import OAI.Dynamics.ThreeTorus.Cutoffs import OAI.Dynamics.ThreeTorus.NeutralSupport namespace OAI /-! Literal retained passage vectors, spectral identities, regularity, old marginals, and quantitative cutoff losses. -/ section noncomputable section open MeasureTheory Set Filter open scoped Topology namespace ThreeTorus lemma integral_passageChart {H : UnitAddCircle → UnitAddCircle} (hH : Measurable H) (f : TripleTorus → ℝ) : (∫ x, f (passageChart H x) ∂tripleVolume) = ∫ x, f x ∂tripleVolume := by let e : TripleTorus ≃ᵐ TripleTorus := { toEquiv := passageChart H measurable_toFun := (measurePreserving_passageChart hH).measurable measurable_invFun := passageChart_inverse_measurable hH } exact (measurePreserving_passageChart hH).integral_comp e.measurableEmbedding f /-- A complete quantitative form of the physical transport estimate (2.9). The sole right-hand quantity is the actual summed Fourier-mass error. -/ lemma physical_retainedPassage_error_le {m : ℤ} {H : UnitAddCircle → UnitAddCircle} {A : UnitAddCircle × UnitAddCircle → ℂ} (hH : Continuous H) (hA : Continuous A) (S : Finset ℤ) (χ W : ℤ → UnitAddCircle → ℝ) (hχc : ∀ j, Continuous (χ j)) (ψ : UnitAddCircle → ℝ) (hψc : Continuous ψ) (hψ : ∀ a, 0 ≤ ψ a ∧ ψ a ≤ 1) (hχ : ∀ j a, 0 ≤ χ j a ∧ χ j a ≤ 1) (hW : ∀ j a, W j a ≠ 0 → j ∈ S ∧ χ j a = 1) (hE : ∀ j, Integrable (fun a => |‖exactPassageCoeff m H A j a‖^2-W j a|) AddCircle.haarAddCircle) (hp : ∀ᵐ a ∂AddCircle.haarAddCircle, Summable (fun j => |‖exactPassageCoeff m H A j a‖^2-W j a|)) (hs : Summable (fun j => ∫ a, |‖exactPassageCoeff m H A j a‖^2-W j a| ∂AddCircle.haarAddCircle)) : (∫ x : TripleTorus, ‖(ψ x.2.2:ℂ)*packet m A x-retainedPassage m H A S χ ψ (passageChart H x)‖^2 ∂tripleVolume) ≤ ∑' j : ℤ, ∫ a, |‖exactPassageCoeff m H A j a‖^2-W j a| ∂AddCircle.haarAddCircle := by have he := integral_passageChart hH.measurable (fun x : TripleTorus => ‖(ψ x.2.1:ℂ)*packet m A ((passageChart H).symm x)-retainedPassage m H A S χ ψ x‖^2) simp only [Equiv.symm_apply_apply] at he change (∫ x : TripleTorus, ‖(ψ x.2.2:ℂ)*packet m A x-retainedPassage m H A S χ ψ (passageChart H x)‖^2 ∂tripleVolume) = _ at he rw [he,integrated_retainedPassage_error hH hA S χ hχc ψ hψc] let E : UnitAddCircle → UnitAddCircle → ℝ := fun a θ => ‖passageFourierInput m H A a θ- ∑ j ∈ S, (χ j a:ℂ)*exactPassageCoeff m H A j a*fourier j θ‖^2 have hEc : Continuous E.uncurry := by apply Continuous.pow apply Continuous.norm apply (continuous_passageFourierInput hH hA).sub apply continuous_finsetSum intro j _hj exact (((Complex.continuous_ofReal.comp (hχc j)).comp continuous_fst).mul ((continuous_exactPassageCoeff hH hA j).comp continuous_fst)).mul ((fourier j).continuous.comp continuous_snd) have hIc : Continuous (fun a => ∫ θ, E a θ ∂AddCircle.haarAddCircle) := by simpa only [setIntegral_univ] using continuous_parametric_integral_of_continuous (μ := AddCircle.haarAddCircle) hEc isCompact_univ have hIi : Integrable (fun a => ∫ θ, E a θ ∂AddCircle.haarAddCircle) AddCircle.haarAddCircle := hIc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) calc _ ≤ ∫ a, ∫ θ, E a θ ∂AddCircle.haarAddCircle ∂AddCircle.haarAddCircle := by apply integral_mono_of_nonneg · exact Filter.Eventually.of_forall (fun a => mul_nonneg (sq_nonneg _) (integral_nonneg (fun _ => sq_nonneg _))) · exact hIi · filter_upwards [] with a exact mul_le_of_le_one_left (integral_nonneg (fun _ => sq_nonneg _)) (by nlinarith [(hψ a).1,(hψ a).2]) _ ≤ _ := integrated_Fourier_cutoff_le_mass_error (fun a => (continuous_passageFourierInput hH hA).comp (continuous_const.prodMk continuous_id)) S χ W hχ hW hE hp hs lemma pointwise_array_mass_error_summable {c : ℤ → UnitAddCircle → ℂ} {W : ℝ → UnitAddCircle → ℝ} (hc : HasCompactSupport W.uncurry) (hcs : ∀ a, Summable (fun j => ‖c j a‖^2)) {n : ℕ} (hn : 0 < n) (a : UnitAddCircle) : Summable (fun j : ℤ => |‖c j a‖^2-(n:ℝ)⁻¹*W ((j:ℝ)/n) a|) := by obtain ⟨S,hS⟩ := compact_lattice_support hc hn 0 have hw : Summable (fun j : ℤ => |(n:ℝ)⁻¹*W ((j:ℝ)/n) a|) := by apply summable_of_ne_finset_zero (s := S) intro j hj have hz : W ((j:ℝ)/n) a = 0 := by simpa using hS j hj a simp [hz] refine Summable.of_nonneg_of_le (f := fun j => ‖c j a‖^2+|(n:ℝ)⁻¹*W ((j:ℝ)/n) a|) (fun _ => abs_nonneg _) (fun j => ?_) ((hcs a).add hw) have hh := abs_sub_le (‖c j a‖^2) 0 ((n:ℝ)⁻¹*W ((j:ℝ)/n) a) have he : |‖c j a‖^2| = ‖c j a‖^2 := abs_of_nonneg (sq_nonneg _) simpa only [sub_zero,zero_sub,abs_neg,he] using hh /-- The actual physical packet cutoff error tends to zero once the unrounded stationary mass theorem is applied. All summability hypotheses are derived. -/ lemma physical_retainedPassage_tendsto {m : ℤ} {H : ℕ → UnitAddCircle → UnitAddCircle} {A : UnitAddCircle × UnitAddCircle → ℂ} {W : ℝ → UnitAddCircle → ℝ} (hH : ∀ n, Continuous (H n)) (hA : Continuous A) (hWc : Continuous W.uncurry) (hWk : HasCompactSupport W.uncurry) (S : ℕ → Finset ℤ) (χ : ℕ → ℤ → UnitAddCircle → ℝ) (hχc : ∀ n j, Continuous (χ n j)) (hχ : ∀ n j a, 0 ≤ χ n j a ∧ χ n j a ≤ 1) (ψ : UnitAddCircle → ℝ) (hψc : Continuous ψ) (hψ : ∀ a, 0 ≤ ψ a ∧ ψ a ≤ 1) (hW : ∀ (n : ℕ), 0 < n → ∀ (j : ℤ) (a : UnitAddCircle), W ((j:ℝ)/n) a ≠ 0 → j ∈ S n ∧ χ n j a = 1) (ht : Tendsto (fun n : ℕ => ∑' j : ℤ, ∫ a : UnitAddCircle, |‖exactPassageCoeff m (H n) A j a‖^2-(n:ℝ)⁻¹*W ((j:ℝ)/n) a| ∂AddCircle.haarAddCircle) atTop (𝓝 0)) : Tendsto (fun n => ∫ x : TripleTorus, ‖(ψ x.2.2:ℂ)*packet m A x- retainedPassage m (H n) A (S n) (χ n) ψ (passageChart (H n) x)‖^2 ∂tripleVolume) atTop (𝓝 0) := by apply squeeze_zero' (Filter.Eventually.of_forall (fun _ => integral_nonneg (fun _ => sq_nonneg _))) _ ht filter_upwards [eventually_gt_atTop 0] with n hn have hEc (j : ℤ) : Continuous (fun a => |‖exactPassageCoeff m (H n) A j a‖^2-(n:ℝ)⁻¹*W ((j:ℝ)/n) a|) := by exact (((continuous_exactPassageCoeff (hH n) hA j).norm.pow 2).sub (continuous_const.mul (hWc.comp (continuous_const.prodMk continuous_id)))).abs apply physical_retainedPassage_error_le (hH n) hA (S n) (χ n) (fun j a => (n:ℝ)⁻¹*W ((j:ℝ)/n) a) (hχc n) ψ hψc hψ (hχ n) · intro j a ha exact hW n hn j a (fun hz => ha (by simp [hz])) · intro j exact (hEc j).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) · apply Filter.Eventually.of_forall intro a exact pointwise_array_mass_error_summable hWk (fun a => continuous_fourier_summable ((continuous_passageFourierInput (hH n) hA).comp (continuous_const.prodMk continuous_id))) hn a · simpa only [sub_zero] using integrated_array_mass_error_summable hWc hWk (continuous_exactPassageCoeff (hH n) hA) (exactPassageCoeff_summable (hH n) hA) hn 0 /-- Physical transport for the actual signed packet. The summed stationary mass error is proved, not an input to this statement. -/ lemma exact_physical_retainedPassage_tendsto {A : UnitAddCircle × UnitAddCircle → ℂ} {A₀ : UnitAddCircle → ℝ → ℂ} {P 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)*(P 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)) (hsupp : ∀ a, Function.support (A₀ a) ⊆ Ioc l u) (hlo : o ≤ l) (huo : u ≤ o+1) (hlu : l ≤ u) (hR : 0 < R) (hP : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) P) (hV : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) V) (hPV : deriv P = V) (hd : ∀ x ∈ Icc (l-2*R) (u+2*R), 0 < deriv V x) (hHc : ∀ n, Continuous (H n)) (hAcont : Continuous A) (hδt : Tendsto δ atTop (𝓝 0)) {χ : ℝ → UnitAddCircle → ℝ} (hχc : Continuous χ.uncurry) (hχ : ∀ s a, 0 ≤ χ s a ∧ χ s a ≤ 1) (ψ : UnitAddCircle → ℝ) (hψc : Continuous ψ) (hψ : ∀ a, 0 ≤ ψ a ∧ ψ 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) (S : ℕ → Finset ℤ) (hS : ∀ n, 0 < n → ∀ j ∉ S n, ∀ a, χ ((j:ℝ)/n) a = 0) : Tendsto (fun n => ∫ x : TripleTorus, ‖(ψ x.2.2:ℂ)*packet m A x- retainedPassage m (H n) A (S n) (fun j => χ ((j:ℝ)/n)) ψ (passageChart (H n) x)‖^2 ∂tripleVolume) atTop (𝓝 0) := by have hbranch := scaled_speed_good_branch hm hR hV hd have hW := stationaryWeight_signed_compact_continuous (contDiff_const.mul hV) hAc (fun a => (hsupp a).trans Ioc_subset_Icc_self) hR hbranch.1 hbranch.2 apply physical_retainedPassage_tendsto (W := fun s a => stationaryWeight (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) (A₀ a) s) hHc hAcont hW.1 hW.2 S (fun n j => χ ((j:ℝ)/n)) (fun _ _ => hχc.comp (continuous_const.prodMk continuous_id)) (fun _ _ _ => hχ _ _) ψ hψc hψ · intro n hn j a hw have he := hχW ((j:ℝ)/n) a hw refine ⟨?_,he⟩ by_contra hj have hz := hS n hn j hj a linarith · exact exactPassageCoeff_summed hm hH hA hA₀ hAc hAd hsupp hlo huo hlu hR hP hV hPV hd hHc hAcont hδt end ThreeTorus noncomputable section open MeasureTheory Set Function Filter open scoped Topology namespace ThreeTorus lemma stationaryWeight_cylinder_support {C : Set ℝ} {V : ℝ → ℝ} {m : ℤ} {A : UnitAddCircle → ℝ → ℂ} : support (fun p : ℝ × UnitAddCircle => stationaryWeight C (fun x => (m:ℝ)*V x) (A p.2) p.1) ⊆ stationaryImage m V (tsupport (fun p : ℝ × UnitAddCircle => A p.2 p.1)) := by intro p hp have hs : p.1 ∈ (fun x => (m:ℝ)*V x) '' C := by by_contra hn exact hp (by simp [stationaryWeight,branchDensity,hn]) have hu := invFunOn_pos hs have hA : A p.2 (invFunOn (fun x => (m:ℝ)*V x) C p.1) ≠ 0 := by intro hz exact hp (by simp [stationaryWeight,branchDensity,hs,hz]) refine ⟨(invFunOn (fun x => (m:ℝ)*V x) C p.1,p.2),?_,Prod.ext hu.2 rfl⟩ exact subset_tsupport _ hA lemma lifted_amplitude_compactSupport {A : UnitAddCircle → ℝ → ℂ} {l u : ℝ} (hA : ∀ a, support (A a) ⊆ Icc l u) : HasCompactSupport (fun p : ℝ × UnitAddCircle => A p.2 p.1) ∧ tsupport (fun p : ℝ × UnitAddCircle => A p.2 p.1) ⊆ Icc l u ×ˢ univ := by have hs : tsupport (fun p : ℝ × UnitAddCircle => A p.2 p.1) ⊆ Icc l u ×ˢ univ := by apply closure_minimal _ (isClosed_Icc.prod isClosed_univ) intro p hp exact ⟨hA p.2 hp,mem_univ _⟩ exact ⟨(isCompact_Icc.prod isCompact_univ).of_isClosed_subset isClosed_closure hs,hs⟩ /-- The output amplitudes from different colors are exactly disjoint at each shared angular index; asymptotic orthogonality is not used here. -/ lemma retained_amplitude_disjoint {χ ξ : ℝ × UnitAddCircle → ℝ} (hd : Disjoint (tsupport χ) (tsupport ξ)) (s : ℝ) (F G : UnitAddCircle × UnitAddCircle → ℂ) : Disjoint (tsupport (fun p : UnitAddCircle × UnitAddCircle => (χ (s,p.1):ℂ)*F p)) (tsupport (fun p : UnitAddCircle × UnitAddCircle => (ξ (s,p.1):ℂ)*G p)) := by have hs (η : ℝ × UnitAddCircle → ℝ) (B : UnitAddCircle × UnitAddCircle → ℂ) : tsupport (fun p : UnitAddCircle × UnitAddCircle => (η (s,p.1):ℂ)*B p) ⊆ (fun p : UnitAddCircle × UnitAddCircle => (s,p.1)) ⁻¹' tsupport η := by apply closure_minimal _ ((isClosed_tsupport η).preimage (continuous_const.prodMk continuous_fst)) intro p hp exact subset_tsupport η (fun hz => hp (by simp [hz])) exact (hd.preimage _).mono (hs χ F) (hs ξ G) lemma norm_spare_character (m : ℤ) (C : UnitAddCircle → ℂ) (a b : UnitAddCircle) : ‖C a * fourier m b‖ = ‖C a‖ := by rw [norm_mul,show ‖fourier m b‖ = 1 from Circle.norm_coe _,mul_one] /-- The source's finite stationary-image support selection, including the zero-frequency exclusion and exact disjointness of the cutoffs. -/ lemma colored_stationary_cutoffs {ι : Type*} [Finite ι] (m : ι → ℤ) (V : ℝ → ℝ) (I : Set ℝ) (K U : ι → Set (ℝ × UnitAddCircle)) (hm : ∀ i, m i ≠ 0) (hV : Continuous V) (hi : InjOn V I) (hK : ∀ i, IsCompact (K i)) (hKI : ∀ i, K i ⊆ I ×ˢ univ) (hzero : ∀ x ∈ I, V x ≠ 0) (hsep : ∀ i j, ∀ x ∈ I, ∀ y ∈ I, (m i:ℝ)*V x = (m j:ℝ)*V y → m i = m j) (hdisj : ∀ i j, i ≠ j → m i = m j → Disjoint (K i) (K j)) (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, stationaryImage (m i) V (K i) ⊆ U i) : ∃ χ : ι → ℝ × UnitAddCircle → ℝ, (∀ i, CylinderSmooth (χ i) ∧ HasCompactSupport (χ i) ∧ tsupport (χ i) ⊆ U i ∧ (∀ p, 0 ≤ χ i p ∧ χ i p ≤ 1) ∧ (∀ a, χ i (0,a) = 0) ∧ ∀ p ∈ stationaryImage (m i) V (K i), χ i =ᶠ[𝓝 p] 1) ∧ Pairwise (fun i j => Disjoint (tsupport (χ i)) (tsupport (χ j))) := by let E i := stationaryImage (m i) V (K i) let O i := U i ∩ {p : ℝ × UnitAddCircle | p.1 ≠ 0} have hO i : IsOpen (O i) := (hU i).inter (isOpen_ne_fun continuous_fst continuous_const) have hEO i : E i ⊆ O i := by intro p hp refine ⟨hKU i hp,?