a x) (hA₀ : ∀ a, ContDiff ℝ 2 (A₀ a)) (hAc : Continuous A₀.uncurry) (hAd : Continuous (fun p : UnitAddCircle × ℝ => deriv (A₀ p.1) p.2)) (hAr : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => A₀ (p.2 : UnitAddCircle) p.1)) (hsupp : ∀ a, support (A₀ a) ⊆ Ioc l u) (hlo : o ≤ l) (huo : u ≤ o+1) (hlu : l ≤ u) (hR : 0 < R) (hPr : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) Pr) (hV : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) V) (hPV : deriv Pr = V) (hdV : ∀ x ∈ Icc (l-2*R) (u+2*R), 0 < deriv V x) (h0 : 0 ∉ (fun x => (m:ℝ)*V x) '' Icc l u) (hHc : ∀ n, Continuous (H n)) (hAcont : Continuous A) (hΔt : Tendsto Δ atTop (𝓝 0)) {v : UnitAddCircle → ℝ} (hv : Measurable v) (hVe : ∀ x : ℝ, v (x:UnitAddCircle) = V x) {B : ℝ} (hB : 0 ≤ B) (hAb : ∀ p, ‖A p‖^2 ≤ B) {δ : ℝ} (c : LawCutoffs δ) (hδ : 0 < δ) (hδ' : δ < 1/8) {χ : ℝ → UnitAddCircle → ℝ} (hχc : Continuous χ.uncurry) (hcχ : HasCompactSupport χ.uncurry) (hzχ : ∀ a, χ 0 a = 0) (hχ : ∀ s a, 0 ≤ χ s a ∧ χ s a ≤ 1) (hχW : ∀ s a, stationaryWeight (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) (A₀ a) s ≠ 0 → χ s a = 1) {Q : ℕ} (hQ : 0 < Q) : let pold := foldCircleDensity (fun s => ∫ a, stationaryWeight (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) (A₀ a) s ∂AddCircle.haarAddCircle) ∃ C : ℝ, 0 ≤ C ∧ ∀ L : ℝ, 0 < L → ∀ β : ℝ, ∀ T : ℕ → Finset ℤ, (∀ n, 0 < n → ∀ j ∈ T n, j ≠ 0) → (∀ n, 0 < n → ∀ j ∉ T n, ∀ a, χ ((j:ℝ)/n) a = 0) → ∀ η : ℝ, 0 < η → ∀ᶠ n : ℕ in atTop, (∑ h : Fin Q, ∫ z, |exactRetainedPacketDensity (Ioo δ (1-δ)) (neutralJitterReal c L β) (neutralCircleCutoff c) χ m H A ((T n).filter (fun j => j % (Q:ℤ) = (h.val:ℤ) % (Q:ℤ))) n z- (Q:ℝ)⁻¹*pold z| ∂AddCircle.haarAddCircle) < C/L+4*δ*B+η := by classical dsimp only let ψ := neutralCircleCutoff c let W := fun s a => stationaryWeight (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) (A₀ a) s let pold := foldCircleDensity (fun s => ∫ a, W s a ∂AddCircle.haarAddCircle) let pcut := foldCircleDensity (fun s => ∫ a, stationaryWeight (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) (fun x => (ψ a:ℂ)*A₀ a x) s ∂AddCircle.haarAddCircle) let F := stationaryCutoffLift (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) (fun p : ℝ × ℝ => A₀ (p.2:UnitAddCircle) p.1) c.ψ have hψc := continuous_neutralCircleCutoff c have hψs : tsupport c.ψ ⊆ Ioc (0:ℝ) (0+1) ∩ Ioo δ (1-δ) := by intro x hx have hh := c.ψ_support hx exact ⟨⟨by linarith [hh.1],by linarith [hh.2]⟩,hh⟩ have heψ : ∀ x ∈ Ioc (0:ℝ) (0+1), c.ψ x = ψ (x:UnitAddCircle) := by intro x hx exact (lawPsi_eq c hδ ⟨hx.1.le,by simpa using hx.2⟩).symm have hcut : (∫ z, |pcut z-pold z| ∂AddCircle.haarAddCircle) ≤ B*(4*δ) := by have hs₁ : ∀ a, support (A₀ a) ⊆ Ioc o (o+1) := by intro a x hx have hh := hsupp a hx exact ⟨lt_of_le_of_lt hlo hh.1,hh.2.trans huo⟩ have hs₂ : ∀ a, support (A₀ a) ⊆ Ioo (l-R) (u+R) := by intro a x hx have hh := hsupp a hx exact ⟨by linarith [hh.1],by linarith [hh.2]⟩ have hb := oldStationaryMarginal_cutoff_l1 hAcont hAc hA hs₁ hs₂ hv hV (fun x _ => hVe x) isOpen_Ioo Ioo_subset_Icc_self (scaled_speed_good_branch hm hR hV hdV).1 (scaled_speed_good_branch hm hR hV hdV).2 hψc (neutralCircleCutoff_bounds c hδ) hB hAb exact hb.trans (mul_le_mul_of_nonneg_left (neutralCircleCutoff_error c hδ hδ') hB) have hmarg : (fun s => ∫ a, F (a,s)) = fun s => ∫ a, stationaryWeight (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) (fun x => (ψ a:ℂ)*A₀ a x) s ∂AddCircle.haarAddCircle := by funext s exact stationaryCutoffLift_marginal heψ (fun x hx => (hψs (subset_tsupport c.ψ hx)).1) _ _ s have hsp (h : Fin Q) := exact_neutral_circle_spectral_estimate hm hH hA hA₀ hAc hAd hAr hsupp hlo huo hlu hR hPr hV hPV hdV h0 hHc hAcont hΔt isOpen_Ioo c.ψ_smooth c.ψ_compact hψs heψ hψc hχc hcχ hzχ (neutralCircleCutoff_bounds c hδ) hχ hχW hQ (h.val:ℤ) choose C hC hrest using hsp refine ⟨∑ h, C h,Finset.sum_nonneg (fun h _ => hC h),?_⟩ intro L hL β T hTN hTZ η hη have hJs := neutralJitterReal_smooth c L β have hJc := continuous_neutralJitter c L β have heJ : ∀ a ∈ Ioo δ (1-δ), neutralJitterReal c L β a = L*a+β := fun a ha => neutralJitterReal_affine c hδ L β ⟨ha.1.le,ha.2.le⟩ have hJinj : InjOn (neutralJitterReal c L β) (Ioo δ (1-δ)) := by intro a ha b hb hab rw [heJ a ha,heJ b hb] at hab nlinarith have hJd : ∀ a ∈ Ioo δ (1-δ), deriv (neutralJitterReal c L β) a ≠ 0 := by intro a ha rw [neutralJitterReal_deriv c hδ L β ha] exact hL.ne' choose S hSN hST ht using fun h => hrest h L hL β (neutralJitterReal c L β) (neutralJitter c L β) heJ hJs hJc.measurable (fun x _ => neutralJitter_coe c L β x) have hpcuti : Integrable pcut AddCircle.haarAddCircle := by have hreg := exact_cutoff_weight_regular hm hR hV hdV hAr (fun a => (hsupp a).trans Ioc_subset_Icc_self) c.ψ_smooth c.ψ_compact h0 have hi := compact_slice_integral_regular ((hreg.1.comp (contDiff_snd.prodMk contDiff_fst)).continuous) (hreg.2.1.comp_homeomorph (Homeomorph.prodComm ℝ ℝ)) dsimp only [pcut] rw [← hmarg] exact integrable_foldCircleDensity hi.1.measurable hi.2 have hpoldi : Integrable pold AddCircle.haarAddCircle := by have hs₁ : ∀ a, support (A₀ a) ⊆ Ioc o (o+1) := fun a x hx => ⟨lt_of_le_of_lt hlo (hsupp a hx).1,(hsupp a hx).2.trans huo⟩ have hs₂ : ∀ a, support (A₀ a) ⊆ Ioo (l-R) (u+R) := by intro a x hx have hh := hsupp a hx exact ⟨by linarith [hh.1],by linarith [hh.2]⟩ exact (oldStationaryMarginal_map hAcont hAc hA hs₁ hs₂ hv hV (fun x _ => hVe x) isOpen_Ioo Ioo_subset_Icc_self (scaled_speed_good_branch hm hR hV hdV).1 (scaled_speed_good_branch hm hR hV hdV).2).1 have hQr : (0:ℝ) < Q := Nat.cast_pos.mpr hQ have hQz : (Q:ℤ) ≠ 0 := by exact_mod_cast hQ.ne' have hqpos : 0 < (Q:ℝ)⁻¹ := inv_pos.mpr hQr have hsall := (Filter.eventually_all).mpr (fun h => ht h (η/Q) (div_pos hη hQr)) filter_upwards [hsall,eventually_gt_atTop (0:ℕ)] with n hnsp hn have hp (h : Fin Q) := exactRetainedPacketDensity_integrable (m := m) hJs isOpen_Ioo hJinj hJd hψc hχc hHc hAcont hn (fun j (hj : j ∈ (T n).filter (fun j => j % (Q:ℤ) = (h.val:ℤ) % (Q:ℤ))) => hTN n hn j (Finset.mem_filter.mp hj).1) have hps (h : Fin Q) : (∫ z, |exactRetainedPacketDensity (Ioo δ (1-δ)) (neutralJitterReal c L β) ψ χ m H A ((T n).filter (fun j => j % (Q:ℤ) = (h.val:ℤ) % (Q:ℤ))) n z- (Q:ℝ)⁻¹*pcut z| ∂AddCircle.haarAddCircle) < C h/L+η/Q := by have he := exactRetainedPacketDensity_residue_rename (Ioo δ (1-δ)) (neutralJitterReal c L β) ψ χ m H A (S h n) (T n) n (Q:ℤ) (h.val:ℤ) hQz (hST h n hn) (hTZ n hn) have hh := hnsp h rw [he,foldCircleDensity_const_mul,hmarg] at hh exact hh have hcscale : (∫ z, |(Q:ℝ)⁻¹*pcut z-(Q:ℝ)⁻¹*pold z| ∂AddCircle.haarAddCircle) ≤ (Q:ℝ)⁻¹*(B*(4*δ)) := by simp_rw [← mul_sub,abs_mul,abs_of_pos hqpos] rw [integral_const_mul] exact mul_le_mul_of_nonneg_left hcut hqpos.le have hfinal (h : Fin Q) : (∫ z, |exactRetainedPacketDensity (Ioo δ (1-δ)) (neutralJitterReal c L β) ψ χ m H A ((T n).filter (fun j => j % (Q:ℤ) = (h.val:ℤ) % (Q:ℤ))) n z- (Q:ℝ)⁻¹*pold z| ∂AddCircle.haarAddCircle) < C h/L+(4*δ*B+η)/Q := by have hh := integral_abs_sub_triangle (hp h) (hpcuti.const_mul (Q:ℝ)⁻¹) (hpoldi.const_mul (Q:ℝ)⁻¹) have hs := hps h calc _ ≤ _ := hh _ < C h/L+η/Q+(Q:ℝ)⁻¹*(B*(4*δ)) := add_lt_add_of_lt_of_le hs hcscale _ = _ := by ring calc _ < ∑ h : Fin Q, (C h/L+(4*δ*B+η)/Q) := by apply Finset.sum_lt_sum · exact fun h _ => (hfinal h).le · exact ⟨⟨0,hQ⟩,Finset.mem_univ _,hfinal ⟨0,hQ⟩⟩ _ = _ := by rw [Finset.sum_add_distrib,← Finset.sum_div] simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] field_simp ring end ThreeTorus end end section /-! The simultaneous neutral passage with an actual integer-rounded common primitive. Its complete geometry and all spectral estimates are supplied by one and the same sequence of smooth Haar-preserving maps. -/ noncomputable section open MeasureTheory Set Function Filter open scoped Topology BigOperators namespace ThreeTorus /-- The jitter, all packet lists and all cutoffs are fixed before the final passage integer. No primitive, convergence, or spectral data are assumed. -/ theorem exists_simultaneous_neutral_residue_passage {ι : Type*} [Fintype ι] {A : ι → UnitAddCircle × UnitAddCircle → ℂ} {A₀ : ι → UnitAddCircle → ℝ → ℂ} {V : ℝ → ℝ} {m : ι → ℤ} {o l u R : ℝ} (hm : ∀ i, m i ≠ 0) (hA : ∀ i a x, x ∈ Ioc o (o+1) → A i ((x : UnitAddCircle),a) = A₀ i a x) (hA₀ : ∀ i a, ContDiff ℝ 2 (A₀ i a)) (hAc : ∀ i, Continuous (A₀ i).uncurry) (hAd : ∀ i, Continuous (fun p : UnitAddCircle × ℝ => deriv (A₀ i p.1) p.2)) (hAr : ∀ i, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => A₀ i (p.2 : UnitAddCircle) p.1)) (hsupp : ∀ i a, support (A₀ i a) ⊆ Ioc l u) (hlo : o ≤ l) (huo : u ≤ o+1) (hlu : l ≤ u) (hR : 0 < R) (hp : Periodic V 1) (hV : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) V) (hdV : ∀ x ∈ Icc (l-2*R) (u+2*R), 0 < deriv V x) (h0 : ∀ i, 0 ∉ (fun x => (m i:ℝ)*V x) '' Icc l u) (hAcont : ∀ i, Continuous (A i)) (hAs : ∀ i, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => A i (p.1,p.2))) {B : ι → ℝ} (hB : ∀ i, 0 ≤ B i) (hAb : ∀ i p, ‖A i p‖^2 ≤ B i) {δ : ℝ} (c : LawCutoffs δ) (hδ : 0 < δ) (hδ' : δ < 1/8) {χ : ι → ℝ → UnitAddCircle → ℝ} (hχs : ∀ i, CylinderSmooth (χ i).uncurry) (hχc : ∀ i, Continuous (χ i).uncurry) (hcχ : ∀ i, HasCompactSupport (χ i).uncurry) (hzχ : ∀ i a, χ i 0 a = 0) (hχ : ∀ i s a, 0 ≤ χ i s a ∧ χ i s a ≤ 1) (hχW : ∀ i s a, stationaryWeight (Icc (l-R) (u+R)) (fun x => (m