_⟩ obtain ⟨q,hq,rfl⟩ := hp exact mul_ne_zero (by exact_mod_cast hm i) (hzero _ (hKI i hq).1) have hEd : Pairwise (Disjoint on E) := by intro i j hij by_cases he : m i = m j · change Disjoint (stationaryImage (m i) V (K i)) (stationaryImage (m j) V (K j)) rw [he] exact stationaryImage_disjoint_same (hm j) hi (hKI i) (hKI j) (hdisj i j hij he) · apply stationaryImage_disjoint_distinct intro p hp q hq hh exact he (hsep i j p.1 (hKI i hp).1 q.1 (hKI j hq).1 hh) obtain ⟨χ,hχ,hχd⟩ := finite_disjoint_cylinder_cutoffs E O (fun i => stationaryImage_compact hV (hK i)) hO hEO hEd refine ⟨χ,fun i => ⟨(hχ i).1,(hχ i).2.1,(hχ i).2.2.1.trans inter_subset_left, (hχ i).2.2.2.1,?_,(hχ i).2.2.2.2⟩,hχd⟩ intro a by_contra ha exact ((hχ i).2.2.1 (subset_tsupport (χ i) ha)).2 rfl lemma simultaneous_retained_indices {ι : Type*} (χ : ι → ℝ × UnitAddCircle → ℝ) (hc : ∀ i, HasCompactSupport (χ i)) (hz : ∀ i a, χ i (0,a) = 0) : ∃ S : ℕ → ι → Finset ℤ, (∀ n i, (0:ℤ) ∉ S n i) ∧ ∀ n, 0 < n → ∀ i j, j ∉ S n i → ∀ a, χ i ((j:ℝ)/n,a) = 0 := by classical have hex (n : ℕ) (i : ι) : ∃ S : Finset ℤ, (0:ℤ) ∉ S ∧ (0 < n → ∀ j ∉ S, ∀ a, χ i ((j:ℝ)/n,a) = 0) := by by_cases hn : 0 < n · obtain ⟨S,hS,hS'⟩ := cylinder_cutoff_finite_indices (hc i) (hz i) n hn exact ⟨S,hS,fun _ => hS'⟩ · exact ⟨∅,Finset.notMem_empty _,fun h => (hn h).elim⟩ choose S hS hS' using hex exact ⟨S,hS,fun n hn i => hS' n i hn⟩ /-- Simultaneous physical transport with constructed, disjoint cylinder cutoffs. The limit is a consequence of the signed stationary-phase theorem for the actual Fourier coefficients, not an assumption on an auxiliary array. -/ theorem colored_physical_passage {ι : Type*} [Finite ι] {A : ι → UnitAddCircle × UnitAddCircle → ℂ} {A₀ : ι → UnitAddCircle → ℝ → ℂ} {P V : ℝ → ℝ} {m : ι → ℤ} {δ : ℕ → ℝ} {H : ℕ → UnitAddCircle → UnitAddCircle} {o l u R : ℝ} (hm : ∀ i, m i ≠ 0) (hH : ∀ n (x : ℝ), H n (x : UnitAddCircle) = ((n:ℝ)*(P x+δ n*x) : UnitAddCircle)) (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)) (hsupp : ∀ i a, Function.support (A₀ i a) ⊆ Ioc l u) (hlo : o ≤ l) (huo : u ≤ o+1) (hlu : l ≤ u) (hR : 0 < R) (hP : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) P) (hV : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) V) (hPV : deriv P = V) (hd : ∀ x ∈ Icc (l-2*R) (u+2*R), 0 < deriv V x) (hHc : ∀ n, Continuous (H n)) (hAcont : ∀ i, Continuous (A i)) (hδt : Tendsto δ atTop (𝓝 0)) (hzero : ∀ x ∈ Icc l u, V x ≠ 0) (hsep : ∀ i j, ∀ x ∈ Icc l u, ∀ y ∈ Icc l u, (m i:ℝ)*V x = (m j:ℝ)*V y → m i = m j) (hdisj : ∀ i j, i ≠ j → m i = m j → Disjoint (tsupport (fun p : ℝ × UnitAddCircle => A₀ i p.2 p.1)) (tsupport (fun p : ℝ × UnitAddCircle => A₀ j p.2 p.1))) (U : ι → Set (ℝ × UnitAddCircle)) (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, stationaryImage (m i) V (tsupport (fun p : ℝ × UnitAddCircle => A₀ i p.2 p.1)) ⊆ U i) (ψ : UnitAddCircle → ℝ) (hψc : Continuous ψ) (hψ : ∀ a, 0 ≤ ψ a ∧ ψ a ≤ 1) : ∃ (χ : ι → ℝ × UnitAddCircle → ℝ) (S : ℕ → ι → Finset ℤ), (∀ i, CylinderSmooth (χ i) ∧ HasCompactSupport (χ i) ∧ tsupport (χ i) ⊆ U i ∧ (∀ p, 0 ≤ χ i p ∧ χ i p ≤ 1)) ∧ Pairwise (fun i j => Disjoint (tsupport (χ i)) (tsupport (χ j))) ∧ (∀ n i, (0:ℤ) ∉ S n i) ∧ (∀ n, 0 < n → ∀ i j, j ∉ S n i → ∀ a, χ i ((j:ℝ)/n,a) = 0) ∧ ∀ i, Tendsto (fun n => ∫ x : TripleTorus, ‖(ψ x.2.2:ℂ)*packet (m i) (A i) x- retainedPassage (m i) (H n) (A i) (S n i) (fun j a => χ i ((j:ℝ)/n,a)) ψ (passageChart (H n) x)‖^2 ∂tripleVolume) atTop (𝓝 0) := by let K i := tsupport (fun p : ℝ × UnitAddCircle => A₀ i p.2 p.1) have hK i := lifted_amplitude_compactSupport (fun a => (hsupp i a).trans Ioc_subset_Icc_self) have hi : InjOn V (Icc l u) := by apply (strictMonoOn_of_deriv_pos (convex_Icc _ _) hV.continuous.continuousOn _).injOn intro x hx have hx' := interior_subset hx exact hd x ⟨by linarith [hx'.1],by linarith [hx'.2]⟩ obtain ⟨χ,hχ,hχd⟩ := colored_stationary_cutoffs m V (Icc l u) K U hm hV.continuous hi (fun i => (hK i).1) (fun i => (hK i).2) hzero hsep hdisj hU hKU obtain ⟨S,hS,hS'⟩ := simultaneous_retained_indices χ (fun i => (hχ i).2.1) (fun i => (hχ i).2.2.2.2.1) refine ⟨χ,S,fun i => ⟨(hχ i).1,(hχ i).2.1,(hχ i).2.2.1,(hχ i).2.2.2.1⟩, hχd,hS,hS',?_⟩ intro i exact exact_physical_retainedPassage_tendsto (χ := fun s a => χ i (s,a)) (hm i) hH (hA i) (hA₀ i) (hAc i) (hAd i) (hsupp i) hlo huo hlu hR hP hV hPV hd hHc (hAcont i) hδt (hχ i).1.continuous (fun s a => (hχ i).2.2.2.1 (s,a)) ψ hψc hψ (fun s a hw => ((hχ i).2.2.2.2.2 (s,a) (stationaryWeight_cylinder_support hw)).self_of_nhds) (fun n => S n i) (fun n hn j hj a => hS' n hn i j hj a) end ThreeTorus noncomputable section open Set Function Filter open scoped Topology namespace ThreeTorus /-- The second support separation in the manuscript uses cutoffs in frequency alone, as opposed to the first passage's frequency/slow-coordinate cutoffs. -/ theorem finite_disjoint_frequency_cutoffs {ι : Type*} [Finite ι] (K U : ι → Set ℝ) (hK : ∀ i, IsCompact (K i)) (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hdisj : Pairwise (Disjoint on K)) : ∃ χ : ι → ℝ → ℝ, (∀ i, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (χ i) ∧ HasCompactSupport (χ i) ∧ tsupport (χ i) ⊆ U i ∧ (∀ s, 0 ≤ χ i s ∧ χ i s ≤ 1) ∧ ∀ s ∈ K i, χ i =ᶠ[𝓝 s] 1) ∧ Pairwise (fun i j => Disjoint (tsupport (χ i)) (tsupport (χ j))) := by have hd : Pairwise (fun i j => Disjoint (𝓝ˢ (K i)) (𝓝ˢ (K j))) := fun i j hij => disjoint_nhdsSet_nhdsSet (hK i).isClosed (hK j).isClosed (hdisj hij) obtain ⟨O,hO,hOd⟩ := hd.exists_mem_filter_of_disjoint have hv i : K i ⊆ interior (O i) ∩ U i := subset_inter (subset_interior_iff_mem_nhdsSet.mpr (hO i)) (hKU i) choose χ hχ using fun i => compact_smooth_cutoff (hK i) (isOpen_interior.inter (hU i)) (hv i) refine ⟨χ,fun i => ⟨(hχ i).1,(hχ i).2.1,(hχ i).2.2.1.trans inter_subset_right, (hχ i).2.2.2⟩,?_⟩ intro i j hij exact (hOd hij).mono ((hχ i).2.2.1.trans (inter_subset_left.trans interior_subset)) ((hχ j).2.2.1.trans (inter_subset_left.trans interior_subset)) lemma compactSupport_frequency_cylinder {χ : ℝ → ℝ} (hc : HasCompactSupport χ) : HasCompactSupport (fun p : ℝ × UnitAddCircle => χ p.1) := by apply (hc.prod isCompact_univ).of_isClosed_subset isClosed_closure apply closure_minimal _ ((isClosed_tsupport χ).prod isClosed_univ) intro p hp exact ⟨subset_tsupport χ hp,mem_univ _⟩ /-- Sampling disjoint frequency supports gives exactly disjoint finite index sets. There is no asymptotic assertion in this exact separation conclusion. -/ theorem disjoint_frequency_indices {ι : Type*} (χ : ι → ℝ → ℝ) (hc : ∀ i, HasCompactSupport (χ i)) (hz : ∀ i, χ i 0 = 0) (hd : Pairwise (fun i j => Disjoint (tsupport (χ i)) (tsupport (χ j)))) : ∃ S : ℕ → ι → Finset ℤ, (∀ n i, (0:ℤ) ∉ S n i) ∧ (∀ 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 := by classical have hex (n : ℕ) (i : ι) : ∃ S : Finset ℤ, (∀ j ∈ S, χ i ((j:ℝ)/n) ≠ 0) ∧ (0 < n → ∀ (j : ℤ), χ i ((j:ℝ)/n) ≠ 0 → j ∈ S) := by by_cases hn : 0 < n · obtain ⟨S,_hS,hS'⟩ := cylinder_cutoff_finite_indices (compactSupport_frequency_cylinder (hc i)) (fun _ => hz i) n hn refine ⟨S.filter (fun j => χ i ((j:ℝ)/n) ≠ 0),?_,?_⟩ · intro j hj exact (Finset.mem_filter.mp hj).2 · intro _ j hj apply Finset.mem_filter.mpr exact ⟨by by_contra hh; exact hj (hS' j hh 0),hj⟩ · exact ⟨∅,by simp,fun hh => (hn hh).elim⟩ choose S hS hS' using hex refine ⟨S,?_,?_,?_⟩ · intro n i hi exact hS n i 0 hi (by simp [hz]) · intro n i j hij rw [Finset.disjoint_left] intro k hi hj exact Set.disjoint_left.mp (hd hij) (subset_tsupport (χ i) (hS n i k hi)) (subset_tsupport (χ j) (hS n j k hj)) · intro n hn i j exact ⟨hS n i j,hS' n i hn j⟩ /-- Compact slow-coordinate supports at equal old index stay disjoint under strictly monotone stationary maps. Distinct indices use the separated ranges. -/ lemma frequency_images_disjoint {ι : Type*} (m : ι → ℤ) (V : ℝ → ℝ) (I : Set ℝ) (K : ι → Set ℝ) (hm : ∀ i, m i ≠ 0) (hi : InjOn V I) (hKI : ∀ i, K i ⊆ I) (hsep : ∀ i j, ∀ x ∈ I, ∀ y ∈ I, (m i:ℝ)*V x = (m j:ℝ)*V y → m i = m j) (hdisj : ∀ i j, i ≠ j → m i = m j → Disjoint (K i) (K j)) : Pairwise (fun i j => Disjoint ((fun x => (m i:ℝ)*V x) '' K i) ((fun x => (m j:ℝ)*V x) '' K j)) := by intro i j hij rw [Set.disjoint_left] intro z hz hz' obtain ⟨x,hx,hxz⟩ := hz obtain ⟨y,hy,hyz⟩ := hz' have he : (m i:ℝ)*V x = (m j:ℝ)*V y := hxz.trans hyz.symm have he' : m i = m j := hsep i j x (hKI i hx) y (hKI j hy) he rw [he'] at he have hm' : (m j:ℝ) ≠ 0 := by exact_mod_cast hm j have hxy := hi (hKI i hx) (hKI j hy) (mul_left_cancel₀ hm' he) exact Set.disjoint_left.mp (hdisj i j hij he') hx (hxy ▸ hy) /-- Full exact producer for the second passage's cutoff and index geometry. -/ theorem second_separation_cutoffs {ι : Type*} [Finite ι] (m : ι → ℤ) (V : ℝ → ℝ) (I : Set ℝ) (K U : ι → Set ℝ) (hm : ∀ i, m i ≠ 0) (hV : Continuous V) (hi : InjOn V I) (hK : ∀ i, IsCompact (K i)) (hKI : ∀ i, K i ⊆ I) (hzero : ∀ x ∈ I, V x ≠ 0) (hsep : ∀ i j, ∀ x ∈ I, ∀ y ∈ I, (m i:ℝ)*V x = (m j:ℝ)*V y → m i = m j) (hdisj : ∀ i j, i ≠ j → m i = m j → Disjoint (K i) (K j)) (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, (fun x => (m i:ℝ)*V x) '' K i ⊆ U i) : ∃ (χ : ι → ℝ → ℝ) (S : ℕ → ι → Finset ℤ), (∀ i, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (χ i) ∧ HasCompactSupport (χ i) ∧ tsupport (χ i) ⊆ U i ∧ (∀ s, 0 ≤ χ i s ∧ χ i s ≤ 1) ∧ ∀ s ∈ (fun x => (m i:ℝ)*V x) '' K i, χ i =ᶠ[𝓝 s] 1) ∧ Pairwise (fun i j => Disjoint (tsupport (χ i)) (tsupport (χ j))) ∧ (∀ n i, (0:ℤ) ∉ S n i) ∧ (∀ 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 := by let E i := (fun x => (m i:ℝ)*V x) '' K i have hE i : IsCompact (E i) := (hK i).image (continuous_const.mul hV) let O i := U i ∩ {x : ℝ | x ≠ 0} have hO i : IsOpen (O i) := (hU i).inter isOpen_compl_singleton have hEO i : E i ⊆ O i := by intro z hz refine ⟨hKU i hz,?_⟩ obtain ⟨x,hx,rfl⟩ := hz exact mul_ne_zero (by exact_mod_cast hm i) (hzero _ (hKI i hx)) obtain ⟨χ,hχ,hχd⟩ := finite_disjoint_frequency_cutoffs E O hE hO hEO (frequency_images_disjoint m V I K hm hi hKI hsep hdisj) have hz i : χ i 0 = 0 := by by_contra h exact ((hχ i).2.2.1 (subset_tsupport (χ i) h)).2 rfl obtain ⟨S,hS,hSd,hS'⟩ := disjoint_frequency_indices χ (fun i => (hχ i).2.1) hz hχd exact ⟨χ,S,fun i => ⟨(hχ i).1,(hχ i).2.1,(hχ i).2.2.1.trans inter_subset_left, (hχ i).2.2.2⟩,hχd,hS,hSd,hS'⟩ end ThreeTorus noncomputable section open MeasureTheory Set Filter Function open scoped Topology namespace ThreeTorus namespace CompactParameter /-- Compact-parameter differentiation, used by the exact passage near epsilon=0. -/ lemma hasFDerivAt_compact_integral {E : Type} {G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [NormedAddCommGroup G] [NormedSpace ℝ G] [CompleteSpace G] {F : E × ℝ → G} (hF : ContDiff ℝ 1 F) (a b : ℝ) (x : E) : HasFDerivAt (fun y => ∫ t in a..b, F (y,t)) (∫ t in a..b, (fderiv ℝ F (x,t)).comp (ContinuousLinearMap.inl ℝ E ℝ)) x := by let D : E × ℝ → E →L[ℝ] G := fun p => (fderiv ℝ F p).comp (ContinuousLinearMap.inl ℝ E ℝ) have hD : Continuous D := hF.continuous_fderiv (by norm_num) |>.clm_comp continuous_const obtain ⟨C,hC⟩ := ((isCompact_closedBall x 1).prod (isCompact_uIcc (a := a) (b := b))).exists_bound_of_continuousOn hD.continuousOn refine hasFDerivAt_integral_of_dominated_of_fderiv_le'' (F := fun y t => F (y,t)) (F' := fun y t => D (y,t)) (bound := fun _ => C) (Metric.closedBall_mem_nhds x (by norm_num : (0:ℝ)<1)) ?_ ?_ ?_ ?_ intervalIntegrable_const ?_ · exact Eventually.of_forall (fun y => (hF.continuous.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable) · exact (hF.continuous.comp (continuous_const.prodMk continuous_id)).intervalIntegrable _ _ · exact (hD.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable · filter_upwards [ae_restrict_mem measurableSet_uIoc] with t ht intro y hy exact hC (y,t) ⟨hy,uIoc_subset_uIcc ht⟩ · exact Eventually.of_forall (fun t y _ => by simpa [D,Function.comp_def] using ((hF.differentiable one_ne_zero ((ContinuousLinearMap.inl ℝ E ℝ) y+(0,t))).hasFDerivAt).comp y ((ContinuousLinearMap.inl ℝ E ℝ).hasFDerivAt.add_const (0,t))) /-- Smooth parameter integration over a fixed compact interval. -/ lemma contDiff_compact_integral {E : Type} {G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [NormedAddCommGroup G] [NormedSpace ℝ G] [CompleteSpace G] (n : ℕ) {F : E × ℝ → G} (hF : ContDiff ℝ n F) (a b : ℝ) : ContDiff ℝ n (fun y => ∫ t in a..b, F (y,t)) := by induction n generalizing G with | zero => apply contDiff_zero.mpr exact intervalIntegral.continuous_parametric_intervalIntegral_of_continuous' (show Continuous (Function.uncurry (fun y t => F (y,t))) from hF.continuous) a b | succ n ih => let D : E × ℝ → E →L[ℝ] G := fun p => (fderiv ℝ F p).comp (ContinuousLinearMap.inl ℝ E ℝ) have hD : ContDiff ℝ n D := (hF.fderiv_right (by simp)).clm_comp contDiff_const have hder (x : E) := hasFDerivAt_compact_integral (hF.of_le (by exact_mod_cast Nat.succ_pos n)) a b x rw [show ((n+1 : ℕ) : WithTop ℕ∞) = (n : WithTop ℕ∞)+1 by simp, contDiff_succ_iff_fderiv] refine ⟨fun x => (hder x).differentiableAt, ?