i:ℝ)*V x) (A₀ i a) s ≠ 0 → χ i s a = 1) {Q : ℕ} (hQ : 0 < Q) : ∃ Δ : ℕ → ℝ, ∃ H : ℕ → UnitAddCircle → UnitAddCircle, (∀ n, 0 < n → |Δ n| ≤ (n:ℝ)⁻¹) ∧ Tendsto Δ atTop (𝓝 0) ∧ (∀ n, Continuous (H n)) ∧ (∀ n (x : ℝ), H n (x:UnitAddCircle) = ((n:ℝ)*((∫ t in 0..x, V t)+Δ n*x):UnitAddCircle)) ∧ ∃ C Cq : ι → ℝ, (∀ i, 0 ≤ C i) ∧ (∀ i, 0 ≤ Cq i) ∧ ∀ L : ℝ, 0 < L → ∀ β : ℝ, let ψ := neutralCircleCutoff c let J := neutralJitter c L β let J₀ := neutralJitterReal c L β let σ := fun (n : ℕ) a => (((1+J a/n)/n:ℝ):UnitAddCircle) (∀ n, Smooth (torusPassage (H n) (σ n)) ∧ Smooth (torusPassage (H n) (σ n)).symm ∧ MeasurePreserving (torusPassage (H n) (σ n)) μ μ) ∧ (∀ n, 0 < n → ∀ x : Cover, torusPassage (H n) (σ n) (project x) = project (passageFamily V J₀ 1 ((n:ℝ)⁻¹,Δ n) x) ∧ (torusPassage (H n) (σ n)).symm (project x) = project (passageFamily V J₀ (-1) ((n:ℝ)⁻¹,Δ n) x)) ∧ (∀ r : ℝ, ∀ K : Set Cover, IsCompact K → ∀ d : ℕ, TendstoUniformlyOn (fun (n : ℕ) x => iteratedFDeriv ℝ d (passageFamily V J₀ r ((n:ℝ)⁻¹,Δ n)) x) (fun x => iteratedFDeriv ℝ d (fun y : Cover => ![y 0+r*V (y 1),y 1,y 2]) x) atTop K) ∧ ∃ S : ℕ → ι → Finset ℤ, (∀ n, 0 < n → ∀ i j, j ∈ S n i → j ≠ 0) ∧ (∀ n, 0 < n → ∀ i j, j ∉ S n i → ∀ a, χ i ((j:ℝ)/n) a = 0) ∧ (∀ n, 0 < n → ∀ i j, j ∈ S n i ↔ ∃ a, χ i ((j:ℝ)/n) a ≠ 0) ∧ ∀ η : ℝ, 0 < η → ∀ᶠ n : ℕ in atTop, 0 < n ∧ ∀ i, let f := retainedPassage (m i) (H n) (A i) (S n i) (fun j => χ i ((j:ℝ)/n)) ψ let p := exactRetainedPacketDensity (Ioo δ (1-δ)) J₀ ψ (χ i) (m i) H (A i) (S n i) n let pold := foldCircleDensity (fun s => ∫ a, stationaryWeight (Icc (l-R) (u+R)) (fun x => (m i:ℝ)*V x) (A₀ i a) s ∂AddCircle.haarAddCircle) (∫ x : TripleTorus, ‖packet (m i) (A i) x-f (passageChart (H n) x)‖^2 ∂tripleVolume) < 8*δ*B i+η ∧ (∫ z, |p z-pold z| ∂AddCircle.haarAddCircle) < C i/L+4*δ*B i+η ∧ ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => p x) ∧ (∀ k : ℤ, (∫ x : TripleTorus, starRingEnd ℂ (f x)*f (twistIter (σ n) k x) ∂tripleVolume) = ∫ z, (p z:ℂ)*fourier k z ∂AddCircle.haarAddCircle) ∧ (∀ j u a, ‖retainedAmplitude ψ (χ i) (m i) H (A i) n j (u,a)‖ = ‖retainedAmplitude ψ (χ i) (m i) H (A i) n j (u,0)‖) ∧ (∑ h : Fin Q, ∫ z, |exactRetainedPacketDensity (Ioo δ (1-δ)) J₀ ψ (χ i) (m i) H (A i) ((S n i).filter (fun j => j % (Q:ℤ) = (h.val:ℤ) % (Q:ℤ))) n z- (Q:ℝ)⁻¹*pold z| ∂AddCircle.haarAddCircle) < Cq i/L+4*δ*B i+η := by classical obtain ⟨Δ,H,hΔ,hΔt,hHc,hHe,hgeo⟩ := exists_rounded_primitive_family hp hV have hPr : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x => ∫ t in 0..x, V t) := contDiff_real_primitive hV have hPV : deriv (fun x => ∫ t in 0..x, V t) = V := deriv_real_primitive hV.continuous have hv : Continuous hp.lift := by apply Continuous.quotient_lift exact hV.continuous have heV (x : ℝ) : hp.lift (x:UnitAddCircle) = V x := rfl have hneutral i := exact_neutral_passage (hm i) hHe (hA i) (hA₀ i) (hAc i) (hAd i) (hAr i) (hsupp i) hlo huo hlu hR hPr hV hPV hdV (h0 i) hHc (hAcont i) hΔt (hAs i) hv.measurable heV (hB i) (hAb i) c hδ hδ' (hχs i) (hχc i) (hcχ i) (hzχ i) (hχ i) (hχW i) choose C hC hrest using hneutral have hresidue i := exact_neutral_residue_refresh (hm i) hHe (hA i) (hA₀ i) (hAc i) (hAd i) (hAr i) (hsupp i) hlo huo hlu hR hPr hV hPV hdV (h0 i) hHc (hAcont i) hΔt hv.measurable heV (hB i) (hAb i) c hδ hδ' (hχc i) (hcχ i) (hzχ i) (hχ i) (hχW i) hQ choose Cq hCq hres using hresidue refine ⟨Δ,H,hΔ,hΔt,hHc,hHe,C,Cq,hC,hCq,?_⟩ intro L hL β dsimp only obtain ⟨hg,hgraph,hlimit⟩ := hgeo (neutralJitterReal c L β) (neutralJitterReal_periodic c L β) (neutralJitterReal_smooth c L β) refine ⟨hg,hgraph,hlimit,?_⟩ choose S hSN hST hgood using fun i => hrest i L hL β obtain ⟨T,hTN,hTE⟩ := cylinder_cutoff_exact_indices χ hcχ hzχ have hTZ n (hn : 0 < n) i j (hj : j ∉ T n i) a : χ i ((j:ℝ)/n) a = 0 := by by_contra hh exact hj ((hTE n hn i j).mpr ⟨a,hh⟩) refine ⟨T,fun n _ i => hTN n i,hTZ,hTE,?_⟩ intro η hη have hgall := (Filter.eventually_all).mpr (fun i => hgood i η hη) have hqall := (Filter.eventually_all).mpr (fun i => hres i L hL β (fun n => T n i) (fun n _ => hTN n i) (fun n hn => hTZ n hn i) η hη) filter_upwards [eventually_gt_atTop (0:ℕ),hgall,hqall] with n hn hall hq refine ⟨hn,fun i => ?_⟩ have ef := retainedPassage_eq_of_cover (m i) (H n) (A i) (S i n) (T n i) (fun j => χ i ((j:ℝ)/n)) (neutralCircleCutoff c) (hST i n hn) (hTZ n hn i) have ep := exactRetainedPacketDensity_eq_of_cover (Ioo δ (1-δ)) (neutralJitterReal c L β) (neutralCircleCutoff c) (χ i) (m i) H (A i) (S i n) (T n i) n (hST i n hn) (hTZ n hn i) have hh := hall i dsimp only at hh rw [ef,ep] at hh exact ⟨hh.1,hh.2.1,hh.2.2.1,hh.2.2.2.1,hh.2.2.2.2,hq i⟩ end ThreeTorus end end section /-! An unconditional neutral refresh on the concrete iterable packet class. All cutoffs, carrier, broad jitter and rounded geometry are constructed, in that order. The new state uses the literal coefficients and its canonical spectra, not abstract model arrays. -/ noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators namespace ThreeTorus /-- Full neutral state producer. Both approximation errors may be arbitrarily small. The same output state has all actual spectral moments, exactly disjoint angular indices, and the residue refresh statement. -/ theorem eventually_neutral_state_family {ι : Type*} [Fintype ι] (F : PassagePackets ι) (hi : Injective F.m) (hmod : F.SpareIndependent) {Q : ℕ} (hQ : 0 < Q) {ε τ : ℝ} (hε : 0 < ε) (hτ : 0 < τ) : ∀ᶠ c : ℕ in atTop, 0 < c ∧ ∃ δ : ℝ, ∃ cut : LawCutoffs δ, ∃ L : ℝ, ∃ χ : ι → ℝ → ℝ, ∃ Δ : ℕ → ℝ, ∃ H : ℕ → UnitAddCircle → UnitAddCircle, 0 < δ ∧ δ < 1/8 ∧ 0 < L ∧ (∀ i, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (χ i) ∧ HasCompactSupport (χ i) ∧ χ i 0 = 0 ∧ (∀ s, 0 ≤ χ i s ∧ χ i s ≤ 1) ∧ ∀ s ∈ tsupport (χ i), ∃ x ∈ F.core i, |s-(F.m i:ℝ)*((c:ℝ)+F.V x)| < τ) ∧ Pairwise (fun i j => Disjoint (tsupport (χ i)) (tsupport (χ j))) ∧ (∀ n, 0 < n → |Δ n| ≤ (n:ℝ)⁻¹) ∧ Tendsto Δ atTop (𝓝 0) ∧ (∀ n, Continuous (H n)) ∧ (∀ n (x : ℝ), H n (x:UnitAddCircle) = ((n:ℝ)*((∫ t in 0..x, (c:ℝ)+F.V t)+Δ n*x):UnitAddCircle)) ∧ let ψ := neutralCircleCutoff cut let J := neutralJitter cut L 0 let J₀ := neutralJitterReal cut L 0 let σ := fun (n : ℕ) a => (((1+J a/n)/n:ℝ):UnitAddCircle) (∀ n, Smooth (torusPassage (H n) (σ n)) ∧ Smooth (torusPassage (H n) (σ n)).symm ∧ MeasurePreserving (torusPassage (H n) (σ n)) μ μ) ∧ (∀ n, 0 < n → ∀ x : Cover, torusPassage (H n) (σ n) (project x) = project (passageFamily (fun t => (c:ℝ)+F.V t) J₀ 1 ((n:ℝ)⁻¹,Δ n) x) ∧ (torusPassage (H n) (σ n)).symm (project x) = project (passageFamily (fun t => (c:ℝ)+F.V t) J₀ (-1) ((n:ℝ)⁻¹,Δ n) x)) ∧ (∀ r : ℝ, ∀ K : Set Cover, IsCompact K → ∀ d : ℕ, TendstoUniformlyOn (fun (n : ℕ) x => iteratedFDeriv ℝ d (passageFamily (fun t => (c:ℝ)+F.V t) J₀ r ((n:ℝ)⁻¹,Δ n)) x) (fun x => iteratedFDeriv ℝ d (fun y : Cover => ![y 0+r*((c:ℝ)+F.V (y 1)),y 1,y 2]) x) atTop K) ∧ ∃ S : ℕ → ι → Finset ℤ, (∀ n, 0 < n → Pairwise (fun i j => Disjoint (S n i) (S n j))) ∧ (∀ n, 0 < n → ∀ i j, j ∈ S n i ↔ χ i ((j:ℝ)/n) ≠ 0) ∧ ∀ᶠ n : ℕ in atTop, 0 < n ∧ (∀ i, let f := retainedPassage (F.m i) (H n) (F.A i) (S n i) (fun j _ => χ i ((j:ℝ)/n)) ψ let p := exactRetainedPacketDensity (Ioo δ (1-δ)) J₀ ψ (fun s _ => χ i s) (F.m i) H (F.A i) (S n i) n (∫ x : TripleTorus, ‖packet (F.m i) (F.A i) x-f (passageChart (H n) x)‖^2 ∂tripleVolume) < ε ∧ (∫ z, |p z-F.density i z| ∂AddCircle.haarAddCircle) < ε ∧ (∀ k : ℤ, (∫ x : TripleTorus, starRingEnd ℂ (f x)*f (twistIter (σ n) k x) ∂tripleVolume) = ∫ z, (p z:ℂ)*fourier k z ∂AddCircle.haarAddCircle) ∧ (∑ h : Fin Q, ∫ z, |exactRetainedPacketDensity (Ioo δ (1-δ)) J₀ ψ (fun s _ => χ i s) (F.m i) H (F.A i) ((S n i).filter (fun j => j % (Q:ℤ) = (h.val:ℤ) % (Q:ℤ))) n z- (Q:ℝ)⁻¹*F.density i z| ∂AddCircle.haarAddCircle) < ε) ∧ ∃ G : PassagePackets (PassageOutputIndex (S n)), Injective G.m ∧ G.SpareIndependent ∧ G.V = (fun x => (1+J₀ x/n)/n) ∧ (∀ i, G.m i = i.2.val) ∧ (∀ i, G.A i = retainedAmplitude ψ (fun s _ => χ i.1 s) (F.m i.1) H (F.A i.1) n i.2.val) ∧ (∀ i, G.density i = exactRetainedPacketDensity (Ioo δ (1-δ)) J₀ ψ (fun s _ => χ i.1 s) (F.m i.1) H (F.A i.1) {i.2.val} n) ∧ (∀ i (E : ℤ → Prop) [DecidablePred E], exactRetainedPacketDensity (Ioo δ (1-δ)) J₀ ψ (fun s _ => χ i s) (F.m i) H (F.A i) ((S n i).filter E) n = fun z => ∑ j : ↥(S n i), if E j.val then G.density ⟨i,j⟩ z else 0) := by classical obtain ⟨B,hB,hAB⟩ := F.amplitude_bound have hden : 0 < 32*(B+1) := by positivity obtain ⟨δ,hδ,hδhi⟩ := exists_between (lt_min (by norm_num : (0:ℝ)<1/8) (div_pos hε hden)) have hδ' : δ < 1/8 := hδhi.trans_le (min_le_left _ _) have hδB : 8*δ*B < ε/4 := by have hh := (lt_div_iff₀ hden).mp (hδhi.trans_le (min_le_right _ _)) nlinarith let cut := lawCutoffs δ hδ hδ' filter_upwards [F.tracked_cutoffs hi hmod hτ, eventually_integral_speed_separation F.m F.periodic.lift F.nonzero F.speed_continuous] with c hc hsep obtain ⟨hcp,χ,hχ,hdisj⟩ := hc let Fc := F.carrier (c:ℤ) let X : ι → ℝ → UnitAddCircle → ℝ := fun i s _ => χ i s have hXs i : CylinderSmooth (X i).uncurry := by change ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => χ i p.1) exact (hχ i).1.comp contDiff_fst have hXc i : Continuous (X i).uncurry := (hχ i).1.continuous.comp continuous_fst have hXcompact i : HasCompactSupport (X i).uncurry := compactSupport_frequency_cylinder (hχ i).2.1 have hXzero i (a : UnitAddCircle) : X i 0 a = 0 := (hχ i).2.2.1 have hXbound i s (a : UnitAddCircle) : 0 ≤ X i s a ∧ X i s a ≤ 1 := (hχ i).2.2.2.1 s have hXW i s a : stationaryWeight (Icc (Fc.l-Fc.R) (Fc.u+Fc.R)) (fun x => (Fc.m i:ℝ)*Fc.V x) (Fc.A₀ i a) s ≠ 0 → X i s a = 1 := by simpa only [Fc,PassagePackets.carrier,Int.cast_natCast] using (hχ i).2.2.2.2.2 (Icc (F.l-F.R) (F.u+F.R)) a s have hzero i : 0 ∉ (fun x => (Fc.m i:ℝ)*Fc.V x) '' Icc Fc.l Fc.u := by rintro ⟨x,_hx,he⟩ apply hsep.2.1 i (x:UnitAddCircle) change (F.m i:ℝ)*((c:ℝ)+F.V x) = 0 simpa only [Fc,PassagePackets.carrier,Int.cast_natCast] using he obtain ⟨Δ,H,hΔ,hΔt,hHc,hHe,C,Cq,_hC,_hCq,hrest⟩ := exists_simultaneous_neutral_residue_passage Fc.nonzero Fc.lift_matches Fc.two Fc.continuous_lift Fc.continuous_deriv Fc.smooth_lift Fc.support_lift Fc.lo Fc.uo Fc.lu Fc.Rpos Fc.periodic Fc.smooth_speed Fc.positive_deriv hzero Fc.continuous_amplitude Fc.smooth_amplitude (B := fun _ => B) (fun _ => hB) hAB cut hδ hδ' hXs hXc hXcompact hXzero hXbound hXW hQ have hsmall : 0 < ε/4 := by positivity have hCT i : ∀ᶠ L : ℝ in atTop, C i/L < ε/4 := (tendsto_const_nhds.div_atTop tendsto_id).eventually (gt_mem_nhds hsmall) have hCqT i : ∀ᶠ L : ℝ in atTop, Cq i/L < ε/4 := (tendsto_const_nhds.div_atTop tendsto_id).eventually (gt_mem_nhds hsmall) obtain ⟨L,hL,hCL,hCqL⟩ := ((eventually_gt_atTop (0:ℝ)).and (((Filter.eventually_all).mpr hCT).and ((Filter.eventually_all).mpr hCqT))).exists obtain ⟨hgeo,hgraph,hlimit,S,hSN,_hST,hTE,hgood⟩ := hrest L hL 0 have hTE' n (hn : 0 < n) i j : j ∈ S n i ↔ χ i ((j:ℝ)/n) ≠ 0 := by simpa only [X,exists_const] using hTE n hn i j have hSd n (hn : 0 < n) : Pairwise (fun i j => Disjoint (S n i) (S n j)) := by intro i j hij rw [Finset.disjoint_left] intro k hki hkj exact Set.disjoint_left.mp (hdisj hij) (subset_tsupport _ ((hTE' n hn i k).mp hki)) (subset_tsupport _ ((hTE' n hn j k).mp hkj)) refine ⟨hcp,δ,cut,L,χ,Δ,H,hδ,hδ',hL,?_,hdisj,hΔ,hΔt,hHc,?_,?_,?_,?_,S,hSd,hTE',?_⟩ · exact fun i => ⟨(hχ i).1,(hχ i).2.1,(hχ i).2.2.1,(hχ i).2.2.2.1,(hχ i).2.2.2.2.1⟩ · simpa only [Fc,PassagePackets.carrier,Int.cast_natCast] using hHe · exact hgeo · simpa only [Fc,PassagePackets.carrier,Int.cast_natCast] using hgraph · simpa only [Fc,PassagePackets.carrier,Int.cast_natCast] using hlimit filter_upwards [hgood (ε/4) hsmall] with n hn have hn0 := hn.1 refine ⟨hn0,?_,?_⟩ · intro i have hh := hn.2 i dsimp only at hh ⊢ have hPM := (Fc.density_marginal (F.carrier_spare (c:ℤ) hmod) i).trans (F.density_carrier hmod (c:ℤ) i) rw [hPM] at hh refine ⟨lt_trans hh.1 (by linarith), lt_trans hh.2.1 (by linarith [hCL i]), hh.2.2.2.1,lt_trans hh.2.2.2.2.2 (by linarith [hCqL i])⟩ · let Hl (n : ℕ) (x : ℝ) := (n:ℝ)*((∫ t in 0..x, Fc.V t)+Δ n*x) have hHl n : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (Hl n) := contDiff_const.mul ((contDiff_real_primitive Fc.smooth_speed).add (contDiff_const.mul contDiff_id)) have hJs := neutralJitterReal_smooth cut L 0 have hJd x (hx : x ∈ Ioo δ (1-δ)) : 0 < deriv (neutralJitterReal cut L 0) x := by rw [neutralJitterReal_deriv cut hδ L 0 hx] exact hL obtain ⟨G,hGm,hGA,_hGA₀,hGV,_hGo,_hGl,_hGu,hGmod⟩ := literal_output_packets F.m F.A H Hl X (S n) (hSN n hn0) F.continuous_amplitude F.smooth_amplitude hXs hXc hHc hHl hHe cut hδ hδ' (neutralJitterReal_periodic cut L 0) hJs hJd hn0 have heψ : ∀ x ∈ Ioc (0:ℝ) (0+1), neutralCircleCutoff cut (x:UnitAddCircle) = cut.ψ x := by intro x hx rw [neutralCircleCutoff_coe] exact periodize_eq_on_unit ((subset_tsupport cut.ψ).trans (cut.ψ_support.trans (by intro z hz; constructor <;> linarith [hz.1,hz.2]))) ⟨hx.1.le,by simpa using hx.2⟩ have hsψ : tsupport cut.ψ ⊆ Ioc (0:ℝ) (0+1) ∩ Ioo δ (1-δ) := by intro x hx have hh := cut.ψ_support hx exact ⟨⟨by linarith [hh.1],by linarith [hh.2]⟩,hh⟩ have hJinj : InjOn (neutralJitterReal cut L 0) (Ioo δ (1-δ)) := by intro x hx y hy he rw [neutralJitterReal_affine cut hδ L 0 ⟨hx.1.le,hx.2.le⟩, neutralJitterReal_affine cut hδ L 0 ⟨hy.1.le,hy.2.le⟩] at he nlinarith have hGden (i : PassageOutputIndex (S n)) : G.density i = exactRetainedPacketDensity (Ioo δ (1-δ)) (neutralJitterReal cut L 0) (neutralCircleCutoff cut) (X i.1) (F.m i.1) H (F.A i.1) {i.2.val} n := by apply literal_packet_canonical_density G hGmod i (hGm i) (hGA i) ?_ (continuous_neutralJitter cut L 0) hJs (fun _ _ => rfl) isOpen_Ioo hJinj (fun x hx => (hJd x hx).ne') (continuous_neutralCircleCutoff cut) cut.ψ_smooth cut.ψ_compact heψ hsψ (hXc i.1) (hXs i.1) hHc hHl hHe (F.continuous_amplitude i.1) (F.smooth_amplitude i.1) hn0 intro a obtain ⟨x,rfl⟩ := Quotient.mk_surjective a change (G.V x:UnitAddCircle) = _ rw [hGV] rfl refine ⟨G,?_,hGmod,hGV,hGm,hGA,hGden,?_⟩ · have he : G.m = fun i : PassageOutputIndex (S n) => i.2.val := funext hGm rw [he] exact output_index_injective (hSd n hn0) · intro i E inst simp_rw [hGden] exact exactRetainedPacketDensity_filtered_singletons hJs isOpen_Ioo hJinj (fun x hx => (hJd x hx).ne') cut.ψ_smooth cut.ψ_compact heψ hsψ (hXs i) hHl hHe (F.smooth_amplitude i) hn0 (hSN n hn0 i) E end ThreeTorus end end section /-! The anchor carrier can be fixed BEFORE all accuracy parameters and the signal law. The source's order of choices is implemented literally. -/ noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators namespace ThreeTorus lemma tracked_cutoffs_fixed_carrier {ι : Type*} [Finite ι] (F : PassagePackets ι) (hi : Injective F.m) (hmod : F.SpareIndependent) {c : ℕ} (hc : 0 < c ∧ (∀ i x, (F.m i:ℝ)*((c:ℝ)+F.periodic.lift x) ≠ 0) ∧ ∀ i j x y, (F.m i:ℝ)*((c:ℝ)+F.periodic.lift x) = (F.m j:ℝ)*((c:ℝ)+F.periodic.lift y) → F.m i = F.m j) {η : ℝ} (hη : 0 < η) : 0 < c ∧ ∃ χ : ι → ℝ → ℝ, (∀ i, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (χ i) ∧ HasCompactSupport (χ i) ∧ χ i 0 = 0 ∧ (∀ s, 0 ≤ χ i s ∧ χ i s ≤ 1) ∧ (∀ s ∈ tsupport (χ i), ∃ x ∈ F.core i, |s-(F.m i:ℝ)*((c:ℝ)+F.V x)| < η) ∧ ∀ (C : Set ℝ) a s, stationaryWeight C (fun x => (F.m i:ℝ)*((c:ℝ)+F.V x)) (F.A₀ i a) s ≠ 0 → χ i s = 1) ∧ Pairwise (fun i j => Disjoint (tsupport (χ i)) (tsupport (χ j))) := by have hzero i : 0 ∉ (fun x => (F.m i:ℝ)*((c:ℝ)+F.V x)) '' F.core i := by rintro ⟨x,_hx,he⟩ exact hc.2.1 i (x:UnitAddCircle) he have hsep : Pairwise (fun i j => Disjoint ((fun x => (F.m i:ℝ)*((c:ℝ)+F.V x)) '' F.core i) ((fun x => (F.m j:ℝ)*((c:ℝ)+F.V x)) '' F.core j)) := by intro i j hij rw [Set.disjoint_left] rintro s ⟨x,_hx,rfl⟩ ⟨y,_hy,he⟩ exact hij (hi (hc.2.2 i j (x:UnitAddCircle) (y:UnitAddCircle) he.symm)) obtain ⟨χ,hχ,hd⟩ := tracked_frequency_cutoffs F.m (fun x => (c:ℝ)+F.V x) F.core (continuous_const.add F.smooth_speed.continuous) F.core_compact hzero hsep hη exact ⟨hc.1,χ,fun i => ⟨(hχ i).1,(hχ i).2.1,(hχ i).2.2.1, (hχ i).2.2.2.1,(hχ i).2.2.2.2.2.1,fun C a s => (hχ i).2.2.2.2.2.2 C (F.A₀ i a) (F.support_subset_core hmod i a) s⟩,hd⟩ theorem eventually_neutral_anchor_family {ι : Type*} [Fintype ι] (F : PassagePackets ι) (hi : Injective F.m) (hmod : F.SpareIndependent) : ∀ᶠ c : ℕ in atTop, ∀ Q : ℕ, 0 < Q → ∀ ε τ κ : ℝ, 0 < ε → 0 < τ → 0 < κ → 0 < c ∧ ∃ δ : ℝ, ∃ cut : LawCutoffs δ, ∃ L : ℝ, ∃ χ : ι → ℝ → ℝ, ∃ Δ : ℕ → ℝ, ∃ H : ℕ → UnitAddCircle → UnitAddCircle, 0 < δ ∧ δ < 1/8 ∧ δ < κ ∧ 0 < L ∧ (∀ i, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (χ i) ∧ HasCompactSupport (χ i) ∧ χ i 0 = 0 ∧ (∀ s, 0 ≤ χ i s ∧ χ i s ≤ 1) ∧ ∀ s ∈ tsupport (χ i), ∃ x ∈ F.core i, |s-(F.m i:ℝ)*((c:ℝ)+F.V x)| < τ) ∧ Pairwise (fun i j => Disjoint (tsupport (χ i)) (tsupport (χ j))) ∧ (∀ n, 0 < n → |Δ n| ≤ (n:ℝ)⁻¹) ∧ Tendsto Δ atTop (𝓝 0) ∧ (∀ n, Continuous (H n)) ∧ (∀ n (x : ℝ), H n (x:UnitAddCircle) = ((n:ℝ)*((∫ t in 0..x, (c:ℝ)+F.V t)+Δ n*x):UnitAddCircle)) ∧ let ψ := neutralCircleCutoff cut let J := neutralJitter cut L 0 let J₀ := neutralJitterReal cut L 0 let σ := fun (n : ℕ) a => (((1+J a/n)/n:ℝ):UnitAddCircle) (∀ n, Smooth (torusPassage (H n) (σ n)) ∧ Smooth (torusPassage (H n) (σ n)).symm ∧ MeasurePreserving (torusPassage (H n) (σ n)) μ μ) ∧ (∀ n, 0 < n → ∀ x : Cover, torusPassage (H n) (σ n) (project x) = project (passageFamily (fun t => (c:ℝ)+F.V t) J₀ 1 ((n:ℝ)⁻¹,Δ n) x) ∧ (torusPassage (H n) (σ n)).symm (project x) = project (passageFamily (fun t => (c:ℝ)+F.V t) J₀ (-1) ((n:ℝ)⁻¹,Δ n) x)) ∧ (∀ r : ℝ, ∀ K : Set Cover, IsCompact K → ∀ d : ℕ, TendstoUniformlyOn (fun (n : ℕ) x => iteratedFDeriv ℝ d (passageFamily (fun t => (c:ℝ)+F.V t) J₀ r ((n:ℝ)⁻¹,Δ n)) x) (fun x => iteratedFDeriv ℝ d (fun y : Cover => ![y 0+r*((c:ℝ)+F.V (y 1)),y 1,y 2]) x) atTop K) ∧ ∃ S : ℕ → ι → Finset ℤ, (∀ n, 0 < n → Pairwise (fun i j => Disjoint (S n i) (S n j))) ∧ (∀ n, 0 < n → ∀ i j, j ∈ S n i ↔ χ i ((j:ℝ)/n) ≠ 0) ∧ ∀ᶠ n : ℕ in atTop, 0 < n ∧ (∀ i, let f := retainedPassage (F.m i) (H n) (F.A i) (S n i) (fun j _ => χ i ((j:ℝ)/n)) ψ let p := exactRetainedPacketDensity (Ioo δ (1-δ)) J₀ ψ (fun s _ => χ i s) (F.m i) H (F.A i) (S n i) n (∫ x : TripleTorus, ‖packet (F.m i) (F.A i) x-f (passageChart (H n) x)‖^2 ∂tripleVolume) < ε ∧ (∫ z, |p z-F.density i z| ∂AddCircle.haarAddCircle) < ε ∧ (∀ k : ℤ, (∫ x : TripleTorus, starRingEnd ℂ (f x)*f (twistIter (σ n) k x) ∂tripleVolume) = ∫ z, (p z:ℂ)*fourier k z ∂AddCircle.haarAddCircle) ∧ (∑ h : Fin Q, ∫ z, |exactRetainedPacketDensity (Ioo δ (1-δ)) J₀ ψ (fun s _ => χ i s) (F.m i) H (F.A i) ((S n i).filter (fun j => j % (Q:ℤ) = (h.val:ℤ) % (Q:ℤ))) n z- (Q:ℝ)⁻¹*F.density i z| ∂AddCircle.haarAddCircle) < ε) ∧ ∃ G : PassagePackets (PassageOutputIndex (S n)), Injective G.m ∧ G.SpareIndependent ∧ G.V = (fun