_, ?_⟩ · simp · have heq : fderiv ℝ (fun y => ∫ t in a..b, F (y,t)) = fun y => ∫ t in a..b, D (y,t) := by funext y exact (hder y).fderiv rw [heq] exact ih hD lemma smooth_compact_integral {E : Type} {G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [NormedAddCommGroup G] [NormedSpace ℝ G] [CompleteSpace G] {F : E × ℝ → G} (hF : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) F) (a b : ℝ) : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun y => ∫ t in a..b, F (y,t)) := by rw [contDiff_infty] intro n exact contDiff_compact_integral n (hF.of_le (by exact_mod_cast (le_top : (n : ℕ∞) ≤ ⊤))) a b end CompactParameter /-- Smoothness of a Fourier coefficient in an arbitrary finite-dimensional parameter, obtained from a compact interval integral of the literal lift. -/ lemma smooth_fourierCoeff_param {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {f : E → UnitAddCircle → ℂ} (hf : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : E × ℝ => f p.1 (p.2 : UnitAddCircle))) (j : ℤ) : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x => fourierCoeff (f x) j) := by have hchar : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : E × ℝ => (fourier (-j) (p.2 : UnitAddCircle) : ℂ)) := by simp only [fourier_coe_apply, Complex.ofReal_one, div_one] exact (contDiff_const.mul (Complex.ofRealCLM.contDiff.comp contDiff_snd)).cexp have h := CompactParameter.smooth_compact_integral (hchar.mul hf) 0 1 have he (x : E) : fourierCoeff (f x) j = ∫ t in (0:ℝ)..1, (fourier (-j) (t : UnitAddCircle) : ℂ) * f x t := by simpa only [one_div_one,one_smul,zero_add,smul_eq_mul] using fourierCoeff_eq_intervalIntegral (f x) j 0 simpa only [he] using h lemma smooth_fourier_lift (m : ℤ) : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => (fourier m (x : UnitAddCircle) : ℂ)) := by simp only [fourier_coe_apply,Complex.ofReal_one,div_one] exact (contDiff_const.mul Complex.ofRealCLM.contDiff).cexp /-- Smoothness of the exact old-amplitude coefficients; no stationary-phase approximation replaces these coefficients. -/ lemma smooth_exactPassageCoeff {m : ℤ} {H : UnitAddCircle → UnitAddCircle} {Hl : ℝ → ℝ} {A : UnitAddCircle × UnitAddCircle → ℂ} (hHl : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) Hl) (hH : ∀ t : ℝ, H (t : UnitAddCircle) = (Hl t : UnitAddCircle)) (hA : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => A (p.1,p.2))) (j : ℤ) : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun a : ℝ => exactPassageCoeff m H A j a) := by apply smooth_fourierCoeff_param change ContDiff ℝ _ (fun p : ℝ × ℝ => A (p.2,p.1)*fourier m (H p.2)) simp_rw [hH] exact (hA.comp (contDiff_snd.prodMk contDiff_fst)).mul ((smooth_fourier_lift m).comp (hHl.comp contDiff_snd)) /-- The new amplitudes retained by the passage lie in the same smooth packet class needed for its next use. -/ lemma smooth_retained_amplitude {m j : ℤ} {H : UnitAddCircle → UnitAddCircle} {Hl : ℝ → ℝ} {A : UnitAddCircle × UnitAddCircle → ℂ} {χ : ℝ × UnitAddCircle → ℝ} {ψ : UnitAddCircle → ℝ} (s : ℝ) (hHl : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) Hl) (hH : ∀ t : ℝ, H (t : UnitAddCircle) = (Hl t : UnitAddCircle)) (hA : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => A (p.1,p.2))) (hχ : CylinderSmooth χ) (hψ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun a : ℝ => ψ a)) : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => (χ (s,p.1):ℂ)*(ψ p.1:ℂ)*exactPassageCoeff m H A j p.1*fourier m (p.2:UnitAddCircle)) := by have hc : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => χ (s,(p.1:UnitAddCircle))) := hχ.comp (contDiff_const.prodMk contDiff_fst) exact (((Complex.ofRealCLM.contDiff.comp hc).mul (Complex.ofRealCLM.contDiff.comp (hψ.comp contDiff_fst))).mul ((smooth_exactPassageCoeff hHl hH hA j).comp contDiff_fst)).mul ((smooth_fourier_lift m).comp contDiff_snd) end ThreeTorus noncomputable section open MeasureTheory Set Function Filter open scoped Topology ENNReal namespace ThreeTorus /-- A nonnegative integrable density identity proved with signed vector measures is the identical ordinary measure identity. -/ lemma positive_density_map_of_vectorMeasure {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] {μ : Measure X} {ν : Measure Y} {f : X → Y} {q : X → ℝ} {p : Y → ℝ} (hf : Measurable f) (hq : Integrable q μ) (hp : Integrable p ν) (hq0 : ∀ x, 0 ≤ q x) (hp0 : ∀ y, 0 ≤ p y) (he : (μ.withDensityᵥ q).map f = ν.withDensityᵥ p) : (μ.withDensity (fun x => ENNReal.ofReal (q x))).map f = ν.withDensity (fun y => ENNReal.ofReal (p y)) := by ext s hs rw [Measure.map_apply hf hs,withDensity_apply _ (hf hs),withDensity_apply _ hs] rw [← ofReal_integral_eq_lintegral_ofReal hq.integrableOn (Eventually.of_forall hq0), ← ofReal_integral_eq_lintegral_ofReal hp.integrableOn (Eventually.of_forall hp0)] congr 1 have h := congrArg (fun v : VectorMeasure Y ℝ => v s) he simpa only [VectorMeasure.map_apply _ hf hs,withDensityᵥ_apply hq (hf hs), withDensityᵥ_apply hp hs] using h lemma branchDensity_nonneg {I : Set ℝ} {J g D q : ℝ → ℝ} (hq : ∀ t, 0 ≤ q t) (y : ℝ) : 0 ≤ branchDensity I J g D q y := by classical simp only [branchDensity,Set.indicator_apply] split_ifs · exact div_nonneg (hq _) (abs_nonneg _) · rfl lemma unfoldedModeDensity_nonneg {I : Set ℝ} {J g D q : ℝ → ℝ} (hq : ∀ t, 0 ≤ q t) {n : ℝ} (hn : 0 ≤ n) (j t : ℝ) : 0 ≤ unfoldedModeDensity I J g D q n j t := by unfold unfoldedModeDensity scaledBranchKernel exact mul_nonneg (inv_nonneg.mpr (abs_nonneg _)) (branchDensity_nonneg (fun a => mul_nonneg hn (hq a)) _) lemma foldCircleDensity_nonneg {q : ℝ → ℝ} (hq : ∀ t, 0 ≤ q t) (z : UnitAddCircle) : 0 ≤ foldCircleDensity q z := by unfold foldCircleDensity periodizedReal exact tsum_nonneg (fun k => hq _) /-- The literal new amplitude at a retained angular mode. -/ def retainedAmplitude (ψ : UnitAddCircle → ℝ) (χ : ℝ → UnitAddCircle → ℝ) (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) (A : UnitAddCircle × UnitAddCircle → ℂ) (n : ℕ) (j : ℤ) (p : UnitAddCircle × UnitAddCircle) : ℂ := (ψ p.1:ℂ)*(χ ((j:ℝ)/n) p.1:ℂ)*exactPassageCoeff m (H n) A j p.1*fourier m p.2 lemma retainedPassage_eq_sum_packets (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) A S (fun j => χ ((j:ℝ)/n)) ψ x = ∑ j ∈ S, packet j (retainedAmplitude ψ χ m H A n j) x := rfl lemma retainedAmplitude_norm_sq (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 χ m H A n j p‖^2 = retainedCircleWeight ψ χ (fun n j => exactPassageCoeff m (H n) A j) n j p.1 := by simp only [retainedAmplitude,norm_mul,show ‖fourier m p.2‖ = 1 from Circle.norm_coe _, mul_one,mul_pow,Complex.norm_real,Real.norm_eq_abs,sq_abs,retainedCircleWeight] lemma retainedAmplitude_modulus (ψ : UnitAddCircle → ℝ) (χ : ℝ → UnitAddCircle → ℝ) (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) (A : UnitAddCircle × UnitAddCircle → ℂ) (n : ℕ) (j : ℤ) (a b : UnitAddCircle) : ‖retainedAmplitude ψ χ m H A n j (a,b)‖ = ‖retainedAmplitude ψ χ m H A n j (a,0)‖ := by simp only [retainedAmplitude,norm_mul,show ‖fourier m b‖ = 1 from Circle.norm_coe _, show ‖fourier m (0:UnitAddCircle)‖ = 1 from Circle.norm_coe _] -- The same elementary identity used in the supplied whitening proof. lemma map_withDensity_comp {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] (μ : Measure X) {ζ : X → Y} {w : Y → ℝ≥0∞} (hζ : Measurable ζ) (hw : Measurable w) : (μ.withDensity (w ∘ ζ)).map ζ = (μ.map ζ).withDensity w := by ext s hs rw [Measure.map_apply hζ hs, withDensity_apply _ (hζ hs), withDensity_apply _ hs] rw [← lintegral_indicator (hζ hs), ← lintegral_indicator hs, lintegral_map (hw.indicator hs) hζ] apply lintegral_congr intro x rfl /-- If a coefficient's modulus is independent of its spare coordinate, its literal packet spectral measure is the single-coordinate weighted image. -/ lemma packetSpectralMeasure_of_modulus {φ : UnitAddCircle → UnitAddCircle} (hφ : Measurable φ) {j : ℤ} {B : UnitAddCircle × UnitAddCircle → ℂ} {q : UnitAddCircle → ℝ} (hq : Measurable q) (hB : ∀ p, ‖B p‖^2 = q p.1) : packetSpectralMeasure φ j B = (AddCircle.haarAddCircle.withDensity (fun a => ENNReal.ofReal (q a))).map (fun a => j • φ a) := by unfold packetSpectralMeasure have he : (fun p : UnitAddCircle × UnitAddCircle => ENNReal.ofReal (‖B p‖^2)) = (fun a => ENNReal.ofReal (q a)) ∘ Prod.fst := funext (fun p => congrArg _ (hB p)) rw [he] change Measure.map ((fun a => j • φ a) ∘ Prod.fst) _ = _ rw [← Measure.map_map (show Measurable (fun a => j • φ a) from hφ.const_smul j) measurable_fst, map_withDensity_comp slowVolume measurable_fst hq.ennreal_ofReal] simp only [slowVolume,Measure.map_fst_prod,measure_univ,one_smul] lemma retainedCircleWeight_nonneg (ψ : UnitAddCircle → ℝ) (χ : ℝ → UnitAddCircle → ℝ) (c : ℕ → ℤ → UnitAddCircle → ℂ) (n : ℕ) (j : ℤ) (a : UnitAddCircle) : 0 ≤ retainedCircleWeight ψ χ c n j a := by unfold retainedCircleWeight; positivity lemma circleDensityLift_nonneg {q : UnitAddCircle → ℝ} (hq : ∀ a, 0 ≤ q a) (o x : ℝ) : 0 ≤ circleDensityLift o q x := by classical simp only [circleDensityLift,Set.indicator_apply] split_ifs · exact hq _ · rfl /-- The folded branch is the ordinary (positive) spectral density, not just an identity of signed integrals. -/ lemma positive_foldedModeDensity_pushforward {q : UnitAddCircle → ℝ} {q₀ J₀ : ℝ → ℝ} {J : UnitAddCircle → ℝ} {o : ℝ} {I : Set ℝ} (hq : Integrable q AddCircle.haarAddCircle) (hqm : Measurable q₀) (hq0 : ∀ a, 0 ≤ q a) (hq00 : ∀ x, 0 ≤ q₀ x) (hq₀ : ∀ x ∈ Ioc o (o+1), q (x : UnitAddCircle) = q₀ x) (hq₀s : support q₀ ⊆ Ioc o (o+1)) (hqs : support q₀ ⊆ I) (hj : Measurable 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) {n j : ℝ} (hn : 0 < n) (hjn : j ≠ 0) : (AddCircle.haarAddCircle.withDensity (fun a => ENNReal.ofReal (q a))).map (fun a => (((j+(j/n)*J a)/n : ℝ) : UnitAddCircle)) = AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal (foldCircleDensity (unfoldedModeDensity I J₀ (invFunOn J₀ I) (deriv J₀) q₀ n j) z)) := by have hqi : Integrable q₀ := by rw [← circleDensityLift_eq hq₀ hq₀s] exact integrable_circleDensityLift hq o have hD : ∀ x ∈ I, HasDerivWithinAt J₀ (deriv J₀ x) I x := fun x _ => (hJ.differentiable (by simp) x).hasDerivAt.hasDerivWithinAt have hmi := measurable_unfoldedModeDensity hJ hIo hinj hd hqm n j have hii := integrable_unfoldedModeDensity hIo.measurableSet (isOpen_image_of_deriv_ne_zero hJ hIo hd).measurableSet hD hinj (fun _ hx => hinj.leftInvOn_invFunOn hx) hd hqi hn hjn exact positive_density_map_of_vectorMeasure (AddCircle.measurable_mk'.comp ((measurable_const.add (hj.const_mul _)).div_const n)) hq (integrable_foldCircleDensity hmi hii) hq0 (foldCircleDensity_nonneg (unfoldedModeDensity_nonneg hq00 hn.le j)) (foldedModeDensity_pushforward hq hqm hq₀ hq₀s hqs hj hJ hJe hIo hinj hd hn hjn) lemma passage_speed_mode (J : UnitAddCircle → ℝ) (n : ℝ) (j : ℤ) (a : UnitAddCircle) : j • (((1+J a/n)/n : ℝ) : UnitAddCircle) = ((((j:ℝ)+((j:ℝ)/n)*J a)/n : ℝ) : UnitAddCircle) := by rw [← AddCircle.coe_zsmul] congr 1 simp only [zsmul_eq_mul] ring /-- Any coefficient with the actual retained modulus has the displayed nonnegative folded density under the actual new twist. -/ lemma packetSpectralMeasure_new_speed {q : UnitAddCircle → ℝ} {q₀ J₀ : ℝ → ℝ} {J : UnitAddCircle → ℝ} {o : ℝ} {I : Set ℝ} {B : UnitAddCircle × UnitAddCircle → ℂ} (hq : Integrable q AddCircle.haarAddCircle) (hqmeas : Measurable q) (hqm : Measurable q₀) (hB : ∀ p, ‖B p‖^2 = q p.1) (hq00 : ∀ x, 0 ≤ q₀ x) (hq₀ : ∀ x ∈ Ioc o (o+1), q (x : UnitAddCircle) = q₀ x) (hq₀s : support q₀ ⊆ Ioc o (o+1)) (hqs : support q₀ ⊆ I) (hj : Measurable 