x => (1+J₀ x/n)/n) ∧ (∀ i, G.m i = i.2.val) ∧ (∀ i, G.A i = retainedAmplitude ψ (fun s _ => χ i.1 s) (F.m i.1) H (F.A i.1) n i.2.val) ∧ (∀ i, G.density i = exactRetainedPacketDensity (Ioo δ (1-δ)) J₀ ψ (fun s _ => χ i.1 s) (F.m i.1) H (F.A i.1) {i.2.val} n) ∧ (∀ i (E : ℤ → Prop) [DecidablePred E], exactRetainedPacketDensity (Ioo δ (1-δ)) J₀ ψ (fun s _ => χ i s) (F.m i) H (F.A i) ((S n i).filter E) n = fun z => ∑ j : ↥(S n i), if E j.val then G.density ⟨i,j⟩ z else 0) := by classical filter_upwards [eventually_integral_speed_separation F.m F.periodic.lift F.nonzero F.speed_continuous] with c hsep intro Q hQ ε τ κ hε hτ hκ obtain ⟨B,hB,hAB⟩ := F.amplitude_bound have hden : 0 < 32*(B+1) := by positivity obtain ⟨δ,hδ,hδall⟩ := exists_between (lt_min (lt_min (by norm_num : (0:ℝ)<1/8) (div_pos hε hden)) hκ) have hδhi := hδall.trans_le (min_le_left _ _) have hδκ := hδall.trans_le (min_le_right _ _) have hδ' : δ < 1/8 := hδhi.trans_le (min_le_left _ _) have hδB : 8*δ*B < ε/4 := by have hh := (lt_div_iff₀ hden).mp (hδhi.trans_le (min_le_right _ _)) nlinarith let cut := lawCutoffs δ hδ hδ' obtain ⟨hcp,χ,hχ,hdisj⟩ := tracked_cutoffs_fixed_carrier F hi hmod hsep hτ let Fc := F.carrier (c:ℤ) let X : ι → ℝ → UnitAddCircle → ℝ := fun i s _ => χ i s have hXs i : CylinderSmooth (X i).uncurry := by change ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => χ i p.1) exact (hχ i).1.comp contDiff_fst have hXc i : Continuous (X i).uncurry := (hχ i).1.continuous.comp continuous_fst have hXcompact i : HasCompactSupport (X i).uncurry := compactSupport_frequency_cylinder (hχ i).2.1 have hXzero i (a : UnitAddCircle) : X i 0 a = 0 := (hχ i).2.2.1 have hXbound i s (a : UnitAddCircle) : 0 ≤ X i s a ∧ X i s a ≤ 1 := (hχ i).2.2.2.1 s have hXW i s a : stationaryWeight (Icc (Fc.l-Fc.R) (Fc.u+Fc.R)) (fun x => (Fc.m i:ℝ)*Fc.V x) (Fc.A₀ i a) s ≠ 0 → X i s a = 1 := by simpa only [Fc,PassagePackets.carrier,Int.cast_natCast] using (hχ i).2.2.2.2.2 (Icc (F.l-F.R) (F.u+F.R)) a s have hzero i : 0 ∉ (fun x => (Fc.m i:ℝ)*Fc.V x) '' Icc Fc.l Fc.u := by rintro ⟨x,_hx,he⟩ apply hsep.2.1 i (x:UnitAddCircle) change (F.m i:ℝ)*((c:ℝ)+F.V x) = 0 simpa only [Fc,PassagePackets.carrier,Int.cast_natCast] using he obtain ⟨Δ,H,hΔ,hΔt,hHc,hHe,C,Cq,_hC,_hCq,hrest⟩ := exists_simultaneous_neutral_residue_passage Fc.nonzero Fc.lift_matches Fc.two Fc.continuous_lift Fc.continuous_deriv Fc.smooth_lift Fc.support_lift Fc.lo Fc.uo Fc.lu Fc.Rpos Fc.periodic Fc.smooth_speed Fc.positive_deriv hzero Fc.continuous_amplitude Fc.smooth_amplitude (B := fun _ => B) (fun _ => hB) hAB cut hδ hδ' hXs hXc hXcompact hXzero hXbound hXW hQ have hsmall : 0 < ε/4 := by positivity have hCT i : ∀ᶠ L : ℝ in atTop, C i/L < ε/4 := (tendsto_const_nhds.div_atTop tendsto_id).eventually (gt_mem_nhds hsmall) have hCqT i : ∀ᶠ L : ℝ in atTop, Cq i/L < ε/4 := (tendsto_const_nhds.div_atTop tendsto_id).eventually (gt_mem_nhds hsmall) obtain ⟨L,hL,hCL,hCqL⟩ := ((eventually_gt_atTop (0:ℝ)).and (((Filter.eventually_all).mpr hCT).and ((Filter.eventually_all).mpr hCqT))).exists obtain ⟨hgeo,hgraph,hlimit,S,hSN,_hST,hTE,hgood⟩ := hrest L hL 0 have hTE' n (hn : 0 < n) i j : j ∈ S n i ↔ χ i ((j:ℝ)/n) ≠ 0 := by simpa only [X,exists_const] using hTE n hn i j have hSd n (hn : 0 < n) : Pairwise (fun i j => Disjoint (S n i) (S n j)) := by intro i j hij rw [Finset.disjoint_left] intro k hki hkj exact Set.disjoint_left.mp (hdisj hij) (subset_tsupport _ ((hTE' n hn i k).mp hki)) (subset_tsupport _ ((hTE' n hn j k).mp hkj)) refine ⟨hcp,δ,cut,L,χ,Δ,H,hδ,hδ',hδκ,hL,?_,hdisj,hΔ,hΔt,hHc,?_,?_,?_,?_,S,hSd,hTE',?_⟩ · exact fun i => ⟨(hχ i).1,(hχ i).2.1,(hχ i).2.2.1,(hχ i).2.2.2.1,(hχ i).2.2.2.2.1⟩ · simpa only [Fc,PassagePackets.carrier,Int.cast_natCast] using hHe · exact hgeo · simpa only [Fc,PassagePackets.carrier,Int.cast_natCast] using hgraph · simpa only [Fc,PassagePackets.carrier,Int.cast_natCast] using hlimit filter_upwards [hgood (ε/4) hsmall] with n hn have hn0 := hn.1 refine ⟨hn0,?_,?_⟩ · intro i have hh := hn.2 i dsimp only at hh ⊢ have hPM := (Fc.density_marginal (F.carrier_spare (c:ℤ) hmod) i).trans (F.density_carrier hmod (c:ℤ) i) rw [hPM] at hh refine ⟨lt_trans hh.1 (by linarith), lt_trans hh.2.1 (by linarith [hCL i]), hh.2.2.2.1,lt_trans hh.2.2.2.2.2 (by linarith [hCqL i])⟩ · let Hl (n : ℕ) (x : ℝ) := (n:ℝ)*((∫ t in 0..x, Fc.V t)+Δ n*x) have hHl n : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (Hl n) := contDiff_const.mul ((contDiff_real_primitive Fc.smooth_speed).add (contDiff_const.mul contDiff_id)) have hJs := neutralJitterReal_smooth cut L 0 have hJd x (hx : x ∈ Ioo δ (1-δ)) : 0 < deriv (neutralJitterReal cut L 0) x := by rw [neutralJitterReal_deriv cut hδ L 0 hx] exact hL obtain ⟨G,hGm,hGA,_hGA₀,hGV,_hGo,_hGl,_hGu,hGmod⟩ := literal_output_packets F.m F.A H Hl X (S n) (hSN n hn0) F.continuous_amplitude F.smooth_amplitude hXs hXc hHc hHl hHe cut hδ hδ' (neutralJitterReal_periodic cut L 0) hJs hJd hn0 have heψ : ∀ x ∈ Ioc (0:ℝ) (0+1), neutralCircleCutoff cut (x:UnitAddCircle) = cut.ψ x := by intro x hx rw [neutralCircleCutoff_coe] exact periodize_eq_on_unit ((subset_tsupport cut.ψ).trans (cut.ψ_support.trans (by intro z hz; constructor <;> linarith [hz.1,hz.2]))) ⟨hx.1.le,by simpa using hx.2⟩ have hsψ : tsupport cut.ψ ⊆ Ioc (0:ℝ) (0+1) ∩ Ioo δ (1-δ) := by intro x hx have hh := cut.ψ_support hx exact ⟨⟨by linarith [hh.1],by linarith [hh.2]⟩,hh⟩ have hJinj : InjOn (neutralJitterReal cut L 0) (Ioo δ (1-δ)) := by intro x hx y hy he rw [neutralJitterReal_affine cut hδ L 0 ⟨hx.1.le,hx.2.le⟩, neutralJitterReal_affine cut hδ L 0 ⟨hy.1.le,hy.2.le⟩] at he nlinarith have hGden (i : PassageOutputIndex (S n)) : G.density i = exactRetainedPacketDensity (Ioo δ (1-δ)) (neutralJitterReal cut L 0) (neutralCircleCutoff cut) (X i.1) (F.m i.1) H (F.A i.1) {i.2.val} n := by apply literal_packet_canonical_density G hGmod i (hGm i) (hGA i) ?_ (continuous_neutralJitter cut L 0) hJs (fun _ _ => rfl) isOpen_Ioo hJinj (fun x hx => (hJd x hx).ne') (continuous_neutralCircleCutoff cut) cut.ψ_smooth cut.ψ_compact heψ hsψ (hXc i.1) (hXs i.1) hHc hHl hHe (F.continuous_amplitude i.1) (F.smooth_amplitude i.1) hn0 intro a obtain ⟨x,rfl⟩ := Quotient.mk_surjective a change (G.V x:UnitAddCircle) = _ rw [hGV] rfl refine ⟨G,?_,hGmod,hGV,hGm,hGA,hGden,?_⟩ · have he : G.m = fun i : PassageOutputIndex (S n) => i.2.val := funext hGm rw [he] exact output_index_injective (hSd n hn0) · intro i E inst simp_rw [hGden] exact exactRetainedPacketDensity_filtered_singletons hJs isOpen_Ioo hJinj (fun x hx => (hJd x hx).ne') cut.ψ_smooth cut.ψ_compact heψ hsψ (hXs i) hHl hHe (F.smooth_amplitude i) hn0 (hSN n hn0 i) E end ThreeTorus end end section /-! Conjugation symmetry survives the literal retained coefficients. Smooth cutoffs can be made reflection-compatible by a product, without losing the already constructed exact disjointness of their supports. -/ noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators ComplexConjugate namespace ThreeTorus def pairedCylinderCutoff {ι : Type*} (τ : ι → ι) (χ : ι → ℝ → UnitAddCircle → ℝ) (i : ι) (s : ℝ) (a : UnitAddCircle) : ℝ := χ i s a * χ (τ i) (-s) a lemma pairedCylinderCutoff_reflect {ι : Type*} {τ : ι → ι} (hτ : Involutive τ) (χ : ι → ℝ → UnitAddCircle → ℝ) (i : ι) (s : ℝ) (a : UnitAddCircle) : pairedCylinderCutoff τ χ (τ i) s a = pairedCylinderCutoff τ χ i (-s) a := by simp only [pairedCylinderCutoff,hτ i,neg_neg,mul_comm] lemma pairedCylinderCutoff_smooth {ι : Type*} (τ : ι → ι) {χ : ι → ℝ → UnitAddCircle → ℝ} (hχ : ∀ i, CylinderSmooth (χ i).uncurry) (i : ι) : CylinderSmooth (pairedCylinderCutoff τ χ i).uncurry := by change ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => χ i p.1 (p.2:UnitAddCircle) * χ (τ i) (-p.1) (p.2:UnitAddCircle)) exact (hχ i).mul ((hχ (τ i)).comp (contDiff_fst.neg.prodMk contDiff_snd)) lemma pairedCylinderCutoff_tsupport {ι : Type*} (τ : ι → ι) (χ : ι → ℝ → UnitAddCircle → ℝ) (i : ι) : tsupport (pairedCylinderCutoff τ χ i).uncurry ⊆ tsupport (χ i).uncurry := tsupport_mul_subset_left lemma pairedCylinderCutoff_compact {ι : Type*} (τ : ι → ι) {χ : ι → ℝ → UnitAddCircle → ℝ} (hχ : ∀ i, HasCompactSupport (χ i).uncurry) (i : ι) : HasCompactSupport (pairedCylinderCutoff τ χ i).uncurry := (hχ i).mul_right lemma pairedCylinderCutoff_disjoint {ι : Type*} (τ : ι → ι) {χ : ι → ℝ → UnitAddCircle → ℝ} (hd : Pairwise (fun i j => Disjoint (tsupport (χ i).uncurry) (tsupport (χ j).uncurry))) : Pairwise (fun i j => Disjoint (tsupport (pairedCylinderCutoff τ χ i).uncurry) (tsupport (pairedCylinderCutoff τ χ j).uncurry)) := by intro i j hij exact (hd hij).mono (pairedCylinderCutoff_tsupport τ χ i) (pairedCylinderCutoff_tsupport τ χ j) lemma pairedCylinderCutoff_bounds {ι : Type*} (τ : ι → ι) {χ : ι → ℝ → UnitAddCircle → ℝ} (hχ : ∀ i s a, 0 ≤ χ i s a ∧ χ i s a ≤ 1) (i : ι) (s : ℝ) (a : UnitAddCircle) : 0 ≤ pairedCylinderCutoff τ χ i s a ∧ pairedCylinderCutoff τ χ i s a ≤ 1 := by refine ⟨mul_nonneg (hχ i s a).1 (hχ (τ i) (-s) a).1,?