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) {n : ℝ} {j : ℤ} (hn : 0 < n) (hjn : j ≠ 0) : packetSpectralMeasure (fun a => (((1+J a/n)/n:ℝ):UnitAddCircle)) j B = AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal (foldCircleDensity (unfoldedModeDensity I J₀ (invFunOn J₀ I) (deriv J₀) q₀ n j) z)) := by have hσ : Measurable (fun a => (((1+J a/n)/n:ℝ):UnitAddCircle)) := AddCircle.measurable_mk'.comp ((measurable_const.add (hj.div_const n)).div_const n) rw [packetSpectralMeasure_of_modulus hσ hqmeas hB] simp_rw [passage_speed_mode] exact positive_foldedModeDensity_pushforward hq hqm (fun a => (hB (a,0)) ▸ sq_nonneg _) hq00 hq₀ hq₀s hqs hj hJ hJe hIo hinj hd hn (by exact_mod_cast hjn) /-- Density identity for the literal retained coefficient used by the passage. The only support hypothesis is that the cutoff lies on the chosen good branch. -/ lemma retainedAmplitude_spectralDensity {ψ : UnitAddCircle → ℝ} {χ : ℝ → UnitAddCircle → ℝ} {m : ℤ} {H : ℕ → UnitAddCircle → UnitAddCircle} {A : UnitAddCircle × UnitAddCircle → ℂ} {J : UnitAddCircle → ℝ} {J₀ : ℝ → ℝ} {o : ℝ} {I : Set ℝ} (hψ : Continuous ψ) (hχ : Continuous χ.uncurry) (hH : ∀ n, Continuous (H n)) (hA : Continuous A) (hψs : ∀ x ∈ Ioc o (o+1), ψ (x:UnitAddCircle) ≠ 0 → x ∈ I) (hj : Measurable 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) {n : ℕ} {j : ℤ} (hn : 0 < n) (hjn : j ≠ 0) : packetSpectralMeasure (fun a => (((1+J a/n)/n:ℝ):UnitAddCircle)) j (retainedAmplitude ψ χ m H A n j) = AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal (foldCircleDensity (unfoldedModeDensity I J₀ (invFunOn J₀ I) (deriv J₀) (circleDensityLift o (retainedCircleWeight ψ χ (fun n j => exactPassageCoeff m (H n) A j) n j)) n j) z)) := by let C n j := exactPassageCoeff m (H n) A j let q := retainedCircleWeight ψ χ C n j have hC n j : Continuous (C n j) := continuous_exactPassageCoeff (hH n) hA j have hq : Continuous q := ((hψ.pow 2).mul ((hχ.comp (continuous_const.prodMk continuous_id)).pow 2)).mul ((hC n j).norm.pow 2) have hqs : support (circleDensityLift o q) ⊆ Ioc o (o+1) ∩ I := by intro x hx have hxI : x ∈ Ioc o (o+1) := by by_contra hh exact hx (by simp [circleDensityLift,hh]) refine ⟨hxI,hψs x hxI ?_⟩ intro hz exact hx (by simp [circleDensityLift,hxI,q,retainedCircleWeight,hz]) exact packetSpectralMeasure_new_speed (retainedCircleWeight_integrable hψ hχ hC n j) hq.measurable (measurable_circleDensityLift hq.measurable o) (retainedAmplitude_norm_sq ψ χ m H A n j) (circleDensityLift_nonneg (retainedCircleWeight_nonneg ψ χ C n j) o) (fun _ hx => by simp only [circleDensityLift,indicator_of_mem hx]; rfl) (hqs.trans inter_subset_left) (hqs.trans inter_subset_right) hj hJ hJe hIo hinj hd (by exact_mod_cast hn) hjn lemma packet_twisted_pair_zero {A B : UnitAddCircle × UnitAddCircle → ℂ} (hA : Continuous A) (hB : Continuous B) {φ : UnitAddCircle → UnitAddCircle} (hφ : Continuous φ) {m k : ℤ} (hdisj : m = k → Disjoint (support A) (support B)) (n : ℤ) : (∫ x : TripleTorus, starRingEnd ℂ (packet m A x)*packet k B (twistIter φ n x) ∂tripleVolume) = 0 := by let B' : UnitAddCircle × UnitAddCircle → ℂ := fun p => fourier n (k • φ p.1)*B p have hb' : Continuous B' := ((fourier n).continuous.comp ((hφ.comp continuous_fst).const_smul k)).mul hB have he (x : TripleTorus) : packet k B (twistIter φ n x) = packet k B' x := by rw [packet_twistIter] dsimp [packet,B'] ring simp_rw [he] rw [integral_packet_pair hA hb'] split_ifs with hmk · apply integral_eq_zero_of_ae filter_upwards [] with p by_cases ha : A p = 0 · simp [ha] · have hb : B p = 0 := by by_contra hh exact Set.disjoint_left.mp (hdisj hmk) ha hh simp [B',hb] · rfl /-- The spectral measure of a smooth disjoint packet list is the sum of the literal packet pushforwards, demonstrated by every bilateral Fourier moment. -/ lemma disjoint_packet_list_correlation {ι : Type*} [Fintype ι] (m : ι → ℤ) (A : ι → UnitAddCircle × UnitAddCircle → ℂ) (hA : ∀ i, Continuous (A i)) {φ : UnitAddCircle → UnitAddCircle} (hφ : Continuous φ) (hd : ∀ i j, i ≠ j → m i = m j → Disjoint (support (A i)) (support (A j))) (n : ℤ) : (∫ x : TripleTorus, starRingEnd ℂ (∑ i, packet (m i) (A i) x)* (∑ i, packet (m i) (A i) (twistIter φ n x)) ∂tripleVolume) = ∑ i, ∫ z, fourier n z ∂packetSpectralMeasure φ (m i) (A i) := by classical have ht : Continuous (twistIter φ n) := (continuous_fst.add ((hφ.comp (continuous_fst.comp continuous_snd)).const_smul n)).prodMk continuous_snd have hc (i j : ι) : Continuous (fun x : TripleTorus => starRingEnd ℂ (packet (m i) (A i) x)*packet (m j) (A j) (twistIter φ n x)) := (Complex.continuous_conj.comp (continuous_packet (hA i))).mul ((continuous_packet (hA j)).comp ht) have hi (i j : ι) : Integrable (fun x : TripleTorus => starRingEnd ℂ (packet (m i) (A i) x)*packet (m j) (A j) (twistIter φ n x)) tripleVolume := (hc i j).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) simp_rw [map_sum,Finset.sum_mul,Finset.mul_sum] rw [integral_finsetSum _ (fun i _ => integrable_finsetSum _ (fun j _ => hi i j))] apply Finset.sum_congr rfl intro i _ rw [integral_finsetSum _ (fun j _ => hi i j),Finset.sum_eq_single i] · exact (packet_spectral_moment hφ (hA i) (m i) n).symm · intro j _ hji exact packet_twisted_pair_zero (hA i) (hA j) hφ (hd i j hji.symm) n · intro hi exact (hi (Finset.mem_univ i)).elim end ThreeTorus end end end end end end section /-! Exact spectral identity for the literal retained packet vector. This closes the interface between the previously proved coefficient-density estimates and the actual Koopman correlations, without an abstract density hypothesis. -/ noncomputable section open MeasureTheory Set Function Filter open scoped Topology BigOperators namespace ThreeTorus lemma foldCircleDensity_add_ae {q r : ℝ → ℝ} (hq : Measurable q) (hr : Measurable r) (hqi : Integrable q) (hri : Integrable r) : foldCircleDensity (fun x => q x+r x) =ᵐ[AddCircle.haarAddCircle] fun z => foldCircleDensity q z+foldCircleDensity r z := by apply (integrable_foldCircleDensity (hq.add hr) (hqi.add hri)).ae_eq_of_withDensityᵥ_eq ((integrable_foldCircleDensity hq hqi).add (integrable_foldCircleDensity hr hri)) rw [← foldCircleDensity_pushforward (hq.add hr) (hqi.add hri), withDensityᵥ_add hqi hri, withDensityᵥ_add (integrable_foldCircleDensity hq hqi) (integrable_foldCircleDensity hr hri)] have he := map_add (VectorMeasure.mapGm (fun x : ℝ => (x:UnitAddCircle))) (volume.withDensityᵥ q) (volume.withDensityᵥ r) change (volume.withDensityᵥ q+volume.withDensityᵥ r).map _ = (volume.withDensityᵥ q).map _+(volume.withDensityᵥ r).map _ at he rw [he,foldCircleDensity_pushforward hq hqi,foldCircleDensity_pushforward hr hri] lemma foldCircleDensity_finsetSum_ae {ι : Type*} (S : Finset ι) (q : ι → ℝ → ℝ) (hq : ∀ i ∈ S, Measurable (q i)) (hqi : ∀ i ∈ S, Integrable (q i)) : foldCircleDensity (fun x => ∑ i ∈ S, q i x) =ᵐ[AddCircle.haarAddCircle] fun z => ∑ i ∈ S, foldCircleDensity (q i) z := by classical induction S using Finset.induction_on with | empty => filter_upwards [] with z simp [foldCircleDensity,periodizedReal,AddCircle.liftIoc,Set.domRestrict] | @insert i S hi ih => simp only [Finset.sum_insert hi] have hh := foldCircleDensity_add_ae (hq i (Finset.mem_insert_self i S)) (Finset.measurable_sum S (fun j hj => hq j (Finset.mem_insert_of_mem hj))) (hqi i (Finset.mem_insert_self i S)) (integrable_finsetSum S (fun j hj => hqi j (Finset.mem_insert_of_mem hj))) have ht := ih (fun j hj => hq j (Finset.mem_insert_of_mem hj)) (fun j hj => hqi j (Finset.mem_insert_of_mem hj)) filter_upwards [hh,ht] with z hz ht rw [hz,ht] lemma moment_withDensity_real {p : UnitAddCircle → ℝ} (hp : Measurable p) (hp0 : ∀ z, 0 ≤ p z) (v : UnitAddCircle → ℂ) : (∫ z, v z ∂AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal (p z))) = ∫ z, (p z:ℂ)*v z ∂AddCircle.haarAddCircle := by rw [integral_withDensity_eq_integral_toReal_smul hp.ennreal_ofReal (by filter_upwards [] with z; finiteness)] apply integral_congr_ae filter_upwards [] with z rw [ENNReal.toReal_ofReal (hp0 z),Complex.real_smul] lemma continuous_retainedAmplitude {ψ : UnitAddCircle → ℝ} {χ : ℝ → UnitAddCircle → ℝ} {m : ℤ} {H : ℕ → UnitAddCircle → UnitAddCircle} {A : UnitAddCircle × UnitAddCircle → ℂ} (hψ : Continuous ψ) (hχ : Continuous χ.uncurry) (hH : ∀ n, Continuous (H n)) (hA : Continuous A) (n : ℕ) (j : ℤ) : Continuous (retainedAmplitude ψ χ m H A n j) := by unfold retainedAmplitude exact ((((Complex.continuous_ofReal.comp hψ).comp continuous_fst).mul (Complex.continuous_ofReal.comp (hχ.comp (continuous_const.prodMk continuous_fst)))).mul ((continuous_exactPassageCoeff (hH n) hA j).comp continuous_fst)).mul ((fourier m).continuous.comp continuous_snd) /-- All bilateral Koopman moments of the actual finite retained vector are exactly those of its explicit nonnegative folded coefficient density. -/ theorem retainedPassage_spectral_moment {ψ : UnitAddCircle → ℝ} {χ : ℝ → UnitAddCircle → ℝ} {m : ℤ} {H : ℕ → UnitAddCircle → UnitAddCircle} {A : UnitAddCircle × UnitAddCircle → ℂ} {J : UnitAddCircle → ℝ} {J₀ : ℝ → ℝ} {I : Set ℝ} (hψ : Continuous ψ) (hχ : Continuous χ.uncurry) (hH : ∀ n, Continuous (H n)) (hA : Continuous A) (hψs : ∀ x ∈ Ioc (0:ℝ) 1, ψ (x:UnitAddCircle) ≠ 0 → x ∈ I) (hj : 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) {n : ℕ} {S : Finset ℤ} (hn : 0 < n) (hS : ∀ j ∈ S, j ≠ 0) (k : ℤ) : let φ := fun a => (((1+J a/n)/n:ℝ):UnitAddCircle) let f := retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) ψ (∫ x : TripleTorus, starRingEnd ℂ (f x)*f (twistIter φ k x) ∂tripleVolume) = ∫ z, (exactRetainedPacketDensity I J₀ ψ χ m H A S n z:ℂ)*fourier k z ∂AddCircle.haarAddCircle := by classical dsimp only let φ := fun a => (((1+J a/n)/n:ℝ):UnitAddCircle) let B := retainedAmplitude ψ χ m H A n let q := 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 have hφ : Continuous φ := (AddCircle.continuous_mk' 1).comp ((continuous_const.add (hj.div_const n)).div_const n) have hB j : Continuous (B j) := continuous_retainedAmplitude hψ hχ hH hA n j have hreg j (hjS : j ∈ S) := retained_unfolded_regular hJ hIo hinj hd hψ hχ (fun n j => continuous_exactPassageCoeff (m := m) (hH n) hA j) 0 hn (hS j hjS) have hqm j (hjS : j ∈ S) : Measurable (q j) := (hreg j hjS).1 have hqi j (hjS : j ∈ S) : Integrable (q j) := (hreg j hjS).2 have hq0 j : ∀ t, 0 ≤ q j t := unfoldedModeDensity_nonneg (circleDensityLift_nonneg (retainedCircleWeight_nonneg ψ χ _ n j) 0) (Nat.cast_nonneg n) j have hds j (hjS : j ∈ S) : packetSpectralMeasure φ j (B j) = AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal (foldCircleDensity (q j) z)) := retainedAmplitude_spectralDensity hψ hχ hH hA (by simpa using hψs) hj.measurable hJ hJe hIo hinj hd hn (hS j hjS) have hfold := foldCircleDensity_finsetSum_ae S q hqm hqi have hint j (hjS : j ∈ S) : Integrable (fun z => (foldCircleDensity (q j) z:ℂ)*fourier k z) AddCircle.haarAddCircle := (integrable_foldCircleDensity (hqm j hjS) (hqi j hjS)).ofReal.mul_bdd (fourier k).continuous.aestronglyMeasurable (ae_of_all _ (fun z => le_of_eq (Circle.norm_coe _))) have hc := disjoint_packet_list_correlation (fun j : S => (j:ℤ)) (fun j : S => B j) (fun j => hB j) hφ (fun i j hij he => (hij (Subtype.ext he)).elim) k have hsum (x : TripleTorus) : (∑ j : S, packet (j:ℤ) (B j) x) = ∑ j ∈ S, packet j (B j) x := Finset.sum_coe_sort S (fun j => packet j (B j) x) simp_rw [hsum] at hc rw [Finset.sum_coe_sort S (fun j => ∫ z, fourier k z ∂packetSpectralMeasure φ j (B j))] at hc simp only [retainedPassage_eq_sum_packets] change (∫ x : TripleTorus, starRingEnd ℂ (∑ j ∈ S, packet j (B j) x)* (∑ j ∈ S, packet j (B j) (twistIter φ k x)) ∂tripleVolume) = _ rw [hc] calc _ = ∑ j ∈ S, ∫ z, (foldCircleDensity (q j) z:ℂ)*fourier k z ∂AddCircle.haarAddCircle := by apply Finset.sum_congr rfl intro j hjS rw [hds j hjS] exact moment_withDensity_real (measurable_foldCircleDensity (hqm j hjS)) (foldCircleDensity_nonneg (hq0 j)) (fourier k) _ = ∫ z, (∑ j ∈ S, (foldCircleDensity (q j) z:ℂ))*fourier k z ∂AddCircle.haarAddCircle := by simp_rw [Finset.sum_mul] exact (integral_finsetSum _ hint).symm _ = _ := by apply integral_congr_ae filter_upwards [hfold] with z hz change (∑ j ∈ S, (foldCircleDensity (q j) z:ℂ))*fourier k z = (foldCircleDensity (fun t => ∑ j ∈ S, q j t) z:ℂ)*fourier k z rw [hz,Complex.ofReal_sum] end ThreeTorus end end section /-! Smoothness at the edge of the good arc is essential: the actual retained spectral densities must be valid smooth inputs to the next stage. -/ noncomputable section open MeasureTheory Set Function Filter open scoped Topology BigOperators namespace ThreeTorus lemma support_branchDensity_subset {J q : ℝ → ℝ} {I : Set ℝ} : support (branchDensity I J (invFunOn J I) (deriv J) q) ⊆ J '' tsupport q := by intro y hy have hyI : y ∈ J '' I := by by_contra hh exact hy (by simp [branchDensity,hh]) have hx := invFunOn_pos hyI refine ⟨invFunOn J I y,subset_tsupport q ?_,hx.2⟩ intro hq exact hy (by simp [branchDensity,hyI,hq]) lemma branchDensity_compact {J q : ℝ → ℝ} {I : Set ℝ} (hJ : Continuous J) (hq : HasCompactSupport q) : HasCompactSupport (branchDensity I J (invFunOn J I) (deriv J) q) := by apply HasCompactSupport.intro (hq.image hJ) intro y hy exact notMem_support.mp (fun hh => hy (support_branchDensity_subset hh)) lemma branchDensity_smooth {J q : ℝ → ℝ} {I : Set ℝ} (hJ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) J) (hIo : IsOpen I) (hinj : InjOn J I) (hd : ∀ x ∈ I, deriv J x ≠ 0) (hq : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) q) (hc : HasCompactSupport q) (hs : tsupport q ⊆ I) : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (branchDensity I J (invFunOn J I) (deriv J) q) := by classical let g := invFunOn J I have hM : IsCompact (J '' tsupport q) := hc.image hJ.continuous have hD : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (deriv J) := hJ.deriv' rw [contDiff_iff_contDiffAt] intro y by_cases hy : y ∈ J '' tsupport q · obtain ⟨x,hx,rfl⟩ := hy have hxI := hs hx have hyI : J x ∈ J '' I := mem_image_of_mem J hxI have hg := contDiffAt_invFunOn_of_deriv_ne_zero hJ hIo hinj hxI ((hJ.differentiable (by simp) x).hasDerivAt) (hd x hxI) have hge : g (J x) = x := hinj (invFunOn_pos hyI).1 hxI (invFunOn_pos hyI).2 have hden : ContDiffAt ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun y => |deriv J (g y)|) (J x) := (contDiffAt_abs (by simpa only [hge] using hd x hxI)).comp (J x) (hD.contDiffAt.comp (J x) hg) have hquot := (hq.contDiffAt.comp (J x) hg).div hden (by simpa only [hge] using abs_ne_zero.mpr (hd x hxI)) apply hquot.congr_of_eventuallyEq filter_upwards [(isOpen_image_of_deriv_ne_zero hJ hIo hd).mem_nhds hyI] with z hz simp only [branchDensity,indicator_of_mem hz,g] rfl · apply (contDiffAt_const (c := (0:ℝ))).congr_of_eventuallyEq filter_upwards [hM.isClosed.isOpen_compl.mem_nhds hy] with z hz exact notMem_support.mp (fun hh => hz (support_branchDensity_subset hh)) lemma unfoldedModeDensity_smooth {J q : ℝ → ℝ} {I : Set ℝ} (hJ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) J) (hIo : IsOpen I) (hinj : InjOn J I) (hd : ∀ x ∈ I, deriv J x ≠ 0) (hq : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) q) (hc : HasCompactSupport q) (hs : tsupport q ⊆ I) (n j : ℝ) : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (unfoldedModeDensity I J (invFunOn J I) (deriv J) q n j) := by have hmul : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x => n*q x) := contDiff_const.mul hq have hcs : HasCompactSupport (fun x => n*q x) := hc.mul_left have hs' : tsupport (fun x => n*q x) ⊆ I := tsupport_mul_subset_right.trans hs have hh := branchDensity_smooth hJ hIo hinj hd hmul hcs hs' unfold unfoldedModeDensity scaledBranchKernel exact contDiff_const.mul (hh.comp ((contDiff_const.mul contDiff_id).sub contDiff_const |>.div_const _)) lemma unfoldedModeDensity_compact {J q : ℝ → ℝ} {I : Set ℝ} (hJ : Continuous J) (hc : HasCompactSupport q) {n j : ℝ} (hn : n ≠ 0) (hj : j ≠ 0) : HasCompactSupport (unfoldedModeDensity I J (invFunOn J I) (deriv J) q n j) := by have hh := branchDensity_compact (I := I) hJ (hc.mul_left (f := fun _ => n)) let e : ℝ ≃ₜ ℝ := (Homeomorph.mulLeft₀ n hn).trans (Homeomorph.addRight (-j)) let e' : ℝ ≃ₜ ℝ := e.trans (Homeomorph.mulRight₀ (n/j) (div_ne_zero hn hj)) have he (t : ℝ) : e' t = (n*t-j)/(j/n) := by dsimp [e',e] field_simp ring have hc' := hh.comp_homeomorph e' have heq : unfoldedModeDensity I J (invFunOn J I) (deriv J) q n j = fun t => |j/n|⁻¹ * branchDensity I J (invFunOn J I) (deriv J) (fun x => n*q x) (e' t) := by funext t rw [he] rfl rw [heq] exact hc'.mul_left lemma periodizedReal_eq_periodize {q : ℝ → ℝ} (hc : HasCompactSupport q) : periodizedReal q = periodize q := by funext x obtain ⟨R,hR⟩ := hc.isBounded.exists_norm_le obtain ⟨N,hN⟩ := exists_nat_gt (R+|x|+1) have hs : support (fun k : ℤ => q (x+k)) ⊆ (Finset.Icc (-(N:ℤ)) N : Set ℤ) := by intro k hk have hr := hR (x+k) (subset_tsupport q hk) have hh : |(k:ℝ)| < N := by have ha := abs_sub_le (x+(k:ℝ)) 0 x simp only [sub_zero,zero_sub,abs_neg,add_sub_cancel_left,Real.norm_eq_abs] at hr ha linarith have hab := abs_lt.mp hh simp only [Finset.mem_coe,Finset.mem_Icc] constructor · exact_mod_cast (by linarith : -(N:ℝ) ≤ (k:ℝ)) · exact_mod_cast (by linarith : (k:ℝ) ≤ (N:ℝ)) exact tsum_eq_finsum ((Finset.finite_toSet _).subset hs) lemma foldCircleDensity_smooth_lift {q : ℝ → ℝ} (hq : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) q) (hc : HasCompactSupport q) : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => foldCircleDensity q (x:UnitAddCircle)) := by have he : (fun x : ℝ => foldCircleDensity q (x:UnitAddCircle)) = periodize q := by funext x rw [foldCircleDensity_coe_all] exact congrFun (periodizedReal_eq_periodize hc) x rw [he] exact contDiff_periodize hq hc /-- The literal real lift of a retained coefficient weight. The smooth cutoff is taken before the final passage integer, as in the source. -/ def realRetainedWeight (ψ₀ : ℝ → ℝ) (χ : ℝ → UnitAddCircle → ℝ) (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) (A : UnitAddCircle × UnitAddCircle → ℂ) (n : ℕ) (j : ℤ) (x : ℝ) : ℝ := ψ₀ x^2*χ ((j:ℝ)/n) x^2*‖exactPassageCoeff m (H n) A j x‖^2 lemma realRetainedWeight_support (ψ₀ : ℝ → ℝ) (χ : ℝ → UnitAddCircle → ℝ) (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) (A : UnitAddCircle × UnitAddCircle → ℂ) (n : ℕ) (j : ℤ) : tsupport (realRetainedWeight ψ₀ χ m H A n j) ⊆ tsupport ψ₀ := by apply closure_mono intro x hx hz exact hx (by simp [realRetainedWeight,hz]) lemma realRetainedWeight_compact {ψ₀ : ℝ → ℝ} (hc : HasCompactSupport ψ₀) (χ : ℝ → UnitAddCircle → ℝ) (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) (A : UnitAddCircle × UnitAddCircle → ℂ) (n : ℕ) (j : ℤ) : HasCompactSupport (realRetainedWeight ψ₀ χ m H A n j) := hc.of_isClosed_subset isClosed_closure (realRetainedWeight_support ψ₀ χ m H A n j) lemma realRetainedWeight_smooth {ψ₀ : ℝ → ℝ} {χ : ℝ → UnitAddCircle → ℝ} {m : ℤ} {H : ℕ → UnitAddCircle → UnitAddCircle} {Hl : ℕ → ℝ → ℝ} {A : UnitAddCircle × UnitAddCircle → ℂ} (hψ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) ψ₀) (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))) (n : ℕ) (j : ℤ) : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (realRetainedWeight ψ₀ χ m H A n j) := by have hχ' : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => χ ((j:ℝ)/n) x) := hχ.comp (contDiff_const.prodMk contDiff_id) exact ((hψ.pow 2).mul (hχ'.pow 2)).mul ((smooth_exactPassageCoeff (hHl n) (hH n) hA j).norm_sq (𝕜 := ℂ)) lemma retainedWeight_lift_eq {ψ : UnitAddCircle → ℝ} {ψ₀ : ℝ → ℝ} (χ : ℝ → UnitAddCircle → ℝ) (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) (A : UnitAddCircle × UnitAddCircle → ℂ) (n : ℕ) (j : ℤ) (he : ∀ x ∈ Ioc (0:ℝ) (0+1), ψ (x:UnitAddCircle) = ψ₀ x) (hs : tsupport ψ₀ ⊆ Ioc (0:ℝ) (0+1)) : circleDensityLift 0 (retainedCircleWeight ψ χ (fun n j => exactPassageCoeff m (H n) A j) n j) = realRetainedWeight ψ₀ χ m H A n j := by apply circleDensityLift_eq · intro x hx simp only [retainedCircleWeight,realRetainedWeight,he x hx] · exact (subset_tsupport _).trans ((realRetainedWeight_support ψ₀ χ m H A n j).trans hs) /-- Actual retained densities are smooth at all branch edges, including their zero set. This supplies the next passage's regularity; it is not an assumption that the density is already smooth. -/ theorem exactRetainedPacketDensity_smooth_lift {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) : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => exactRetainedPacketDensity I J ψ χ m H A S n x) := by unfold exactRetainedPacketDensity simp_rw [retainedWeight_lift_eq χ m H A n _ he (hs.trans inter_subset_left)] apply foldCircleDensity_smooth_lift · apply ContDiff.sum 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 · have hh (j : ℤ) (hj : j ∈ S) : HasCompactSupport (unfoldedModeDensity I J (invFunOn J I) (deriv J) (realRetainedWeight ψ₀ χ m H A n j) n j) := 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) convert HasCompactSupport.finset_sum hh using 1 first | rfl | (funext time; simp only [Finset.sum_apply]) lemma continuous_of_continuous_lift {Y : Type*} [TopologicalSpace Y] {f : UnitAddCircle → Y} (hf : Continuous (fun x : ℝ => f (x:UnitAddCircle))) : Continuous f := isQuotientMap_quotient_mk'.continuous_iff.mpr hf end ThreeTorus end end section /-! Identification of the reference stationary marginal with the actual old packet spectrum. In particular integer carrier speeds do not change it. -/ noncomputable section open MeasureTheory Set Function Filter open scoped Topology BigOperators namespace ThreeTorus lemma positive_circle_branch_fold {q : UnitAddCircle → ℝ} {q₀ J₀ : ℝ → ℝ} {J : UnitAddCircle → ℝ} {o : ℝ} {I : Set ℝ} (hq : Integrable q AddCircle.haarAddCircle) (hqm : Measurable q₀) (hq0 : ∀ a, 0 ≤ q a) (hq00 : ∀ x, 0 ≤ q₀ x) (hq₀ : ∀ x ∈ Ioc o (o+1), q (x:UnitAddCircle) = q₀ x) (hq₀s : support q₀ ⊆ Ioc o (o+1)) (hqs : support q₀ ⊆ I) (hj : Measurable 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) : (AddCircle.haarAddCircle.withDensity (fun a => ENNReal.ofReal (q a))).map (fun a => (J a:UnitAddCircle)) = AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal (foldCircleDensity (branchDensity I J₀ (invFunOn J₀ I) (deriv J₀) q₀) z)) := by let w := branchDensity I J₀ (invFunOn J₀ I) (deriv J₀) q₀ have hqi : Integrable q₀ := by rw [← circleDensityLift_eq hq₀ hq₀s] exact integrable_circleDensityLift hq o have hD : ∀ x ∈ I, HasDerivWithinAt J₀ (deriv J₀ x) I x := fun x _ => (hJ.differentiable (by simp) x).hasDerivAt.hasDerivWithinAt have hIm := (isOpen_image_of_deriv_ne_zero hJ hIo hd).measurableSet have hwi : Integrable w := integrable_branchDensity hIo.measurableSet hIm hD hinj (fun _ hx => hinj.leftInvOn_invFunOn hx) hd hqi.integrableOn have hwm : Measurable w := measurable_branchDensity_smooth hJ hIo hinj hd hqm have hmap := circle_branch_pushforward hq hq₀ hq₀s hqs hj hJ.continuous.measurable hJe hIo.measurableSet hIm hD hinj (fun _ hx => hinj.leftInvOn_invFunOn hx) hd apply positive_density_map_of_vectorMeasure (AddCircle.measurable_mk'.comp hj) hq (integrable_foldCircleDensity hwm hwi) hq0 (foldCircleDensity_nonneg (branchDensity_nonneg hq00)) rw [← foldCircleDensity_pushforward hwm hwi,← hmap] exact (vectorMeasure_map_comp _ hj AddCircle.measurable_mk').symm lemma stationaryWeight_change_branch {A : ℝ → ℂ} {J : ℝ → ℝ} {C I : Set ℝ} (hIC : I ⊆ C) (hi : InjOn J C) (hs : support A ⊆ I) : stationaryWeight C J A = stationaryWeight I J A := by have h := stationaryWeight_family_of_branch (Ω := Unit) hIC hi (fun _ => hs) funext s have hh := congrFun h (s,()) exact hh /-- The density of a literal old angular packet, not an abstract probability model. The integer speed carrier vanishes on the temporal circle. -/ theorem oldPacket_spectralDensity_of_modulus {A : UnitAddCircle × UnitAddCircle → ℂ} {A₀ : ℝ → ℂ} {v : UnitAddCircle → ℝ} {V : ℝ → ℝ} {m c : ℤ} {o : ℝ} {C I : Set ℝ} (hA : Continuous A) (hA₀ : Continuous A₀) (hmod : ∀ u a, ‖A (u,a)‖ = ‖A (u,0)‖) (hAe : ∀ x ∈ Ioc o (o+1), A ((x:UnitAddCircle),0) = A₀ x) (has : support A₀ ⊆ Ioc o (o+1)) (hAs : support A₀ ⊆ I) (hv : Measurable v) (hV : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) V) (hVe : ∀ x ∈ I, v (x:UnitAddCircle) = V x) (hIo : IsOpen I) (hIC : I ⊆ C) (hinj : InjOn (fun x => (m:ℝ)*V x) C) (hd : ∀ x ∈ I, deriv (fun x => (m:ℝ)*V x) x ≠ 0) : packetSpectralMeasure (fun a => ((v