_⟩ exact (mul_le_mul (hχ i s a).2 (hχ (τ i) (-s) a).2 (hχ (τ i) (-s) a).1 zero_le_one).trans (by norm_num) lemma pairedCylinderCutoff_one {ι : Type*} (τ : ι → ι) {χ : ι → ℝ → UnitAddCircle → ℝ} {K : ι → Set (ℝ × UnitAddCircle)} (hK : ∀ i p, p ∈ K i → (-p.1,p.2) ∈ K (τ i)) (hχ : ∀ i p, p ∈ K i → (χ i).uncurry =ᶠ[𝓝 p] 1) (i : ι) (p : ℝ × UnitAddCircle) (hp : p ∈ K i) : (pairedCylinderCutoff τ χ i).uncurry =ᶠ[𝓝 p] 1 := by have he₁ := hχ i p hp have he₂ := (hχ (τ i) (-p.1,p.2) (hK i p hp)).comp_tendsto ((continuous_fst.neg.prodMk continuous_snd).tendsto p) filter_upwards [he₁,he₂] with q hq₁ hq₂ change χ i q.1 q.2 * χ (τ i) (-q.1) q.2 = 1 change χ i q.1 q.2 = 1 at hq₁ change χ (τ i) (-q.1) q.2 = 1 at hq₂ rw [hq₁,hq₂,one_mul] lemma symmetric_exact_indices {ι : Type*} {τ : ι → ι} {χ : ι → ℝ → UnitAddCircle → ℝ} (hc : ∀ i, HasCompactSupport (χ i).uncurry) (hz : ∀ i a, χ i 0 a = 0) (hsym : ∀ i s a, χ (τ i) s a = χ i (-s) a) : ∃ S : ℕ → ι → Finset ℤ, (∀ n i j, j ∈ S n i → j ≠ 0) ∧ (∀ n, 0 < n → ∀ i j, j ∈ S n i ↔ ∃ a, χ i ((j:ℝ)/n) a ≠ 0) ∧ ∀ n, 0 < n → ∀ i j, j ∈ S n (τ i) ↔ -j ∈ S n i := by obtain ⟨S,hS0,hS⟩ := cylinder_cutoff_exact_indices χ hc hz refine ⟨S,hS0,hS,?_⟩ intro n hn i j rw [hS n hn (τ i) j,hS n hn i (-j)] simp only [hsym,Int.cast_neg,neg_div] lemma retainedAmplitude_reflect (ψ : UnitAddCircle → ℝ) (χ : ℝ → UnitAddCircle → ℝ) (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) (A : UnitAddCircle × UnitAddCircle → ℂ) (n : ℕ) (j : ℤ) (p : UnitAddCircle × UnitAddCircle) : retainedAmplitude ψ (fun s a => χ (-s) a) (-m) H (fun x => conj (A x)) n j p = conj (retainedAmplitude ψ χ m H A n (-j) p) := by simp only [retainedAmplitude,exactPassageCoeff_neg,Int.cast_neg,neg_div,map_mul, Complex.conj_ofReal,fourier_neg] lemma packet_reflect (m : ℤ) (A : UnitAddCircle × UnitAddCircle → ℂ) (x : TripleTorus) : packet (-m) (fun p => conj (A p)) x = conj (packet m A x) := by simp only [packet,map_mul,fourier_neg] lemma retainedPassage_reflect (ψ : UnitAddCircle → ℝ) (χ : ℝ → UnitAddCircle → ℝ) (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) (A : UnitAddCircle × UnitAddCircle → ℂ) (S : Finset ℤ) (n : ℕ) (x : TripleTorus) : retainedPassage (-m) (H n) (fun p => conj (A p)) (S.image Neg.neg) (fun j a => χ (-(j:ℝ)/n) a) ψ x = conj (retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) ψ x) := by classical have he : (fun (j : ℤ) a => χ (-(j:ℝ)/n) a) = (fun (j : ℤ) a => (fun s a => χ (-s) a) ((j:ℝ)/n) a) := by funext j a rw [neg_div] rw [he] change (∑ j ∈ S.image Neg.neg, packet j (retainedAmplitude ψ (fun s a => χ (-s) a) (-m) H (fun p => conj (A p)) n j) x) = conj (∑ j ∈ S, packet j (retainedAmplitude ψ χ m H A n j) x) rw [Finset.sum_image (fun _ _ _ _ h => neg_injective h),map_sum] apply Finset.sum_congr rfl intro j _hj have ha : retainedAmplitude ψ (fun s a => χ (-s) a) (-m) H (fun p => conj (A p)) n (-j) = fun p => conj (retainedAmplitude ψ χ m H A n j p) := by funext p simpa only [neg_neg] using retainedAmplitude_reflect ψ χ m H A n (-j) p rw [ha,packet_reflect] end ThreeTorus end end section /-! Actual conjugation data, invariant cutoff choices, and canonical spectral reflection. No symmetry of an approximate model is substituted for reality of the packets. -/ noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators ComplexConjugate namespace ThreeTorus namespace PassagePackets variable {ι : Type*} (F : PassagePackets ι) /-- The entire increasing good arc, including its relation to the lifted support interval. The excluded length is exactly `2*δ`. -/ def GoodArc (δ : ℝ) : Prop := F.o = 0 ∧ δ < F.l ∧ F.u < 1-δ ∧ ∀ x ∈ Ioo δ (1-δ), 0 < deriv F.V x structure Reflection (τ : ι → ι) : Prop where index : ∀ i, F.m (τ i) = -F.m i amplitude : ∀ i p, F.A (τ i) p = conj (F.A i p) lift : ∀ i a x, F.A₀ (τ i) a x = conj (F.A₀ i a x) variable {F} lemma Reflection.involutive {τ : ι → ι} (h : F.Reflection τ) (hi : Injective F.m) : Involutive τ := fun i => hi (by rw [h.index,h.index,neg_neg]) lemma Reflection.core {τ : ι → ι} (h : F.Reflection τ) (i : ι) : F.core (τ i) = F.core i := by unfold PassagePackets.core tsupport congr 1 ext x simp only [mem_support,h.lift,map_ne_zero] lemma Reflection.weight {τ : ι → ι} (h : F.Reflection τ) (i : ι) (s : ℝ) : F.weight (τ i) s = F.weight i (-s) := by have hinj := (scaled_speed_good_branch (F.nonzero i) F.Rpos F.smooth_speed F.positive_deriv).1 have ha : F.A₀ (τ i) 0 = fun x => conj (F.A₀ i 0 x) := funext (h.lift i 0) simp only [PassagePackets.weight,h.index,ha,Int.cast_neg,neg_mul] exact stationaryWeight_reflect hinj _ s end PassagePackets lemma foldCircleDensity_reflect (q : ℝ → ℝ) (z : UnitAddCircle) : foldCircleDensity (fun s => q (-s)) z = foldCircleDensity q (-z) := by obtain ⟨x,rfl⟩ := Quotient.mk_surjective z rw [← AddCircle.coe_neg,foldCircleDensity_coe_all,foldCircleDensity_coe_all] calc (∑' k : ℤ, q (-(x+k))) = ∑' k : ℤ, q (-x+(-k:ℤ)) := by apply tsum_congr intro k congr 1 simp only [Int.cast_neg] ring _ = ∑' k : ℤ, q (-x+k) := (Equiv.neg ℤ).tsum_eq (fun k => q (-x+k)) lemma PassagePackets.Reflection.density {ι : Type*} {F : PassagePackets ι} {τ : ι → ι} (h : F.Reflection τ) (i : ι) (z : UnitAddCircle) : F.density (τ i) z = F.density i (-z) := by unfold PassagePackets.density rw [show F.weight (τ i) = (fun s => F.weight i (-s)) from funext (h.weight i)] exact foldCircleDensity_reflect _ z /-- Reflection-compatible, exactly disjoint cutoffs with the full tracking and stationary identity used by the actual neutral and signal producers. -/ theorem tracked_symmetric_cutoffs_fixed_carrier {ι : Type*} [Finite ι] (F : PassagePackets ι) (hi : Injective F.m) (hmod : F.SpareIndependent) {τ : ι → ι} (hr : F.Reflection τ) {c : ℕ} (hc : 0 < c ∧ (∀ i x, (F.m i:ℝ)*((c:ℝ)+F.periodic.lift x) ≠ 0) ∧ ∀ i j x y, (F.m i:ℝ)*((c:ℝ)+F.periodic.lift x) = (F.m j:ℝ)*((c:ℝ)+F.periodic.lift y) → F.m i = F.m j) {η : ℝ} (hη : 0 < η) : 0 < c ∧ ∃ χ : ι → ℝ → ℝ, (∀ i, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (χ i) ∧ HasCompactSupport (χ i) ∧ χ i 0 = 0 ∧ (∀ s, 0 ≤ χ i s ∧ χ i s ≤ 1) ∧ (∀ s ∈ tsupport (χ i), ∃ x ∈ F.core i, |s-(F.m i:ℝ)*((c:ℝ)+F.V x)| < η) ∧ ∀ (C : Set ℝ) a s, stationaryWeight C (fun x => (F.m i:ℝ)*((c:ℝ)+F.V x)) (F.A₀ i a) s ≠ 0 → χ i s = 1) ∧ Pairwise (fun i j => Disjoint (tsupport (χ i)) (tsupport (χ j))) ∧ (∀ i s, χ (τ i) s = χ i (-s)) := by have hzero i : 0 ∉ (fun x => (F.m i:ℝ)*((c:ℝ)+F.V x)) '' F.core i := by rintro ⟨x,_hx,he⟩ exact hc.2.1 i (x:UnitAddCircle) he have hsep : Pairwise (fun i j => Disjoint ((fun x => (F.m i:ℝ)*((c:ℝ)+F.V x)) '' F.core i) ((fun x => (F.m j:ℝ)*((c:ℝ)+F.V x)) '' F.core j)) := by intro i j hij rw [Set.disjoint_left] rintro s ⟨x,_hx,rfl⟩ ⟨y,_hy,he⟩ exact hij (hi (hc.2.2 i j (x:UnitAddCircle) (y:UnitAddCircle) he.symm)) obtain ⟨χ,hχ,hd⟩ := tracked_frequency_cutoffs F.m (fun x => (c:ℝ)+F.V x) F.core (continuous_const.add F.smooth_speed.continuous) F.core_compact hzero hsep hη let ψ : ι → ℝ → ℝ := fun i s => χ i s*χ (τ i) (-s) have hsub i : tsupport (ψ i) ⊆ tsupport (χ i) := tsupport_mul_subset_left refine ⟨hc.1,ψ,fun i => ⟨(hχ i).1.mul ((hχ (τ i)).1.comp contDiff_neg), (hχ i).2.1.mul_right,?_,?_,?_,?_⟩,?_,?_⟩ · simp only [ψ,(hχ i).2.2.1,zero_mul] · intro s exact ⟨mul_nonneg ((hχ i).2.2.2.1 s).1 ((hχ (τ i)).2.2.2.1 (-s)).1, (mul_le_mul ((hχ i).2.2.2.1 s).2 ((hχ (τ i)).2.2.2.1 (-s)).2 ((hχ (τ i)).2.2.2.1 (-s)).1 zero_le_one).trans (by norm_num)⟩ · intro s hs exact (hχ i).2.2.2.2.2.1 s (hsub i hs) · intro C a s hs have hsi := stationaryWeight_support_subset (F.support_subset_core hmod i a) hs obtain ⟨x,hx,hxs⟩ := hsi have hre : -s ∈ (fun x => (F.m (τ i):ℝ)*((c:ℝ)+F.V x)) '' F.core (τ i) := by refine ⟨x,(hr.core i).symm ▸ hx,?_⟩ simp only [hr.index,Int.cast_neg,neg_mul,hxs] have hleft := (hχ i).2.2.2.2.2.2 C (F.A₀ i a) (F.support_subset_core hmod i a) s hs have hright := ((hχ (τ i)).2.2.2.2.1 (-s) hre).self_of_nhds change χ (τ i) (-s) = 1 at hright simp only [ψ,hleft,hright,one_mul] · intro i j hij exact (hd hij).mono (hsub i) (hsub j) · intro i s simp only [ψ,hr.involutive hi i,neg_neg,mul_comm] end ThreeTorus end end section noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators ComplexConjugate namespace ThreeTorus namespace PassagePackets variable {ι κ : Type*} (F : PassagePackets ι) def cylinderCore (i : ι) : Set (ℝ × UnitAddCircle) := tsupport (fun p : ℝ × UnitAddCircle => F.A₀ i p.2 p.1) lemma cylinderCore_subset (i : ι) : F.cylinderCore i ⊆ Icc F.l F.u ×ˢ univ := by apply closure_minimal · intro p hp exact ⟨⟨(F.support_lift i p.2 hp).1.le,(F.support_lift i p.2 hp).2⟩,mem_univ _⟩ · exact isClosed_Icc.prod isClosed_univ lemma cylinderCore_compact (i : ι) : IsCompact (F.cylinderCore i) := (isCompact_Icc.prod isCompact_univ).of_isClosed_subset (isClosed_tsupport _) (F.cylinderCore_subset i) lemma support_subset_cylinderCore (i : ι) (a : UnitAddCircle) {x : ℝ} (hx : x ∈ support (F.A₀ i a)) : (x,a) ∈ F.cylinderCore i := subset_tsupport _ hx lemma reflection_cylinderCore {τ : ι → ι} (hr : F.Reflection τ) (i : ι) : F.cylinderCore (τ i) = F.cylinderCore i := by unfold cylinderCore tsupport congr 1 ext p simp only [mem_support,hr.lift,map_ne_zero] /-- Multiply the literal torus amplitudes by smooth disjoint cylinder cutoffs and retain their actual lifts. All passage regularity is derived. -/ def clipped (q : κ → ι) (χ : κ → ℝ × UnitAddCircle → ℝ) (hχ : ∀ k, CylinderSmooth (χ k)) (hc : ∀ k, HasCompactSupport (χ k)) (hs : ∀ k, support (χ k) ⊆ Ioo F.o (F.o+1) ×ˢ univ) : PassagePackets κ where m := fun k => F.m (q k) nonzero := fun k => F.nonzero (q k) A := fun k p => (cylinderTorusCutoff (χ k) p:ℂ)*F.A (q k) p A₀ := fun k a x => (χ k (x,a):ℂ)*F.A₀ (q k) a x o := F.o l := F.l u := F.u R := F.R lo := F.lo uo := F.uo lu := F.lu Rpos := F.Rpos lift_matches := by intro k a x hx rw [cylinderTorusCutoff_eq (hs k) hx,F.lift_matches (q k) a x hx] two := by intro k a have hh := (cylinder_smooth_regular (hχ k)).1 a exact ((Complex.ofRealCLM.contDiff.comp hh).of_le (WithTop.coe_le_coe.mpr (le_top : (2:ℕ∞) ≤ ⊤))).mul (F.two (q k) a) continuous_lift := by intro k exact (Complex.continuous_ofReal.comp ((hχ k).continuous.comp continuous_swap)).mul (F.continuous_lift (q k)) continuous_deriv := by intro k have he (p : UnitAddCircle × ℝ) : deriv (fun x => (χ k (x,p.1):ℂ)*F.A₀ (q k) p.1 x) p.2 = Complex.ofReal (deriv (fun x => χ k (x,p.1)) p.2)*F.A₀ (q k) p.1 p.2 + (χ k (p.2,p.1):ℂ)*deriv (F.A₀ (q k) p.1) p.2 := by have hχd := ((cylinder_smooth_regular (hχ k)).1 p.1).differentiable (by simp) p.2 have hFd := (F.two (q k) p.1).differentiable (by norm_num) p.2 have hx : HasDerivAt (fun x => (χ k (x,p.1):ℂ)) (Complex.ofReal (deriv (fun x => χ k (x,p.1)) p.2)) p.2 := Complex.ofRealCLM.hasFDerivAt.comp_hasDerivAt _ hχd.hasDerivAt exact (hx.mul hFd.hasDerivAt).deriv simp_rw [he] exact ((Complex.continuous_ofReal.comp ((cylinder_smooth_regular (hχ k)).2.2.comp continuous_swap)).mul (F.continuous_lift (q k))).add ((Complex.continuous_ofReal.comp ((hχ k).continuous.comp continuous_swap)).mul (F.continuous_deriv (q k))) smooth_lift := fun k => (Complex.ofRealCLM.contDiff.comp (hχ k)).mul (F.smooth_lift (q k)) support_lift := by intro k a x hx apply F.support_lift (q k) a intro hz exact hx (by simp [hz]) continuous_amplitude := fun k => (Complex.continuous_ofReal.comp (cylinderTorusCutoff_continuous (hχ k) (hc k))).mul (F.continuous_amplitude (q k)) smooth_amplitude := fun k => (Complex.ofRealCLM.contDiff.comp (cylinderTorusCutoff_smooth (hχ k) (hc k))).mul (F.smooth_amplitude (q k)) V := F.V periodic := F.periodic smooth_speed := F.smooth_speed positive_deriv := F.positive_deriv lemma clipped_core_subset (q : κ → ι) (χ : κ → ℝ × UnitAddCircle → ℝ) (hχ : ∀ k, CylinderSmooth (χ k)) (hc : ∀ k, HasCompactSupport (χ k)) (hs : ∀ k, support (χ k) ⊆ Ioo F.o (F.o+1) ×ˢ univ) (k : κ) : (F.clipped q χ hχ hc hs).cylinderCore k ⊆ tsupport (χ k) := by apply closure_minimal · intro p hp apply subset_tsupport intro hz exact hp (by simp [clipped,hz]) · exact isClosed_tsupport _ end PassagePackets end ThreeTorus end end section noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators ComplexConjugate namespace ThreeTorus /-- First of the two neutral separations: equal angular modes are separated using the ACTUAL two-dimensional compact coefficient supports. In particular no spare-independence or injectivity of the old mode list is assumed. -/ theorem first_separation_cutoffs {ι : Type*} [Finite ι] (F : PassagePackets ι) (hd : ∀ i j, i ≠ j → F.m i = F.m j → Disjoint (F.cylinderCore i) (F.cylinderCore j)) (τ : ι → ι) (hτ : Involutive τ) (hr : F.Reflection τ) : ∃ c : ℕ, 0 < c ∧ (∀ x : ℝ, (c:ℝ)+F.V x ≠ 0) ∧ ∃ χ : ι → ℝ → UnitAddCircle → ℝ, (∀ i, CylinderSmooth (χ i).uncurry ∧ HasCompactSupport (χ i).uncurry ∧ (∀ a, χ i 0 a = 0) ∧ (∀ s a, 0 ≤ χ i s a ∧ χ i s a ≤ 1) ∧ ∀ (C : Set ℝ) a s, stationaryWeight C (fun x => (F.m i:ℝ)*((c:ℝ)+F.V x)) (F.A₀ i a) s ≠ 0 → χ i s a = 1) ∧ Pairwise (fun i j => Disjoint (tsupport (χ i).uncurry) (tsupport (χ j).uncurry)) ∧ (∀ i s a, χ (τ i) s a = χ i (-s) a) := by classical let m : Option ι → ℤ := fun i => i.elim 1 F.m have hm (i : Option ι) : m i ≠ 0 := by cases i with | none => norm_num [m] | some i => exact F.nonzero i obtain ⟨c,hc,hc0,hsep⟩ := exists_integral_speed_separation m F.periodic.lift hm F.speed_continuous let V : ℝ → ℝ := fun x => (c:ℝ)+F.V x have h0 (x : ℝ) : V x ≠ 0 := by simpa [m,V] using hc0 none (x:UnitAddCircle) have hV : Continuous V := continuous_const.add F.smooth_speed.continuous have hi : InjOn V (Icc F.l F.u) := by apply StrictMonoOn.injOn apply strictMonoOn_of_deriv_pos (convex_Icc _ _) hV.continuousOn intro x hx have hx' := interior_subset hx dsimp [V] rw [deriv_const_add] exact F.positive_deriv x ⟨by linarith [hx'.1,F.Rpos],by linarith [hx'.2,F.Rpos]⟩ let E (i : ι) := stationaryImage (F.m i) V (F.cylinderCore i) have hEc i : IsCompact (E i) := stationaryImage_compact hV (F.cylinderCore_compact i) have hEd : Pairwise (Disjoint on E) := by intro i j hij by_cases he : F.m i = F.m j · change Disjoint (stationaryImage (F.m i) V (F.cylinderCore i)) (stationaryImage (F.m j) V (F.cylinderCore j)) rw [← he] exact stationaryImage_disjoint_same (F.nonzero i) hi (F.cylinderCore_subset i) (F.cylinderCore_subset j) (hd i j hij he) · apply stationaryImage_disjoint_distinct intro p hp q hq h exact he (hsep (some i) (some j) (p.1:UnitAddCircle) (q.1:UnitAddCircle) h) let U : Set (ℝ × UnitAddCircle) := {p | p.1 ≠ 0} have hU : IsOpen U := isOpen_ne_fun continuous_fst continuous_const have hEU i : E i ⊆ U := by rintro _ ⟨p,hp,rfl⟩ exact mul_ne_zero (by exact_mod_cast F.nonzero i) (h0 p.1) obtain ⟨ξ,hξ,hξd⟩ := finite_disjoint_cylinder_cutoffs E (fun _ => U) hEc (fun _ => hU) hEU hEd let χ : ι → ℝ → UnitAddCircle → ℝ := fun i s a => ξ i (s,a)*ξ (τ i) (-s,a) have hsub i : tsupport (χ i).uncurry ⊆ tsupport (ξ i) := tsupport_mul_subset_left have hre i {s : ℝ} {a : UnitAddCircle} (hs : (s,a) ∈ E i) : (-s,a) ∈ E (τ i) := by obtain ⟨p,hp,he⟩ := hs have he1 := congrArg Prod.fst he have he2 := congrArg Prod.snd he dsimp only at he1 he2 refine ⟨p,(F.reflection_cylinderCore hr i).symm ▸ hp,?_⟩ apply Prod.ext · change (F.m (τ i):ℝ)*V p.1 = -s rw [hr.index,Int.cast_neg,neg_mul,he1] · exact he2 have hsE (i : ι) (C : Set ℝ) (a : UnitAddCircle) (s : ℝ) (hs : stationaryWeight C (fun x => (F.m i:ℝ)*V x) (F.A₀ i a) s ≠ 0) : (s,a) ∈ E i := by obtain ⟨x,hx,hxs⟩ := stationaryWeight_support_subset (show support (F.A₀ i a) ⊆ {x | (x,a) ∈ F.cylinderCore i} from fun x hx => F.support_subset_cylinderCore i a hx) hs exact ⟨(x,a),hx,Prod.ext hxs rfl⟩ refine ⟨c,hc,h0,χ,fun i => ⟨?_,?_,?_,?_,?_⟩,?_,?_⟩ · exact (hξ i).1.mul ((hξ (τ i)).1.comp (contDiff_fst.neg.prodMk contDiff_snd)) · exact (hξ i).2.1.mul_right · intro a have hz : ξ i (0,a) = 0 := by by_contra hh exact (hξ i).2.2.1 (subset_tsupport _ hh) rfl simp [χ,hz] · intro s a exact ⟨mul_nonneg ((hξ i).2.2.2.1 (s,a)).1 ((hξ (τ i)).2.2.2.1 (-s,a)).1, (mul_le_mul ((hξ i).2.2.2.1 (s,a)).2 ((hξ (τ i)).2.2.2.1 (-s,a)).2 ((hξ (τ i)).2.2.2.1 (-s,a)).1 zero_le_one).trans (by norm_num)⟩ · intro C a s hs have hs' := hsE i C a s hs have h1 := ((hξ i).2.2.2.2 (s,a) hs').self_of_nhds have h2 := ((hξ (τ i)).2.2.2.2 (-s,a) (hre i hs')).self_of_nhds change ξ i (s,a) = 1 at h1 change ξ (τ i) (-s,a) = 1 at h2 simp only [χ,h1,h2,one_mul] · intro i j hij exact (hξd hij).mono (hsub i) (hsub j) · intro i s a simp only [χ,hτ i,neg_neg,mul_comm] end ThreeTorus end end section noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators namespace ThreeTorus namespace PassagePackets variable {ι : Type*} (F : PassagePackets ι) def marginalWeight (i : ι) (s : ℝ) : ℝ := ∫ a, stationaryWeight (Icc (F.l-F.R) (F.u+F.R)) (fun x => (F.m i:ℝ)*F.V x) (F.A₀ i a) s ∂AddCircle.haarAddCircle def marginalDensity (i : ι) : UnitAddCircle → ℝ := foldCircleDensity (F.marginalWeight i) lemma marginalWeight_regular (i : ι) : Continuous (F.marginalWeight i) ∧ HasCompactSupport (F.marginalWeight i) := by let w : ℝ × UnitAddCircle → ℝ := fun p => stationaryWeight (Icc (F.l-F.R) (F.u+F.R)) (fun x => (F.m i:ℝ)*F.V x) (F.A₀ i p.2) p.1 have hb := scaled_speed_good_branch (F.nonzero i) F.Rpos F.smooth_speed F.positive_deriv have hw := stationaryWeight_signed_compact_continuous (contDiff_const.mul F.smooth_speed) (F.continuous_lift i) (fun a => (F.support_lift i a).trans Ioc_subset_Icc_self) F.Rpos hb.1 hb.2 have hc : Continuous (F.marginalWeight i) := by have hh : Continuous (uncurry (fun (s : ℝ) (a : UnitAddCircle) => w (s,a))) := hw.1 change Continuous (fun s => ∫ a, w (s,a) ∂AddCircle.haarAddCircle) simpa only [setIntegral_univ] using continuous_parametric_integral_of_continuous (X := ℝ) (Y := UnitAddCircle) (μ := AddCircle.haarAddCircle) hh isCompact_univ refine ⟨hc, (hw.2.image continuous_fst).of_isClosed_subset (isClosed_tsupport _) ?_⟩ apply closure_minimal ?