a-c:ℝ):UnitAddCircle)) m A = AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal (foldCircleDensity (stationaryWeight C (fun x => (m:ℝ)*V x) A₀) z)) := by have hqc : Continuous (fun u : UnitAddCircle => ‖A (u,0)‖^2) := ((hA.comp (continuous_id.prodMk continuous_const)).norm.pow 2) have hps := packetSpectralMeasure_of_modulus (φ := fun a => (((v a-(c:ℝ)) : ℝ):UnitAddCircle)) (j := m) (B := A) (q := fun u => ‖A (u,0)‖^2) (AddCircle.measurable_mk'.comp (hv.sub measurable_const)) hqc.measurable (fun p => by rw [hmod]) rw [hps] have hmode (a : UnitAddCircle) : m • ((v a-c:ℝ):UnitAddCircle) = ((m:ℝ)*v a:UnitAddCircle) := by rw [← AddCircle.coe_zsmul] simp only [zsmul_eq_mul] rw [mul_sub] have he : ((m:ℝ)*(c:ℝ):UnitAddCircle) = 0 := by rw [← Int.cast_mul] exact (AddCircle.coe_eq_zero_iff 1).mpr ⟨m*c,by simp⟩ rw [AddCircle.coe_sub,he,sub_zero] simp_rw [hmode] rw [stationaryWeight_change_branch hIC hinj hAs] apply positive_circle_branch_fold (q₀ := fun x => ‖A₀ x‖^2) (hqc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) ((hA₀.norm.pow 2).measurable) (fun _ => sq_nonneg _) (fun _ => sq_nonneg _) (fun x hx => by rw [hAe x hx]) (by intro x hx; apply has; intro hz; apply hx; simp [hz]) (by intro x hx; apply hAs; intro hz; apply hx; simp [hz]) (hv.const_mul (m:ℝ)) (contDiff_const.mul hV) (fun x hx => by rw [hVe x hx]) hIo (hinj.mono hIC) hd end ThreeTorus end end section /-! Actual old densities for amplitudes depending on both slow coordinates. -/ noncomputable section open MeasureTheory Set Function Filter open scoped Topology BigOperators ENNReal namespace ThreeTorus lemma map_fst_withDensity_marginal {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] {μ : Measure X} {ν : Measure Y} [SFinite μ] [SFinite ν] {q : X × Y → ℝ≥0∞} (hq : Measurable q) : ((μ.prod ν).withDensity q).map Prod.fst = μ.withDensity (fun x => ∫⁻ y, q (x,y) ∂ν) := by ext s hs rw [Measure.map_apply measurable_fst hs,withDensity_apply _ (measurable_fst hs), withDensity_apply _ hs,← lintegral_indicator (measurable_fst hs), ← lintegral_indicator hs,lintegral_prod _ (hq.indicator (measurable_fst hs)).aemeasurable] apply lintegral_congr intro x by_cases hx : x ∈ s · simp only [Set.indicator_of_mem hx] apply lintegral_congr intro y exact Set.indicator_of_mem (show (x,y) ∈ Prod.fst ⁻¹' s from hx) q · simp only [Set.indicator_of_notMem hx] have hz (y : Y) : ((Prod.fst ⁻¹' s).indicator q) (x,y) = 0 := Set.indicator_of_notMem (show (x,y) ∉ Prod.fst ⁻¹' s from hx) q simp only [hz,lintegral_zero] lemma continuous_slowMarginal {A : UnitAddCircle × UnitAddCircle → ℂ} (hA : Continuous A) : Continuous (fun u => ∫ a, ‖A (u,a)‖^2 ∂AddCircle.haarAddCircle) := by have hc : Continuous (uncurry (fun (u a : UnitAddCircle) => ‖A (u,a)‖^2)) := hA.norm.pow 2 simpa only [setIntegral_univ] using continuous_parametric_integral_of_continuous (X := UnitAddCircle) (Y := UnitAddCircle) (μ := AddCircle.haarAddCircle) hc isCompact_univ lemma packetSpectralMeasure_marginal {φ : UnitAddCircle → UnitAddCircle} {A : UnitAddCircle × UnitAddCircle → ℂ} {m : ℤ} (hφ : Measurable φ) (hA : Continuous A) : packetSpectralMeasure φ m A = (AddCircle.haarAddCircle.withDensity (fun u => ENNReal.ofReal (∫ a, ‖A (u,a)‖^2 ∂AddCircle.haarAddCircle))).map (fun u => m • φ u) := by unfold packetSpectralMeasure slowVolume change Measure.map ((fun u => m • φ u) ∘ Prod.fst) _ = _ rw [← Measure.map_map (show Measurable (fun u => m • φ u) from hφ.const_smul m) measurable_fst, map_fst_withDensity_marginal (q := fun p => ENNReal.ofReal (‖A p‖^2)) (by simpa only [Pi.pow_apply] using (hA.norm.pow 2).measurable.ennreal_ofReal)] congr 2 funext u exact (ofReal_integral_eq_lintegral_ofReal (((hA.comp (continuous_const.prodMk continuous_id)).norm.pow 2).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) (ae_of_all _ (fun a => sq_nonneg _))).symm lemma branchDensity_integral {Ω : Type*} [MeasurableSpace Ω] {ν : Measure Ω} (I : Set ℝ) (J g D : ℝ → ℝ) (q : ℝ × Ω → ℝ) (s : ℝ) : branchDensity I J g D (fun u => ∫ a, q (u,a) ∂ν) s = ∫ a, branchDensity I J g D (fun u => q (u,a)) s ∂ν := by classical unfold branchDensity by_cases hs : s ∈ J '' I · simp only [Set.indicator_of_mem hs,integral_div] · simp only [Set.indicator_of_notMem hs,integral_zero] /-- An arbitrary smooth/continuous old packet has the stationary marginal as its temporal density. Spare-coordinate independence is not required. -/ theorem oldPacket_spectralDensity {A : UnitAddCircle × UnitAddCircle → ℂ} {A₀ : UnitAddCircle → ℝ → ℂ} {v : UnitAddCircle → ℝ} {V : ℝ → ℝ} {m c : ℤ} {o : ℝ} {C I : Set ℝ} (hA : Continuous A) (hA₀ : Continuous A₀.uncurry) (hAe : ∀ a x, x ∈ Ioc o (o+1) → A ((x:UnitAddCircle),a) = A₀ a x) (has : ∀ a, support (A₀ a) ⊆ Ioc o (o+1)) (hAs : ∀ a, support (A₀ a) ⊆ I) (hv : Measurable v) (hV : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) V) (hVe : ∀ x ∈ I, v (x:UnitAddCircle) = V x) (hIo : IsOpen I) (hIC : I ⊆ C) (hinj : InjOn (fun x => (m:ℝ)*V x) C) (hd : ∀ x ∈ I, deriv (fun x => (m:ℝ)*V x) x ≠ 0) : packetSpectralMeasure (fun a => ((v a-c:ℝ):UnitAddCircle)) m A = AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal (foldCircleDensity (fun s => ∫ a, stationaryWeight C (fun x => (m:ℝ)*V x) (A₀ a) s ∂AddCircle.haarAddCircle) z)) := by let q : UnitAddCircle → ℝ := fun u => ∫ a, ‖A (u,a)‖^2 ∂AddCircle.haarAddCircle let q₀ : ℝ → ℝ := fun x => ∫ a, ‖A₀ a x‖^2 ∂AddCircle.haarAddCircle have hqc : Continuous q := continuous_slowMarginal hA have hq₀c : Continuous q₀ := by have hc : Continuous (uncurry (fun (x : ℝ) (a : UnitAddCircle) => ‖A₀ a x‖^2)) := (hA₀.comp continuous_swap).norm.pow 2 simpa only [q₀,setIntegral_univ] using continuous_parametric_integral_of_continuous (X := ℝ) (Y := UnitAddCircle) (μ := AddCircle.haarAddCircle) hc isCompact_univ have hs (K : Set ℝ) (hK : ∀ a, support (A₀ a) ⊆ K) : support q₀ ⊆ K := by intro x hx by_contra hnot apply hx have hz (a : UnitAddCircle) : A₀ a x = 0 := by by_contra hn exact hnot (hK a hn) simp only [q₀,hz,norm_zero,zero_pow (by decide : 2 ≠ 0),integral_zero] have hmap := packetSpectralMeasure_marginal (m := m) (A := A) (φ := fun a => (((v a-(c:ℝ)) : ℝ):UnitAddCircle)) (AddCircle.measurable_mk'.comp (hv.sub measurable_const)) hA rw [hmap] have hmode (a : UnitAddCircle) : m • ((v a-c:ℝ):UnitAddCircle) = (((m:ℝ)*v a:ℝ):UnitAddCircle) := by rw [← AddCircle.coe_zsmul] simp only [zsmul_eq_mul] rw [mul_sub] have he : ((m:ℝ)*(c:ℝ):UnitAddCircle) = 0 := by rw [← Int.cast_mul] exact (AddCircle.coe_eq_zero_iff 1).mpr ⟨m*c,by simp⟩ rw [AddCircle.coe_sub,he,sub_zero] simp_rw [hmode] have hb : branchDensity I (fun x => (m:ℝ)*V x) (invFunOn (fun x => (m:ℝ)*V x) I) (deriv (fun x => (m:ℝ)*V x)) q₀ = fun s => ∫ a, stationaryWeight C (fun x => (m:ℝ)*V x) (A₀ a) s ∂AddCircle.haarAddCircle := by funext s rw [show q₀ = fun x => ∫ a, ‖A₀ a x‖^2 ∂AddCircle.haarAddCircle from rfl, branchDensity_integral (q := fun p : ℝ × UnitAddCircle => ‖A₀ p.2 p.1‖^2)] apply integral_congr_ae filter_upwards [] with a exact congrFun (stationaryWeight_change_branch hIC hinj (hAs a)).symm s rw [← hb] exact positive_circle_branch_fold (hqc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) hq₀c.measurable (fun _ => integral_nonneg (fun _ => sq_nonneg _)) (fun _ => integral_nonneg (fun _ => sq_nonneg _)) (fun x hx => integral_congr_ae (ae_of_all _ (fun a => by dsimp only; rw [hAe a x hx]))) (hs _ has) (hs _ hAs) (hv.const_mul (m:ℝ)) (contDiff_const.mul hV) (fun x hx => by rw [hVe x hx]) hIo (hinj.mono hIC) hd end ThreeTorus end end section /-! The unchanged old spectral marginal, as a signed pushforward, for applying L¹ contraction to physical cutoffs. -/ noncomputable section open MeasureTheory Set Function Filter open scoped Topology BigOperators ENNReal namespace ThreeTorus lemma signed_circle_branch_fold {q : UnitAddCircle → ℝ} {q₀ J₀ : ℝ → ℝ} {J : UnitAddCircle → ℝ} {o : ℝ} {I : Set ℝ} (hq : Integrable q AddCircle.haarAddCircle) (hqm : Measurable q₀) (hq₀ : ∀ x ∈ Ioc o (o+1), q (x:UnitAddCircle) = q₀ x) (hq₀s : support q₀ ⊆ Ioc o (o+1)) (hqs : support q₀ ⊆ I) (hj : Measurable 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) : Integrable (foldCircleDensity (branchDensity I J₀ (invFunOn J₀ I) (deriv J₀) q₀)) AddCircle.haarAddCircle ∧ (AddCircle.haarAddCircle.withDensityᵥ q).map (fun a => (J a:UnitAddCircle)) = AddCircle.haarAddCircle.withDensityᵥ (foldCircleDensity (branchDensity I J₀ (invFunOn J₀ I) (deriv J₀) q₀)) := by let w := branchDensity I J₀ (invFunOn J₀ I) (deriv J₀) q₀ have hqi : Integrable q₀ := by rw [← circleDensityLift_eq hq₀ hq₀s] exact integrable_circleDensityLift hq o have hD : ∀ x ∈ I, HasDerivWithinAt J₀ (deriv J₀ x) I x := fun x _ => (hJ.differentiable (by simp) x).hasDerivAt.hasDerivWithinAt have hIm := (isOpen_image_of_deriv_ne_zero hJ hIo hd).measurableSet have hwi : Integrable w := integrable_branchDensity hIo.measurableSet hIm hD hinj (fun _ hx => hinj.leftInvOn_invFunOn hx) hd hqi.integrableOn have hwm : Measurable w := measurable_branchDensity_smooth hJ hIo hinj hd hqm have hmap := circle_branch_pushforward hq hq₀ hq₀s hqs hj hJ.continuous.measurable hJe hIo.measurableSet hIm hD hinj (fun _ hx => hinj.leftInvOn_invFunOn hx) hd refine ⟨integrable_foldCircleDensity hwm hwi,?_⟩ rw [← foldCircleDensity_pushforward hwm hwi,← hmap] exact (vectorMeasure_map_comp _ hj AddCircle.measurable_mk').symm theorem oldStationaryMarginal_map {A : UnitAddCircle × UnitAddCircle → ℂ} {A₀ : UnitAddCircle → ℝ → ℂ} {v : UnitAddCircle → ℝ} {V : ℝ → ℝ} {m : ℤ} {o : ℝ} {C I : Set ℝ} (hA : Continuous A) (hA₀ : Continuous A₀.uncurry) (hAe : ∀ a x, x ∈ Ioc o (o+1) → A ((x:UnitAddCircle),a) = A₀ a x) (has : ∀ a, support (A₀ a) ⊆ Ioc o (o+1)) (hAs : ∀ a, support (A₀ a) ⊆ I) (hv : Measurable v) (hV : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) V) (hVe : ∀ x ∈ I, v (x:UnitAddCircle) = V x) (hIo : IsOpen I) (hIC : I ⊆ C) (hinj : InjOn (fun x => (m:ℝ)*V x) C) (hd : ∀ x ∈ I, deriv (fun x => (m:ℝ)*V x) x ≠ 0) : let p := foldCircleDensity (fun s => ∫ a, stationaryWeight C (fun x => (m:ℝ)*V x) (A₀ a) s ∂AddCircle.haarAddCircle) Integrable p AddCircle.haarAddCircle ∧ (AddCircle.haarAddCircle.withDensityᵥ (fun u => ∫ a, ‖A (u,a)‖^2 ∂AddCircle.haarAddCircle)).map (fun u => (((m:ℝ)*v u : ℝ):UnitAddCircle)) = AddCircle.haarAddCircle.withDensityᵥ p := by dsimp only let q : UnitAddCircle → ℝ := fun u => ∫ a, ‖A (u,a)‖^2 ∂AddCircle.haarAddCircle let q₀ : ℝ → ℝ := fun x => ∫ a, ‖A₀ a x‖^2 ∂AddCircle.haarAddCircle have hqc : Continuous q := continuous_slowMarginal hA have hq₀c : Continuous q₀ := by have hc : Continuous (uncurry (fun (x : ℝ) (a : UnitAddCircle) => ‖A₀ a x‖^2)) := (hA₀.comp continuous_swap).norm.pow 2 simpa only [q₀,setIntegral_univ] using continuous_parametric_integral_of_continuous (X := ℝ) (Y := UnitAddCircle) (μ := AddCircle.haarAddCircle) hc isCompact_univ have hs (K : Set ℝ) (hK : ∀ a, support (A₀ a) ⊆ K) : support q₀ ⊆ K := by intro x hx by_contra hnot apply hx have hz (a : UnitAddCircle) : A₀ a x = 0 := by by_contra hn exact hnot (hK a hn) simp only [q₀,hz,norm_zero,zero_pow (by decide : 2 ≠ 0),integral_zero] have hb : branchDensity I (fun x => (m:ℝ)*V x) (invFunOn (fun x => (m:ℝ)*V x) I) (deriv (fun x => (m:ℝ)*V x)) q₀ = fun s => ∫ a, stationaryWeight C (fun x => (m:ℝ)*V x) (A₀ a) s ∂AddCircle.haarAddCircle := by funext s rw [show q₀ = fun x => ∫ a, ‖A₀ a x‖^2 ∂AddCircle.haarAddCircle from rfl, branchDensity_integral (q := fun p : ℝ × UnitAddCircle => ‖A₀ p.2 p.1‖^2)] apply integral_congr_ae filter_upwards [] with a exact congrFun (stationaryWeight_change_branch hIC hinj (hAs a)).symm s rw [← hb] exact signed_circle_branch_fold (hqc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) hq₀c.measurable (fun x hx => integral_congr_ae (ae_of_all _ (fun a => by dsimp only; rw [hAe a x hx]))) (hs _ has) (hs _ hAs) (hv.const_mul (m:ℝ)) (contDiff_const.mul hV) (fun x hx => by rw [hVe x hx]) hIo (hinj.mono hIC) hd end ThreeTorus end end section /-! The neutral passage's old cutoff reference differs from the actual old packet density by at most its physical missing mass. -/ noncomputable section open MeasureTheory Set Function Filter open scoped Topology BigOperators namespace ThreeTorus lemma cutoff_slowMarginal_l1 {A : UnitAddCircle × UnitAddCircle → ℂ} {ψ : UnitAddCircle → ℝ} (hA : Continuous A) (hψc : Continuous ψ) (hψ : ∀ a, 0 ≤ ψ a ∧ ψ a ≤ 1) {B : ℝ} (_hB : 0 ≤ B) (hAb : ∀ p, ‖A p‖^2 ≤ B) : (∫ u, |(∫ a, ‖A (u,a)‖^2 ∂AddCircle.haarAddCircle)- (∫ a, ‖(ψ a:ℂ)*A (u,a)‖^2 ∂AddCircle.haarAddCircle)| ∂AddCircle.haarAddCircle) ≤ B*(∫ a, (1-ψ a^2) ∂AddCircle.haarAddCircle) := by let D := B*(∫ a, (1-ψ a^2) ∂AddCircle.haarAddCircle) have hd : ∀ u, |(∫ a, ‖A (u,a)‖^2 ∂AddCircle.haarAddCircle)- (∫ a, ‖(ψ a:ℂ)*A (u,a)‖^2 ∂AddCircle.haarAddCircle)| ≤ D := by intro u have ha : Continuous (fun a => ‖A (u,a)‖^2) := (hA.comp (continuous_const.prodMk continuous_id)).norm.pow 2 have hap : Continuous (fun a => ‖(ψ a:ℂ)*A (u,a)‖^2) := ((Complex.continuous_ofReal.comp hψc).mul (hA.comp (continuous_const.prodMk continuous_id))).norm.pow 2 have hc : Continuous (fun a => ‖A (u,a)‖^2-‖(ψ a:ℂ)*A (u,a)‖^2) := ha.sub hap have he (a : UnitAddCircle) : ‖A (u,a)‖^2-‖(ψ a:ℂ)*A (u,a)‖^2 = (1-ψ a^2)*‖A (u,a)‖^2 := by simp only [norm_mul,mul_pow,Complex.norm_real,Real.norm_eq_abs,sq_abs] ring have hz (a : UnitAddCircle) : 0 ≤ ‖A (u,a)‖^2-‖(ψ a:ℂ)*A (u,a)‖^2 := by rw [he] exact mul_nonneg (by nlinarith [(hψ a).1,(hψ a).2]) (sq_nonneg _) rw [← integral_sub (ha.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) (hap.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)), abs_of_nonneg (integral_nonneg hz)] have hbound : Continuous (fun a => B*(1-ψ a^2)) := continuous_const.mul (continuous_const.sub (hψc.pow 2)) calc _ ≤ ∫ a, B*(1-ψ a^2) ∂AddCircle.haarAddCircle := by apply integral_mono (hc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) (hbound.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) intro a dsimp only rw [he,mul_comm B] exact mul_le_mul_of_nonneg_left (hAb _) (by nlinarith [(hψ a).1,(hψ a).2]) _ = D := by rw [integral_const_mul] calc _ ≤ ∫ _ : UnitAddCircle, D ∂AddCircle.haarAddCircle := by exact integral_mono_of_nonneg (ae_of_all _ (fun _ => abs_nonneg _)) (integrable_const D) (ae_of_all _ hd) _ = _ := by simp [D] /-- The reference in the neutral Riemann estimate is genuinely the cutoff old spectrum. Its loss is bounded independently of slope, residue modulus or passage integer. -/ theorem oldStationaryMarginal_cutoff_l1 {A : UnitAddCircle × UnitAddCircle → ℂ} {A₀ : UnitAddCircle → ℝ → ℂ} {v : UnitAddCircle → ℝ} {V : ℝ → ℝ} {m : ℤ} {o : ℝ} {C I : Set ℝ} (hA : Continuous A) (hA₀ : Continuous A₀.uncurry) (hAe : ∀ a x, x ∈ Ioc o (o+1) → A ((x:UnitAddCircle),a) = A₀ a x) (has : ∀ a, support (A₀ a) ⊆ Ioc o (o+1)) (hAs : ∀ a, support (A₀ a) ⊆ I) (hv : Measurable v) (hV : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) V) (hVe : ∀ x ∈ I, v (x:UnitAddCircle) = V x) (hIo : IsOpen I) (hIC : I ⊆ C) (hinj : InjOn (fun x => (m:ℝ)*V x) C) (hd : ∀ x ∈ I, deriv (fun x => (m:ℝ)*V x) x ≠ 0) {ψ : UnitAddCircle → ℝ} (hψc : Continuous ψ) (hψ : ∀ a, 0 ≤ ψ a ∧ ψ a ≤ 1) {B : ℝ} (hB : 0 ≤ B) (hAb : ∀ p, ‖A p‖^2 ≤ B) : let p := foldCircleDensity (fun s => ∫ a, stationaryWeight C (fun x => (m:ℝ)*V x) (A₀ a) s ∂AddCircle.haarAddCircle) let pψ := foldCircleDensity (fun s => ∫ a, stationaryWeight C (fun x => (m:ℝ)*V x) (fun x => (ψ a:ℂ)*A₀ a x) s ∂AddCircle.haarAddCircle) (∫ z, |pψ z-p z| ∂AddCircle.haarAddCircle) ≤ B*(∫ a, (1-ψ a^2) ∂AddCircle.haarAddCircle) := by dsimp only let Aψ : UnitAddCircle × UnitAddCircle → ℂ := fun p => (ψ p.2:ℂ)*A p let A₀ψ : UnitAddCircle → ℝ → ℂ := fun a x => (ψ a:ℂ)*A₀ a x have hAψ : Continuous Aψ := ((Complex.continuous_ofReal.comp hψc).comp continuous_snd).mul hA have hA₀ψ : Continuous A₀ψ.uncurry := ((Complex.continuous_ofReal.comp hψc).comp continuous_fst).mul hA₀ have hs (K : Set ℝ) (hK : ∀ a, support (A₀ a) ⊆ K) : ∀ a, support (A₀ψ a) ⊆ K := by intro a x hx exact hK a (right_ne_zero_of_mul hx) have he : ∀ a x, x ∈ Ioc o (o+1) → Aψ ((x:UnitAddCircle),a) = A₀ψ a x := by intro a x hx dsimp only [Aψ,A₀ψ] rw [hAe a x hx] obtain ⟨hpi,hmap⟩ := oldStationaryMarginal_map hA hA₀ hAe has hAs hv hV hVe hIo hIC hinj hd obtain ⟨hψi,hψmap⟩ := oldStationaryMarginal_map hAψ hA₀ψ he (hs _ has) (hs _ hAs) hv hV hVe hIo hIC hinj hd have hq := continuous_slowMarginal hA have hqψ := continuous_slowMarginal hAψ have hb := l1_pushforward_difference (hqψ.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) (hq.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) hψi hpi (AddCircle.measurable_mk'.comp (hv.const_mul (m:ℝ))) hψmap hmap refine hb.trans ?_ simp only [Aψ,abs_sub_comm] exact cutoff_slowMarginal_l1 hA hψc hψ hB hAb end ThreeTorus end end section /-! Quantitative physical and spectral conclusions for the same actual packet. This is the analytic (not yet geometric finite-stage) transmission boundary. -/ noncomputable section open MeasureTheory Set Function Filter open scoped Topology BigOperators FourierTransform namespace ThreeTorus lemma sq_norm_sub_le_two (x y z : ℂ) : ‖x-z‖^2 ≤ 2*‖x-y‖^2+2*‖y-z‖^2 := by have hh : ‖x-z‖ ≤ ‖x-y‖+‖y-z‖ := by convert norm_add_le (x-y) (y-z) using 1 congr 1 ring have h0 := norm_nonneg (x-z) have h1 := norm_nonneg (x-y) have h2 := norm_nonneg (y-z) nlinarith [sq_nonneg (‖x-y‖-‖y-z‖)] lemma continuous_passageChart {H : UnitAddCircle → UnitAddCircle} (hH : Continuous H) : Continuous (passageChart H) := by change Continuous (fun x : TripleTorus => (x.2.1,x.2.2,x.1+(-H x.2.1))) exact continuous_snd.fst.prodMk (continuous_snd.snd.prodMk (continuous_fst.add (hH.comp continuous_snd.fst).neg)) lemma continuous_retainedPassage {ψ : UnitAddCircle → ℝ} {χ : ℝ → UnitAddCircle → ℝ} {m : ℤ} {H : ℕ → UnitAddCircle → UnitAddCircle} {A : UnitAddCircle × UnitAddCircle → ℂ} (hψ : Continuous ψ) (hχ : Continuous χ.uncurry) (hH : ∀ n, Continuous (H n)) (hA : Continuous A) (n : ℕ) (S : Finset ℤ) : Continuous (retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) ψ) := by change Continuous (fun x => ∑ j ∈ S, packet j (retainedAmplitude ψ χ m H A n j) x) exact continuous_finsetSum _ (fun j _ => continuous_packet (continuous_retainedAmplitude hψ hχ hH hA n j)) lemma cutoff_packet_error_eq {A : UnitAddCircle × UnitAddCircle → ℂ} (hmod : ∀ u a, ‖A (u,a)‖ = ‖A (u,0)‖) (ψ : UnitAddCircle → ℝ) (m : ℤ) : (∫ x : TripleTorus, ‖packet m A x-(ψ x.2.2:ℂ)*packet m A x‖^2 ∂tripleVolume) = (∫ u, ‖A (u,0)‖^2 ∂AddCircle.haarAddCircle)* (∫ a, (1-ψ a)^2 ∂AddCircle.haarAddCircle) := by have he (x : TripleTorus) : ‖packet m A x-(ψ x.2.2:ℂ)*packet m A x‖^2 = ‖A (x.2.1,0)‖^2*(1-ψ x.2.2)^2 := by rw [show packet m A x-(ψ x.2.2:ℂ)*packet m A x = (1-(ψ x.2.2:ℂ))*packet m A x by ring] simp only [packet,norm_mul,fourier_apply,Circle.norm_coe,mul_one] rw [← Complex.ofReal_one,← Complex.ofReal_sub,Complex.norm_real,Real.norm_eq_abs, mul_pow,sq_abs,hmod] ring simp_rw [he] change (∫ x : UnitAddCircle × (UnitAddCircle × UnitAddCircle), ‖A (x.2.1,0)‖^2*(1-ψ x.2.2)^2 ∂AddCircle.haarAddCircle.prod slowVolume) = _ rw [integral_fun_snd (μ := AddCircle.haarAddCircle) (ν := slowVolume) (fun p : UnitAddCircle × UnitAddCircle => ‖A (p.1,0)‖^2*(1-ψ p.2)^2)] simpa only [probReal_univ,one_smul] using (integral_prod_mul (μ := AddCircle.haarAddCircle) (ν := AddCircle.haarAddCircle) (fun u => ‖A (u,0)‖^2) (fun a => (1-ψ a)^2)) lemma cutoff_packet_error_le {A : UnitAddCircle × UnitAddCircle → ℂ} (hmod : ∀ u a, ‖A (u,a)‖ = ‖A (u,0)‖) {δ : ℝ} (c : LawCutoffs δ) (hδ : 0 < δ) (hδ' : δ < 1/8) (m : ℤ) : (∫ x : TripleTorus, ‖packet m A x-(neutralCircleCutoff c x.2.2:ℂ)*packet m A x‖^2 ∂tripleVolume) ≤ (∫ u, ‖A (u,0)‖^2 ∂AddCircle.haarAddCircle)*(4*δ) := by rw [cutoff_packet_error_eq hmod] apply mul_le_mul_of_nonneg_left _ (integral_nonneg (fun _ => sq_nonneg _)) have hc := continuous_neutralCircleCutoff c have hi : Integrable (fun a => (1-neutralCircleCutoff c a^2)) AddCircle.haarAddCircle := (continuous_const.sub (hc.pow 2)).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) calc _ ≤ ∫ a, (1-neutralCircleCutoff c a^2) ∂AddCircle.haarAddCircle := by apply integral_mono_of_nonneg (ae_of_all _ (fun _ => sq_nonneg _)) hi exact ae_of_all _ (fun a => by have h := neutralCircleCutoff_bounds c hδ a nlinarith) _ ≤ _ := neutralCircleCutoff_error c hδ hδ' /-- Removing the good-arc cutoff adds no assumed physical approximation. Its missing mass is explicitly paid. -/ lemma uncut_retainedPassage_error_le {A : UnitAddCircle × UnitAddCircle → ℂ} {m : ℤ} {H : ℕ → UnitAddCircle → UnitAddCircle} {χ : ℝ → UnitAddCircle → ℝ} (hA : Continuous A) (hH : ∀ n, Continuous (H n)) (hχ : Continuous χ.uncurry) (hmod : ∀ u a, ‖A (u,a)‖ = ‖A (u,0)‖) {δ : ℝ} (c : LawCutoffs δ) (hδ : 0 < δ) (hδ' : δ < 1/8) (n : ℕ) (S : Finset ℤ) : (∫ x : TripleTorus, ‖packet m A x-retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) (neutralCircleCutoff c) (passageChart (H n) x)‖^2 ∂tripleVolume) ≤ 8*δ*(∫ u, ‖A (u,0)‖^2 ∂AddCircle.haarAddCircle)+ 2*(∫ x : TripleTorus, ‖(neutralCircleCutoff c x.2.2:ℂ)*packet m A x- retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) (neutralCircleCutoff c) (passageChart (H n) x)‖^2 ∂tripleVolume) := by let f := packet m A let g := fun x : TripleTorus => (neutralCircleCutoff c x.2.2:ℂ)*f x let h := fun x : TripleTorus => retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) (neutralCircleCutoff c) (passageChart (H n) x) have hf : Continuous f := continuous_packet hA have hg : Continuous g := (Complex.continuous_ofReal.comp ((continuous_neutralCircleCutoff c).comp continuous_snd.snd)).mul hf have hh : Continuous h := (continuous_retainedPassage (continuous_neutralCircleCutoff c) hχ hH hA n S).comp (continuous_passageChart (hH n)) have hi (v w : TripleTorus → ℂ) (hv : Continuous v) (hw : Continuous w) : Integrable (fun x => ‖v x-w x‖^2) tripleVolume := ((hv.sub hw).norm.pow 2).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) have hfg := hi f g hf hg have hgh := hi g h hg hh calc _ ≤ ∫ x, (2*‖f x-g x‖^2+2*‖g x-h x‖^2) ∂tripleVolume := by apply integral_mono (hi f h hf hh) ((hfg.const_mul 2).add (hgh.const_mul 2)) intro x exact sq_norm_sub_le_two (f x) (g x) (h x) _ = 2*(∫ x, ‖f x-g x‖^2 ∂tripleVolume)+2*(∫ x, ‖g x-h x‖^2 ∂tripleVolume) := by rw [integral_add (hfg.const_mul 2) (hgh.const_mul 2),integral_const_mul,integral_const_mul] _ ≤ _ := by have hh := cutoff_packet_error_le hmod c hδ hδ' m change (∫ x, ‖f x-g x‖^2 ∂tripleVolume) ≤ _ at hh dsimp only [g,h] nlinarith /-- A single actual quantile passage simultaneously satisfies the physical, spectral, smoothness and all bilateral-moment conclusions. The passage integer remains arbitrarily large after the law, cutoffs and amplitudes are fixed. -/ theorem exact_quantile_passage {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)) (hmod : ∀ a x, ‖A₀ a x‖ = ‖A₀ 0 x‖) (hmodA : ∀ u a, ‖A (u,a)‖ = ‖A (u,0)‖) (hAs : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => A (p.1,p.2))) {ρ : ℝ → ℝ} (hρ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) ρ) (hi : Integrable ρ) (hp : ∀ y, 0 < ρ y) (hρm : ∫ y, ρ y = 1) {C : ℝ} (hC : 0 ≤ C) (hb : ∀ y, |ρ y| ≤ C/(1+y^2)) {D : ℝ} (hF : ∀ v, D < |v| → 𝓕 (fun y => (ρ y : ℂ)) v = 0) {δ : ℝ} (c : LawCutoffs δ) (hδ : 0 < δ) (hδ' : δ < 1/8) {χ : ℝ → UnitAddCircle → ℝ} (hχs : CylinderSmooth χ.uncurry) (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) {g : UnitAddCircle → ℝ} (hg : Continuous g) {ε : ℝ} (hideal : ∀ s ∈ support (stationaryWeight (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) (A₀ 0)), ∀ y, |foldCircleDensity (scaledDensity ρ s) y-g y| ≤ ε) {η : ℝ} (hη : 0 < η) : let w := stationaryWeight (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) (A₀ 0) ∃ S : ℕ → Finset ℤ, (∀ n, 0 < n → ∀ j ∈ S n, j ≠ 0) ∧ (∀ n, 0 < n → ∀ j ∉ S n, ∀ a, χ ((j:ℝ)/n) a = 0) ∧ ∀ᶠ n : ℕ in atTop, let ψ := neutralCircleCutoff c let J := quantileCircleJitter ρ c let φ := fun a => (((1+J a/n)/n:ℝ):UnitAddCircle) let f := retainedPassage m (H n) A (S n) (fun j => χ ((j:ℝ)/n)) ψ let p := exactRetainedPacketDensity (Ioo δ (1-δ)) (lawJitter ρ c) ψ χ m H A (S n) n (∫ x : TripleTorus, ‖packet m A x-f (passageChart (H n) x)‖^2 ∂tripleVolume) < 8*δ*(∫ u, ‖A (u,0)‖^2 ∂AddCircle.haarAddCircle)+η ∧ (∫ z, |p z-foldCircleDensity w z*g ((n:ℤ) • z)| ∂AddCircle.haarAddCircle) < (∫ s, w s)*(4*δ+ε)+η ∧ ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => p x) ∧ (∀ k : ℤ, (∫ x : TripleTorus, starRingEnd ℂ (f x)*f (twistIter φ k x) ∂tripleVolume) = ∫ z, (p z:ℂ)*fourier k z ∂AddCircle.haarAddCircle) ∧ (∀ j u a, ‖retainedAmplitude ψ χ m H A n j (u,a)‖ = ‖retainedAmplitude ψ χ m H A n j (u,0)‖) := by dsimp only obtain ⟨S,hSN,hST,hsp⟩ := exact_quantile_circle_spectral_estimate hm hH hA hA₀ hAc hAd hAr hsupp hlo huo hlu hR hPr hV hPV hdV h0 hHc hAcont hΔt hmod hρ hi hp hρm hC hb hF c hδ hδ' hχc hcχ hzχ hχ hχW hg hideal hη refine ⟨S,hSN,hST,?