_ (hw.2.image continuous_fst).isClosed intro s hs by_contra hn have hz (a : UnitAddCircle) : w (s,a) = 0 := by by_contra hh exact hn ⟨(s,a),subset_tsupport w hh,rfl⟩ apply hs simp only [marginalWeight,show ∀ a, stationaryWeight (Icc (F.l-F.R) (F.u+F.R)) (fun x => (F.m i:ℝ)*F.V x) (F.A₀ i a) s = 0 from hz,integral_zero] lemma marginalDensity_nonneg (i : ι) (z : UnitAddCircle) : 0 ≤ F.marginalDensity i z := foldCircleDensity_nonneg (fun s => integral_nonneg (fun _a => branchDensity_nonneg (fun _ => sq_nonneg _) s)) z lemma marginalDensity_measurable (i : ι) : Measurable (F.marginalDensity i) := measurable_foldCircleDensity (F.marginalWeight_regular i).1.measurable lemma marginalDensity_integrable (i : ι) : Integrable (F.marginalDensity i) AddCircle.haarAddCircle := integrable_foldCircleDensity (F.marginalWeight_regular i).1.measurable ((F.marginalWeight_regular i).1.integrable_of_hasCompactSupport (F.marginalWeight_regular i).2) /-- The actual old packet spectrum, without spare independence. -/ theorem marginalDensity_spectral (i : ι) : packetSpectralMeasure (fun u => (F.periodic.lift u:UnitAddCircle)) (F.m i) (F.A i) = AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal (F.marginalDensity i z)) := by have hb := scaled_speed_good_branch (F.nonzero i) F.Rpos F.smooth_speed F.positive_deriv have hs a : support (F.A₀ i a) ⊆ Ioo (F.l-F.R) (F.u+F.R) := by intro x hx have hh := F.support_lift i a hx constructor <;> linarith [F.Rpos,hh.1,hh.2] have he := oldPacket_spectralDensity (m := F.m i) (c := 0) (v := F.periodic.lift) (F.continuous_amplitude i) (F.continuous_lift i) (F.lift_matches i) (fun a => (F.support_lift i a).trans (by intro x hx; exact ⟨lt_of_le_of_lt F.lo hx.1,hx.2.trans F.uo⟩)) hs F.speed_continuous.measurable F.smooth_speed (fun _ _ => rfl) isOpen_Ioo Ioo_subset_Icc_self hb.1 hb.2 change _ = AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal (foldCircleDensity (fun s => ∫ a, stationaryWeight (Icc (F.l-F.R) (F.u+F.R)) (fun x => (F.m i:ℝ)*F.V x) (F.A₀ i a) s ∂AddCircle.haarAddCircle) z)) simpa only [Int.cast_zero,sub_zero] using he lemma marginalDensity_carrier (c : ℤ) (i : ι) : (F.carrier c).marginalDensity i =ᵐ[AddCircle.haarAddCircle] F.marginalDensity i := by have he : AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal ((F.carrier c).marginalDensity i z)) = AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal (F.marginalDensity i z)) := by rw [← (F.carrier c).marginalDensity_spectral i,← F.marginalDensity_spectral i] congr 1 funext z obtain ⟨x,rfl⟩ := Quotient.mk_surjective z exact F.carrier_circle c x have h := (withDensity_eq_iff_of_sigmaFinite ((F.carrier c).marginalDensity_measurable i).ennreal_ofReal.aemeasurable (F.marginalDensity_measurable i).ennreal_ofReal.aemeasurable).mp he filter_upwards [h] with z hz have hh := congrArg ENNReal.toReal hz simpa only [ENNReal.toReal_ofReal ((F.carrier c).marginalDensity_nonneg i z), ENNReal.toReal_ofReal (F.marginalDensity_nonneg i z)] using hh lemma marginalDensity_eq (hmod : F.SpareIndependent) (i : ι) : F.marginalDensity i = F.density i := F.density_marginal hmod i end PassagePackets end ThreeTorus end end section noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators namespace ThreeTorus namespace PassagePackets variable {ι : Type*} (F : PassagePackets ι) lemma weight_integrable (i : ι) : Integrable (F.weight i) := (F.weight_regular i).1.continuous.integrable_of_hasCompactSupport (F.weight_regular i).2 lemma density_mass (i : ι) : (∫ z, F.density i z ∂AddCircle.haarAddCircle) = ∫ s, F.weight i s := integral_foldCircleDensity (F.weight_regular i).1.continuous.measurable (F.weight_integrable i) lemma weight_mass_carrier (hmod : F.SpareIndependent) (c : ℤ) (i : ι) : (∫ s, (F.carrier c).weight i s) = ∫ s, F.weight i s := by rw [← (F.carrier c).density_mass i,F.density_carrier hmod c i,F.density_mass i] lemma weight_mass_bound [Finite ι] : ∃ B : ℝ, 0 ≤ B ∧ ∀ i, (∫ s, F.weight i s) ≤ B := by obtain ⟨B,hB⟩ := (Set.finite_range (fun i => ∫ s, F.weight i s)).isCompact.isBounded.exists_norm_le exact ⟨max B 0,le_max_right _ _,fun i => (le_abs_self _).trans ((hB _ (mem_range_self i)).trans (le_max_left _ _))⟩ end PassagePackets end ThreeTorus end end section /-! Signal transmission on actual iterable packet states. The common prime modulus and strength bound are fixed before every later finite index type, scale, state, cutoff, error budget and passage integer. -/ noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators namespace ThreeTorus universe u theorem exists_signal_state_family {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) : ∃ α : ℝ, ∃ L : ℕ, ∃ γ₀ : ℝ, 0 < α ∧ α < b/2 ∧ 2 ≤ L ∧ D/b < L ∧ 0 < γ₀ ∧ ∀ γ : ℝ, 0 < γ → γ ≤ γ₀ → (∀ v, 2*γ*‖d v‖ ≤ 1/2) ∧ ∀ (ι : Type u) [Fintype ι], ∀ (F : PassagePackets ι), Injective F.m → F.SpareIndependent → ∀ B : ℝ, 0 < B → (∀ i, b ≤ |(F.m i:ℝ)/B|) → ∀ ε τ : ℝ, 0 < ε → 0 < τ → ∀ᶠ c : ℕ in atTop, 0 < c ∧ ∃ δ : ℝ, ∃ cut : LawCutoffs δ, ∃ χ : ι → ℝ → ℝ, ∃ Δ : ℕ → ℝ, ∃ H : ℕ → UnitAddCircle → UnitAddCircle, 0 < δ ∧ δ < 1/8 ∧ (∀ i, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (χ i) ∧ HasCompactSupport (χ i) ∧ χ i 0 = 0 ∧ (∀ s, 0 ≤ χ i s ∧ χ i s ≤ 1) ∧ ∀ s ∈ tsupport (χ i), ∃ x ∈ F.core i, |s-(F.m i:ℝ)*((c:ℝ)+F.V x)| < τ) ∧ Pairwise (fun i j => Disjoint (tsupport (χ i)) (tsupport (χ j))) ∧ (∀ n, 0 < n → |Δ n| ≤ (n:ℝ)⁻¹) ∧ Tendsto Δ atTop (𝓝 0) ∧ (∀ n, Continuous (H n)) ∧ (∀ n (x : ℝ), H n (x:UnitAddCircle) = ((n:ℝ)*((∫ t in 0..x, (c:ℝ)+F.V t)+Δ n*x):UnitAddCircle)) ∧ let ρ := signalJitter α γ d (signalPrimeProduct L) B c let ψ := neutralCircleCutoff cut let J := quantileCircleJitter ρ cut let J₀ := lawJitter ρ cut let σ := fun (n : ℕ) a => (((1+J a/n)/n:ℝ):UnitAddCircle) ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) ρ ∧ Integrable ρ ∧ (∀ y, 0 < ρ y) ∧ (∫ y, ρ y = 1) ∧ (∀ n, Smooth (torusPassage (H n) (σ n)) ∧ Smooth (torusPassage (H n) (σ n)).symm ∧ MeasurePreserving (torusPassage (H n) (σ n)) μ μ) ∧ (∀ n, 0 < n → ∀ x : Cover, torusPassage (H n) (σ n) (project x) = project (passageFamily (fun t => (c:ℝ)+F.V t) J₀ 1 ((n:ℝ)⁻¹,Δ n) x) ∧ (torusPassage (H n) (σ n)).symm (project x) = project (passageFamily (fun t => (c:ℝ)+F.V t) J₀ (-1) ((n:ℝ)⁻¹,Δ n) x)) ∧ (∀ r : ℝ, ∀ K : Set Cover, IsCompact K → ∀ e : ℕ, TendstoUniformlyOn (fun (n : ℕ) x => iteratedFDeriv ℝ e (passageFamily (fun t => (c:ℝ)+F.V t) J₀ r ((n:ℝ)⁻¹,Δ n)) x) (fun x => iteratedFDeriv ℝ e (fun y : Cover => ![y 0+r*((c:ℝ)+F.V (y 1)),y 1,y 2]) x) atTop K) ∧ ∃ S : ℕ → ι → Finset ℤ, (∀ n, 0 < n → Pairwise (fun i j => Disjoint (S n i) (S n j))) ∧ (∀ n, 0 < n → ∀ i j, j ∈ S n i ↔ χ i ((j:ℝ)/n) ≠ 0) ∧ ∀ᶠ n : ℕ in atTop, 0 < n ∧ (∀ i, let f := retainedPassage (F.m i) (H n) (F.A i) (S n i) (fun j _ => χ i ((j:ℝ)/n)) ψ let p := exactRetainedPacketDensity (Ioo δ (1-δ)) J₀ ψ (fun s _ => χ i s) (F.m i) H (F.A i) (S n i) n (∫ x : TripleTorus, ‖packet (F.m i) (F.A i) x-f (passageChart (H n) x)‖^2 ∂tripleVolume) < ε ∧ (∫ z, |p z-F.density i z*signalCircleModel (signalPrimeProduct L) γ B d (F.m i) ((n:ℤ) • z)| ∂AddCircle.haarAddCircle) < ε ∧ (∀ k : ℤ, (∫ x : TripleTorus, starRingEnd ℂ (f x)*f (twistIter (σ n) k x) ∂tripleVolume) = ∫ z, (p z:ℂ)*fourier k z ∂AddCircle.haarAddCircle)) ∧ ∃ G : PassagePackets (PassageOutputIndex (S n)), Injective G.m ∧ G.SpareIndependent ∧ G.V = (fun x => (1+J₀ x/n)/n) ∧ (∀ i, G.m i = i.2.val) ∧ (∀ i, G.A i = retainedAmplitude ψ (fun s _ => χ i.1 s) (F.m i.1) H (F.A i.1) n i.2.val) ∧ (∀ i, G.density i = exactRetainedPacketDensity (Ioo δ (1-δ)) J₀ ψ (fun s _ => χ i.1 s) (F.m i.1) H (F.A i.1) {i.2.val} n) ∧ (∀ i (E : ℤ → Prop) [DecidablePred E], exactRetainedPacketDensity (Ioo δ (1-δ)) J₀ ψ (fun s _ => χ i s) (F.m i) H (F.A i) ((S n i).filter E) n = fun z => ∑ j : ↥(S n i), if E j.val then G.density ⟨i,j⟩ z else 0) := by classical obtain ⟨α,L,γ₀,hα,hαb,hL,hLD,hγ₀,hγ⟩ := exists_actual_signal_family hb hbD hd hdc hdr hd0 hds refine ⟨α,L,γ₀,hα,hαb,hL,hLD,hγ₀,?_⟩ intro γ hγp hγs obtain ⟨hdb,hsignal⟩ := hγ γ hγp hγs refine ⟨hdb,?_⟩ intro ι inst F hi hmod B hB hm ε τ hε hτ obtain ⟨M,hM,hAM⟩ := F.amplitude_bound obtain ⟨W,hW,hWM⟩ := F.weight_mass_bound let K := M+W+1 have hK : 0 < K := by dsimp [K]; positivity have hMK : M ≤ K := by dsimp [K]; linarith have hWK : W ≤ K := by dsimp [K]; linarith let ε₀ := ε/(8*K) have hε₀ : 0 < ε₀ := div_pos hε (by positivity) obtain ⟨δ,hδ,hδhi⟩ := exists_between (lt_min (by norm_num : (0:ℝ)<1/8) (div_pos hε (show 0 < 32*K by positivity))) have hδ' : δ < 1/8 := hδhi.trans_le (min_le_left _ _) have hδK : 32*δ*K < ε := by have hh := (lt_div_iff₀ (show 0 < 32*K by positivity)).mp (hδhi.trans_le (min_le_right _ _)) nlinarith have hδM : 8*δ*M < ε/4 := by nlinarith [mul_le_mul_of_nonneg_left hMK hδ.le] have hε₀K : ε₀*K = ε/8 := by dsimp [ε₀]; field_simp have hcoef : 0 ≤ 4*δ+ε₀ := by positivity have hCK : K*(4*δ+ε₀) < ε/4 := by nlinarith let cut := lawCutoffs δ hδ hδ' have hsignal' := hsignal ι B hB F.m hm F.A F.A₀ F.V F.o F.l F.u F.R F.lift_matches F.two F.continuous_lift F.continuous_deriv F.smooth_lift F.support_lift F.lo F.uo F.lu F.Rpos F.periodic F.smooth_speed F.positive_deriv F.continuous_amplitude F.smooth_amplitude hmod.1 hmod.2 ε₀ hε₀ filter_upwards [F.tracked_cutoffs hi hmod hτ,hsignal'] with c hc hs obtain ⟨hcp,χ,hχ,hdisj⟩ := hc let Fc := F.carrier (c:ℤ) let X : ι → ℝ → UnitAddCircle → ℝ := fun i s _ => χ i s have hXs i : CylinderSmooth (X i).uncurry := by change ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => χ i p.1) exact (hχ i).1.comp contDiff_fst have hXc i : Continuous (X i).uncurry := (hχ i).1.continuous.comp continuous_fst have hXcompact i : HasCompactSupport (X i).uncurry := compactSupport_frequency_cylinder (hχ i).2.1 have hXzero i (a : UnitAddCircle) : X i 0 a = 0 := (hχ i).2.2.1 have hXbound i s (a : UnitAddCircle) : 0 ≤ X i s a ∧ X i s a ≤ 1 := (hχ i).2.2.2.1 s have hXW i s a : stationaryWeight (Icc (F.l-F.R) (F.u+F.R)) (fun x => (F.m i:ℝ)*((c:ℝ)+F.V x)) (F.A₀ i a) s ≠ 0 → X i s a = 1 := (hχ i).2.2.2.2.2 _ a s obtain ⟨hρs,hρi,hρp,hρm,Δ,H,hΔ,hΔt,hHc,hHe,hgeo,hgraph,hlimit,S,hSN,_hST,hTE,hgood⟩ := hs.2.2 δ cut hδ hδ' X hXs hXc hXcompact hXzero hXbound hXW have hTE' n (hn : 0 < n) i j : j ∈ S n i ↔ χ i ((j:ℝ)/n) ≠ 0 := by simpa only [X,exists_const] using hTE n hn i j have hSd n (hn : 0 < n) : Pairwise (fun i j => Disjoint (S n i) (S n j)) := by intro i j hij rw [Finset.disjoint_left] intro k hki hkj exact Set.disjoint_left.mp (hdisj hij) (subset_tsupport _ ((hTE' n hn i k).mp hki)) (subset_tsupport _ ((hTE' n hn j k).mp hkj)) refine ⟨hcp,δ,cut,χ,Δ,H,hδ,hδ',?_,hdisj,hΔ,hΔt,hHc,hHe,hρs,hρi,hρp,hρm,hgeo,hgraph,hlimit,S,hSd,hTE',?