_⟩ have hψc := continuous_neutralCircleCutoff c have hph := exact_physical_retainedPassage_tendsto hm hH hA hA₀ hAc hAd hsupp hlo huo hlu hR hPr hV hPV hdV hHc hAcont hΔt hχc hχ (neutralCircleCutoff c) hψc (neutralCircleCutoff_bounds c hδ) hχW S hST have hJs := lawJitter_smooth c hρ hi hp hρm have hJinj := lawJitter_injOn c hρ.continuous hi hp hρm hδ have hJd : ∀ x ∈ Ioo δ (1-δ), deriv (lawJitter ρ c) x ≠ 0 := by intro x hx obtain ⟨hd,hdp⟩ := lawJitter_deriv_pos c hρ hi hp hρm hδ hx rw [hd.deriv] exact hdp.ne' 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), neutralCircleCutoff c (x:UnitAddCircle) = c.ψ x := by intro x hx exact lawPsi_eq c hδ ⟨hx.1.le,by simpa using hx.2⟩ have hψsupport : ∀ x ∈ Ioc (0:ℝ) 1, neutralCircleCutoff c (x:UnitAddCircle) ≠ 0 → x ∈ Ioo δ (1-δ) := by intro x hx hn rw [heψ x (by simpa using hx)] at hn exact (hψs (subset_tsupport c.ψ hn)).2 have hHl : ∀ n : ℕ, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x => (n:ℝ)*(Pr x+Δ n*x)) := fun _ => contDiff_const.mul (hPr.add (contDiff_const.mul contDiff_id)) filter_upwards [hsp,hph.eventually (eventually_lt_nhds (half_pos hη)), eventually_gt_atTop (0:ℕ)] with n hnsp hnph hn refine ⟨?_,hnsp,?_,?_,?_⟩ · have hh := uncut_retainedPassage_error_le (m := m) hAcont hHc hχc hmodA c hδ hδ' n (S n) exact hh.trans_lt (by linarith) · exact exactRetainedPacketDensity_smooth_lift hJs isOpen_Ioo hJinj hJd c.ψ_smooth c.ψ_compact heψ hψs hχs hHl hH hAs hn (hSN n hn) · intro k exact retainedPassage_spectral_moment hψc hχc hHc hAcont hψsupport (continuous_quantileCircleJitter c hρ hi hp hρm) hJs (fun x _ => quantileCircleJitter_coe ρ c x) isOpen_Ioo hJinj hJd hn (hSN n hn) k · exact fun _ _ _ => retainedAmplitude_modulus _ _ _ _ _ _ _ _ _ end ThreeTorus end end section /-! Quantitative good-arc cutoff losses for genuine two-coordinate amplitudes. This does not presume spare-coordinate independence. -/ noncomputable section open MeasureTheory Set Function Filter open scoped Topology BigOperators namespace ThreeTorus lemma continuous_stationaryWeight_parameter {Ω : Type*} [TopologicalSpace Ω] {A : Ω → ℝ → ℂ} (hA : Continuous A.uncurry) (C : Set ℝ) (J : ℝ → ℝ) (s : ℝ) : Continuous (fun a => stationaryWeight C J (A a) s) := by classical unfold stationaryWeight branchDensity by_cases hs : s ∈ J '' C · simp only [Set.indicator_of_mem hs] exact ((hA.comp (continuous_id.prodMk continuous_const)).norm.pow 2).div_const _ · simp only [Set.indicator_of_notMem hs] exact continuous_const lemma stationaryCutoffLift_marginal {A : UnitAddCircle → ℝ → ℂ} {ψ : UnitAddCircle → ℝ} {ψ₀ : ℝ → ℝ} {o : ℝ} (heψ : ∀ x ∈ Ioc o (o+1), ψ₀ x = ψ (x:UnitAddCircle)) (hsψ : support ψ₀ ⊆ Ioc o (o+1)) (C : Set ℝ) (J : ℝ → ℝ) (s : ℝ) : (∫ a : ℝ, stationaryCutoffLift C J (fun p : ℝ × ℝ => A (p.2:UnitAddCircle) p.1) ψ₀ (a,s)) = ∫ a, stationaryWeight C J (fun x => (ψ a:ℂ)*A a x) s ∂AddCircle.haarAddCircle := by let q : UnitAddCircle → ℝ := fun a => ψ a^2*stationaryWeight C J (A a) s have he : circleDensityLift o q = fun a => stationaryCutoffLift C J (fun p : ℝ × ℝ => A (p.2:UnitAddCircle) p.1) ψ₀ (a,s) := by apply circleDensityLift_eq · intro x hx simp only [q,stationaryCutoffLift,heψ x hx] · intro x hx apply hsψ intro hz exact hx (by simp [stationaryCutoffLift,hz]) rw [← he,circle_integral_lift] apply integral_congr_ae filter_upwards [] with a rw [stationaryWeight_mul] simp [q,Complex.norm_real,Real.norm_eq_abs,sq_abs] lemma packet_cutoff_error_uniform {A : UnitAddCircle × UnitAddCircle → ℂ} {ψ : UnitAddCircle → ℝ} (hψc : Continuous ψ) (hψ : ∀ a, 0 ≤ ψ a ∧ ψ a ≤ 1) {B : ℝ} (hB : 0 ≤ B) (hAb : ∀ p, ‖A p‖^2 ≤ B) (m : ℤ) : (∫ x : TripleTorus, ‖packet m A x-(ψ x.2.2:ℂ)*packet m A x‖^2 ∂tripleVolume) ≤ B*(∫ a, (1-ψ a^2) ∂AddCircle.haarAddCircle) := by have hc : Continuous (fun x : TripleTorus => B*(1-ψ x.2.2^2)) := continuous_const.mul (continuous_const.sub ((hψc.comp continuous_snd.snd).pow 2)) have hi := hc.integrable_of_hasCompactSupport (μ := tripleVolume) (HasCompactSupport.of_compactSpace _) calc _ ≤ ∫ x : TripleTorus, B*(1-ψ x.2.2^2) ∂tripleVolume := by apply integral_mono_of_nonneg (ae_of_all _ (fun _ => sq_nonneg _)) hi filter_upwards [] with x have he : ‖packet m A x-(ψ x.2.2:ℂ)*packet m A x‖^2 = ‖A (x.2.1,x.2.2)‖^2*(1-ψ x.2.2)^2 := by rw [show packet m A x-(ψ x.2.2:ℂ)*packet m A x = (1-(ψ x.2.2:ℂ))*packet m A x by ring] simp only [packet,norm_mul,fourier_apply,Circle.norm_coe,mul_one] rw [← Complex.ofReal_one,← Complex.ofReal_sub,Complex.norm_real,Real.norm_eq_abs, mul_pow,sq_abs] ring rw [he] have h := hψ x.2.2 calc _ ≤ B*(1-ψ x.2.2)^2 := mul_le_mul_of_nonneg_right (hAb _) (sq_nonneg _) _ ≤ _ := mul_le_mul_of_nonneg_left (by nlinarith) hB _ = _ := by change (∫ x : UnitAddCircle × (UnitAddCircle × UnitAddCircle), B*(1-ψ x.2.2^2) ∂AddCircle.haarAddCircle.prod slowVolume) = _ rw [integral_fun_snd (μ := AddCircle.haarAddCircle (T := (1:ℝ))) (ν := slowVolume) (fun p : UnitAddCircle × UnitAddCircle => B*(1-ψ p.2^2))] simp only [probReal_univ,one_smul] change (∫ x : UnitAddCircle × UnitAddCircle, B*(1-ψ x.2^2) ∂AddCircle.haarAddCircle.prod AddCircle.haarAddCircle) = _ rw [integral_fun_snd (μ := AddCircle.haarAddCircle (T := (1:ℝ))) (ν := AddCircle.haarAddCircle) (fun a : UnitAddCircle => B*(1-ψ a^2))] simp only [probReal_univ,one_smul,integral_const_mul] lemma packet_neutral_cutoff_error {A : UnitAddCircle × UnitAddCircle → ℂ} {B : ℝ} (hB : 0 ≤ B) (hAb : ∀ p, ‖A p‖^2 ≤ B) {δ : ℝ} (c : LawCutoffs δ) (hδ : 0 < δ) (hδ' : δ < 1/8) (m : ℤ) : (∫ x : TripleTorus, ‖packet m A x-(neutralCircleCutoff c x.2.2:ℂ)*packet m A x‖^2 ∂tripleVolume) ≤ B*(4*δ) := (packet_cutoff_error_uniform (continuous_neutralCircleCutoff c) (neutralCircleCutoff_bounds c hδ) hB hAb m).trans (mul_le_mul_of_nonneg_left (neutralCircleCutoff_error c hδ hδ') hB) lemma uncut_retainedPassage_uniform_error {A : UnitAddCircle × UnitAddCircle → ℂ} {m : ℤ} {H : ℕ → UnitAddCircle → UnitAddCircle} {χ : ℝ → UnitAddCircle → ℝ} (hA : Continuous A) (hH : ∀ n, Continuous (H n)) (hχ : Continuous χ.uncurry) {B : ℝ} (hB : 0 ≤ B) (hAb : ∀ p, ‖A p‖^2 ≤ B) {δ : ℝ} (c : LawCutoffs δ) (hδ : 0 < δ) (hδ' : δ < 1/8) (n : ℕ) (S : Finset ℤ) : (∫ x : TripleTorus, ‖packet m A x-retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) (neutralCircleCutoff c) (passageChart (H n) x)‖^2 ∂tripleVolume) ≤ 8*δ*B+2*(∫ x : TripleTorus, ‖(neutralCircleCutoff c x.2.2:ℂ)*packet m A x- retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) (neutralCircleCutoff c) (passageChart (H n) x)‖^2 ∂tripleVolume) := by let f := packet m A let g := fun x : TripleTorus => (neutralCircleCutoff c x.2.2:ℂ)*f x let h := fun x : TripleTorus => retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) (neutralCircleCutoff c) (passageChart (H n) x) have hf : Continuous f := continuous_packet hA have hg : Continuous g := (Complex.continuous_ofReal.comp ((continuous_neutralCircleCutoff c).comp continuous_snd.snd)).mul hf have hh : Continuous h := (continuous_retainedPassage (continuous_neutralCircleCutoff c) hχ hH hA n S).comp (continuous_passageChart (hH n)) have hi (v w : TripleTorus → ℂ) (hv : Continuous v) (hw : Continuous w) : Integrable (fun x => ‖v x-w x‖^2) tripleVolume := ((hv.sub hw).norm.pow 2).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) have hfg := hi f g hf hg have hgh := hi g h hg hh calc _ ≤ ∫ x, (2*‖f x-g x‖^2+2*‖g x-h x‖^2) ∂tripleVolume := by apply integral_mono (hi f h hf hh) ((hfg.const_mul 2).add (hgh.const_mul 2)) intro x exact sq_norm_sub_le_two (f x) (g x) (h x) _ = 2*(∫ x, ‖f x-g x‖^2 ∂tripleVolume)+2*(∫ x, ‖g x-h x‖^2 ∂tripleVolume) := by rw [integral_add (hfg.const_mul 2) (hgh.const_mul 2),integral_const_mul,integral_const_mul] _ ≤ _ := by have hh := packet_neutral_cutoff_error hB hAb c hδ hδ' m change (∫ x, ‖f x-g x‖^2 ∂tripleVolume) ≤ _ at hh dsimp only [g,h] nlinarith end ThreeTorus end end section /-! The actual neutral refresh: a single retained vector simultaneously approximates the old physical packet and its old temporal spectrum. The amplitude may depend on both slow coordinates. -/ noncomputable section open MeasureTheory Set Function Filter open scoped Topology BigOperators namespace ThreeTorus theorem exact_neutral_passage {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)) (hAs : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => A (p.1,p.2))) {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} (hχs : CylinderSmooth χ.uncurry) (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) : 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, ∃ S : ℕ → Finset ℤ, (∀ n, 0 < n → ∀ j ∈ S n, j ≠ 0) ∧ (∀ n, 0 < n → ∀ j ∉ S n, ∀ a, χ ((j:ℝ)/n) a = 0) ∧ ∀ η : ℝ, 0 < η → ∀ᶠ n : ℕ in atTop, let ψ := neutralCircleCutoff c let J := neutralJitter c L β let φ := fun a => (((1+J a/n)/n:ℝ):UnitAddCircle) let f := retainedPassage m (H n) A (S n) (fun j => χ ((j:ℝ)/n)) ψ let p := exactRetainedPacketDensity (Ioo δ (1-δ)) (neutralJitterReal c L β) ψ χ m H A (S n) n (∫ x : TripleTorus, ‖packet m A x-f (passageChart (H n) x)‖^2 ∂tripleVolume) < 8*δ*B+η ∧ (∫ z, |p z-pold z| ∂AddCircle.haarAddCircle) < C/L+4*δ*B+η ∧ ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => p x) ∧ (∀ k : ℤ, (∫ x : TripleTorus, starRingEnd ℂ (f x)*f (twistIter φ k x) ∂tripleVolume) = ∫ z, (p z:ℂ)*fourier k z ∂AddCircle.haarAddCircle) ∧ (∀ j u a, ‖retainedAmplitude ψ χ m H A n j (u,a)‖ = ‖retainedAmplitude ψ χ m H A n j (u,0)‖) := by 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 hψsupport : ∀ x ∈ Ioc (0:ℝ) 1, ψ (x:UnitAddCircle) ≠ 0 → x ∈ Ioo δ (1-δ) := by intro x hx hn rw [← heψ x (by simpa using hx)] at hn exact (hψs (subset_tsupport c.ψ hn)).2 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 obtain ⟨C,hC,hsp⟩ := 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 (Q := 1) (by decide) 0 refine ⟨C,hC,?_⟩ intro L hL β 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' obtain ⟨S,hSN,hST,ht⟩ := hsp L hL β (neutralJitterReal c L β) (neutralJitter c L β) heJ hJs hJc.measurable (fun x _ => neutralJitter_coe c L β x) simp only [Nat.cast_one,mul_one,zero_add,inv_one,one_mul] at hSN hST ht refine ⟨S,hSN,hST,?_⟩ intro η hη have hph := exact_physical_retainedPassage_tendsto hm hH hA hA₀ hAc hAd hsupp hlo huo hlu hR hPr hV hPV hdV hHc hAcont hΔt hχc hχ ψ hψc (neutralCircleCutoff_bounds c hδ) hχW S hST have hpi (n : ℕ) (hn : 0 < n) := exactRetainedPacketDensity_integrable (m := m) hJs isOpen_Ioo hJinj hJd hψc hχc hHc hAcont hn (hSN n hn) 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 hspec := ht (η/2) (half_pos hη) change ∀ᶠ n : ℕ in atTop, (∫ z, |exactRetainedPacketDensity (Ioo δ (1-δ)) (neutralJitterReal c L β) ψ χ m H A (S n) n z-foldCircleDensity (fun s => ∫ a, F (a,s)) z| ∂AddCircle.haarAddCircle) < C/L+η/2 at hspec rw [hmarg] at hspec have hHl : ∀ n : ℕ, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x => (n:ℝ)*(Pr x+Δ n*x)) := fun _ => contDiff_const.mul (hPr.add (contDiff_const.mul contDiff_id)) filter_upwards [hspec,hph.eventually (eventually_lt_nhds (half_pos hη)), eventually_gt_atTop (0:ℕ)] with n hnsp hnph hn refine ⟨?_,?_,?_,?_,?_⟩ · have hh := uncut_retainedPassage_uniform_error (m := m) hAcont hHc hχc hB hAb c hδ hδ' n (S n) exact hh.trans_lt (by linarith) · exact (integral_abs_sub_triangle (hpi n hn) hpcuti hpoldi).trans_lt (by linarith) · exact exactRetainedPacketDensity_smooth_lift hJs isOpen_Ioo hJinj hJd c.ψ_smooth c.ψ_compact (fun x hx => (heψ x hx).symm) hψs hχs hHl hH hAs hn (hSN n hn) · intro k exact retainedPassage_spectral_moment hψc hχc hHc hAcont hψsupport hJc hJs (fun x _ => neutralJitter_coe c L β x) isOpen_Ioo hJinj hJd hn (hSN n hn) k · exact fun _ _ _ => retainedAmplitude_modulus _ _ _ _ _ _ _ _ _ end ThreeTorus end end end OAI