_⟩ · exact fun i => ⟨(hχ i).1,(hχ i).2.1,(hχ i).2.2.1,(hχ i).2.2.2.1,(hχ i).2.2.2.2.1⟩ have hsmall : 0 < ε/4 := by positivity filter_upwards [hgood (ε/4) hsmall] with n hn have hn0 := hn.1 refine ⟨hn0,?_,?_⟩ · intro i have hh := hn.2 i dsimp only at hh ⊢ have hden : foldCircleDensity (stationaryWeight (Icc (F.l-F.R) (F.u+F.R)) (fun x => (F.m i:ℝ)*((c:ℝ)+F.V x)) (F.A₀ i 0)) = F.density i := by simpa only [Fc,PassagePackets.carrier,Int.cast_natCast,PassagePackets.density, PassagePackets.weight] using F.density_carrier hmod (c:ℤ) i have hmass : (∫ s, stationaryWeight (Icc (F.l-F.R) (F.u+F.R)) (fun x => (F.m i:ℝ)*((c:ℝ)+F.V x)) (F.A₀ i 0) s)*(4*δ+ε₀) < ε/4 := by have he := F.weight_mass_carrier hmod (c:ℤ) i dsimp only [PassagePackets.weight,PassagePackets.carrier] at he simp only [Int.cast_natCast] at he rw [he] exact (mul_le_mul_of_nonneg_right ((hWM i).trans hWK) hcoef).trans_lt hCK have hAint : Integrable (fun u : UnitAddCircle => ‖F.A i (u,0)‖^2) AddCircle.haarAddCircle := (((F.continuous_amplitude i).comp (continuous_id.prodMk continuous_const)).norm.pow 2).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) have hMint : (∫ u, ‖F.A i (u,0)‖^2 ∂AddCircle.haarAddCircle) ≤ M := by calc _ ≤ ∫ _u : UnitAddCircle, M ∂AddCircle.haarAddCircle := integral_mono hAint (integrable_const _) (fun u => hAM i (u,0)) _ = M := by simp rw [hden] at hh refine ⟨lt_trans hh.1 ?_,lt_trans hh.2.1 (by linarith),hh.2.2.2.1⟩ have hhM := mul_le_mul_of_nonneg_left hMint (show 0 ≤ 8*δ by positivity) linarith · let ρ := signalJitter α γ d (signalPrimeProduct L) B c let Hl (n : ℕ) (x : ℝ) := (n:ℝ)*((∫ t in 0..x, (c:ℝ)+F.V t)+Δ n*x) have hHl n : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (Hl n) := contDiff_const.mul ((contDiff_real_primitive (contDiff_const.add F.smooth_speed)).add (contDiff_const.mul contDiff_id)) have hJs := lawJitter_smooth cut hρs hρi hρp hρm have hJd x (hx : x ∈ Ioo δ (1-δ)) : 0 < deriv (lawJitter ρ cut) x := by obtain ⟨he,hp⟩ := lawJitter_deriv_pos cut hρs hρi hρp hρm hδ hx rwa [he.deriv] obtain ⟨G,hGm,hGA,_hGA₀,hGV,_hGo,_hGl,_hGu,hGmod⟩ := literal_output_packets F.m F.A H Hl X (S n) (hSN n hn0) F.continuous_amplitude F.smooth_amplitude hXs hXc hHc hHl hHe cut hδ hδ' (periodize_periodic _) hJs hJd hn0 have heψ : ∀ x ∈ Ioc (0:ℝ) (0+1), neutralCircleCutoff cut (x:UnitAddCircle) = cut.ψ x := by intro x hx rw [neutralCircleCutoff_coe] exact periodize_eq_on_unit ((subset_tsupport cut.ψ).trans (cut.ψ_support.trans (by intro z hz; constructor <;> linarith [hz.1,hz.2]))) ⟨hx.1.le,by simpa using hx.2⟩ have hsψ : tsupport cut.ψ ⊆ Ioc (0:ℝ) (0+1) ∩ Ioo δ (1-δ) := by intro x hx have hh := cut.ψ_support hx exact ⟨⟨by linarith [hh.1],by linarith [hh.2]⟩,hh⟩ have hGden (i : PassageOutputIndex (S n)) : G.density i = exactRetainedPacketDensity (Ioo δ (1-δ)) (lawJitter ρ cut) (neutralCircleCutoff cut) (X i.1) (F.m i.1) H (F.A i.1) {i.2.val} n := by apply literal_packet_canonical_density G hGmod i (hGm i) (hGA i) ?_ (continuous_quantileCircleJitter cut hρs hρi hρp hρm) hJs (fun _ _ => rfl) isOpen_Ioo (lawJitter_injOn cut hρs.continuous hρi hρp hρm hδ) (fun x hx => (hJd x hx).ne') (continuous_neutralCircleCutoff cut) cut.ψ_smooth cut.ψ_compact heψ hsψ (hXc i.1) (hXs i.1) hHc hHl hHe (F.continuous_amplitude i.1) (F.smooth_amplitude i.1) hn0 intro a obtain ⟨x,rfl⟩ := Quotient.mk_surjective a change (G.V x:UnitAddCircle) = _ rw [hGV] rfl refine ⟨G,?_,hGmod,hGV,hGm,hGA,hGden,?_⟩ · have he : G.m = fun i : PassageOutputIndex (S n) => i.2.val := funext hGm rw [he] exact output_index_injective (hSd n hn0) · intro i E inst simp_rw [hGden] exact exactRetainedPacketDensity_filtered_singletons hJs isOpen_Ioo (lawJitter_injOn cut hρs.continuous hρi hρp hρm hδ) (fun x hx => (hJd x hx).ne') cut.ψ_smooth cut.ψ_compact heψ hsψ (hXs i) hHl hHe (F.smooth_amplitude i) hn0 (hSN n hn0 i) E end ThreeTorus end end section /-! Support tracking for the actual next packets. Canonical density support is linked to a genuine old-amplitude point; the same point controls both normalized angular indices and temporal frequencies. -/ noncomputable section open Set Function Filter MeasureTheory open scoped Topology BigOperators namespace ThreeTorus namespace PassagePackets variable {ι : Type*} (F : PassagePackets ι) lemma speed_bound : ∃ P : ℝ, 0 ≤ P ∧ ∀ x, |F.V x| ≤ P := by obtain ⟨P,hP⟩ := (isCompact_range F.speed_continuous).isBounded.exists_norm_le exact ⟨max P 0,le_max_right _ _,fun x => (hP _ (mem_range_self (x:UnitAddCircle))).trans (le_max_left _ _)⟩ lemma angular_bound [Finite ι] : ∃ M : ℝ, 0 ≤ M ∧ ∀ i, |(F.m i:ℝ)| ≤ M := by obtain ⟨M,hM⟩ := (Set.finite_range (fun i => (F.m i:ℝ))).isCompact.isBounded.exists_norm_le exact ⟨max M 0,le_max_right _ _,fun i => (hM _ (mem_range_self i)).trans (le_max_left _ _)⟩ end PassagePackets lemma periodic_real_bound {J : ℝ → ℝ} (hs : Continuous J) (hp : Periodic J 1) : ∃ B : ℝ, 0 ≤ B ∧ ∀ x, |J x| ≤ B := by have hc : Continuous hp.lift := by apply Continuous.quotient_lift exact hs obtain ⟨B,hB⟩ := (isCompact_range hc).isBounded.exists_norm_le exact ⟨max B 0,le_max_right _ _,fun x => (hB _ (mem_range_self (x:UnitAddCircle))).trans (le_max_left _ _)⟩ /-- The tracked frequency cutoff already controls every retained index, including zero-amplitude packets. The temporal inequality holds for every new speed point. -/ theorem literal_output_tracking {ι : Type*} {F : PassagePackets ι} {S : ι → Finset ℤ} {G : PassagePackets (PassageOutputIndex S)} {χ : ι → ℝ → ℝ} {c n : ℕ} {B D P M Q τ : ℝ} {J : ℝ → ℝ} (hc : 0 < c) (hn : 0 < n) (hB : 0 < B) (hD : ∀ i, |(F.m i:ℝ)/B| ≤ D) (hP : ∀ x, |F.V x| ≤ P) (hM : ∀ i, |(F.m i:ℝ)| ≤ M) (hQ : ∀ x, |J x| ≤ Q) (hS : ∀ i j, j ∈ S i → χ i ((j:ℝ)/n) ≠ 0) (hχ : ∀ i s, s ∈ tsupport (χ i) → ∃ x ∈ F.core i, |s-(F.m i:ℝ)*((c:ℝ)+F.V x)| < τ) (hm : ∀ i, G.m i = i.2.val) (hV : G.V = fun x => (1+J x/n)/n) : ∀ i, ∃ x ∈ F.core i.1, |(G.m i:ℝ)/((n:ℝ)*c*B)-(F.m i.1:ℝ)/B| ≤ D*P/c+τ/((c:ℝ)*B) ∧ ∀ y, dist ((G.m i:ℝ)*G.V y:UnitAddCircle) ((F.m i.1:ℝ)*F.V x:UnitAddCircle) ≤ τ+(M*((c:ℝ)+P)+τ)*Q/n := by intro i obtain ⟨x,hx,htrack⟩ := hχ i.1 _ (subset_tsupport _ (hS i.1 i.2.val i.2.property)) have hcR : 0 < (c:ℝ) := by exact_mod_cast hc have hnR : 0 < (n:ℝ) := by exact_mod_cast hn have hP0 : 0 ≤ P := (abs_nonneg _).trans (hP 0) have hs := passage_retained_size hcR.le (hP x) htrack.le have hsM : |(i.2.val:ℝ)/n| ≤ M*((c:ℝ)+P)+τ := hs.trans (by gcongr exact hM i.1) refine ⟨x,hx,?_,?_⟩ · rw [hm i] have heq : (i.2.val:ℝ)/((n:ℝ)*c*B) = ((i.2.val:ℝ)/n)/((c:ℝ)*B) := by ring rw [heq] exact passage_index_drift hcR hB (hD i.1) (hP x) htrack.le · intro y rw [hm i,hV] have heq : (i.2.val:ℝ)*((1+J y/n)/n) = (i.2.val:ℝ)/n+((i.2.val:ℝ)/n)*J y/n := by ring rw [heq] have ht := passage_temporal_drift (J := J y) (c := (c:ℤ)) hn (by simpa only [Int.cast_natCast] using htrack.le) apply ht.trans exact add_le_add le_rfl (div_le_div_of_nonneg_right (mul_le_mul hsM (hQ y) (abs_nonneg _) ((abs_nonneg _).trans hsM)) hnR.le) /-- Every nonzero output density is close to an OLD support point. This uses literal angular indices and speed, not weak convergence of the densities. -/ theorem canonical_output_tracking {ι : Type*} {F : PassagePackets ι} {S : ι → Finset ℤ} {G : PassagePackets (PassageOutputIndex S)} {χ : ι → ℝ → ℝ} {c n : ℕ} {B D P M Q τ : ℝ} {J : ℝ → ℝ} (hc : 0 < c) (hn : 0 < n) (hB : 0 < B) (hD : ∀ i, |(F.m i:ℝ)/B| ≤ D) (hP : ∀ x, |F.V x| ≤ P) (hM : ∀ i, |(F.m i:ℝ)| ≤ M) (hQ : ∀ x, |J x| ≤ Q) (hS : ∀ i j, j ∈ S i → χ i ((j:ℝ)/n) ≠ 0) (hχ : ∀ i s, s ∈ tsupport (χ i) → ∃ x ∈ F.core i, |s-(F.m i:ℝ)*((c:ℝ)+F.V x)| < τ) (hm : ∀ i, G.m i = i.2.val) (hV : G.V = fun x => (1+J x/n)/n) : ∀ i z, G.density i z ≠ 0 → ∃ x ∈ F.core i.1, |(G.m i:ℝ)/((n:ℝ)*c*B)-(F.m i.1:ℝ)/B| ≤ D*P/c+τ/((c:ℝ)*B) ∧ dist z ((F.m i.1:ℝ)*F.V x:UnitAddCircle) ≤ τ+(M*((c:ℝ)+P)+τ)*Q/n := by intro i z hz obtain ⟨x,hx,hidx,ht⟩ := literal_output_tracking hc hn hB hD hP hM hQ hS hχ hm hV i obtain ⟨y,_hy,he⟩ := G.density_support i hz exact ⟨x,hx,hidx,he ▸ ht y⟩ /-- Large carrier first, small cutoff enlargement second, and final integer last. The bounded jitter only affects this LAST choice. -/ lemma support_drift_budgets {D P M B e : ℝ} (hB : 0 < B) (he : 0 < e) : ∀ᶠ c : ℕ in atTop, 0 < c ∧ ∀ τ : ℝ, 0 < τ → τ < e/2 → D*P/c+τ/((c:ℝ)*B) < e ∧ ∀ Q : ℝ, ∀ᶠ n : ℕ in atTop, 0 < n ∧ τ+(M*((c:ℝ)+P)+τ)*Q/n < e := by have hlim : Tendsto (fun c : ℕ => D*P/(c:ℝ)+(e/2)/((c:ℝ)*B)) atTop (𝓝 0) := by have hc := tendsto_natCast_atTop_atTop (R := ℝ) have h₁ : Tendsto (fun c : ℕ => D*P/(c:ℝ)) atTop (𝓝 0) := tendsto_const_nhds.div_atTop hc have h₂ : Tendsto (fun c : ℕ => (e/2)/((c:ℝ)*B)) atTop (𝓝 0) := tendsto_const_nhds.div_atTop (hc.atTop_mul_const hB) simpa using h₁.add h₂ filter_upwards [eventually_gt_atTop (0:ℕ),hlim.eventually (gt_mem_nhds he)] with c hc hh refine ⟨hc,fun τ _hτ hτe => ⟨?_,?_⟩⟩ · have hcR : 0 < (c:ℝ) := by exact_mod_cast hc exact (add_le_add_right (div_le_div_of_nonneg_right hτe.le (mul_pos hcR hB).le) _).trans_lt hh · intro Q have hlimn : Tendsto (fun n : ℕ => τ+(M*((c:ℝ)+P)+τ)*Q/(n:ℝ)) atTop (𝓝 τ) := by simpa using tendsto_const_nhds.add (tendsto_const_nhds.div_atTop (tendsto_natCast_atTop_atTop (R := ℝ)) : Tendsto (fun n : ℕ => (M*((c:ℝ)+P)+τ)*Q/(n:ℝ)) atTop (𝓝 0)) exact (eventually_gt_atTop (0:ℕ)).and (hlimn.eventually (gt_mem_nhds (by linarith))) end ThreeTorus end end end OAI