| import OAI.Analysis.CircleDomains.Rigidity.CodeRealFreeTrace |
| import OAI.Analysis.CircleDomains.Selection.RealDomainIntersections |
| import OAI.Analysis.CircleDomains.Selection.FiniteFlagIntersections |
| import OAI.Analysis.CircleDomains.Probability.BandMeasure |
| import OAI.Analysis.CircleDomains.Selection.MeanLevelAssembly |
|
|
| namespace OAI |
|
|
| /-! Generic prior-level intersections for real coordinates under a base-code |
| law, with physical, flagged, and free contributions. -/ |
|
|
| open Set Function Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| attribute [local instance] sphereMetric sphereBorel |
| local instance : BorelSpace Sphere := ⟨rfl⟩ |
| variable [MeasurableSpace C(PathTime, Sphere)] [BorelSpace C(PathTime, Sphere)] |
|
|
| theorem mean_code_real_level_intersections_finite |
| (hcoarea : PlanarLipschitzFiberCoarea) |
| (hsard : SmoothPlanarSard) (hsch : PlanarSchoenflies) |
| (d : PathCodeData Sphere Sphere) |
| (hG : IsOpen d.physicalDomain) (hU : IsOpen d.quotientDomain) |
| (hdense : Dense d.quotientDomain) (hdust : d.dust = d.quotientDomainᶜ) |
| (π : Sphere → Sphere) (hπ : Continuous π) (hclosed : IsClosedMap π) (hsurj : Surjective π) |
| (hr : ∀ x y, π x = π y ↔ x = y ∨ |
| (x ∈ d.physicalDomainᶜ ∧ y ∈ connectedComponentIn d.physicalDomainᶜ x)) |
| (he : ∀ y : d.physicalDomain, (d.domainEquiv y).val = π y.val) |
| (hTD : IsTotallyDisconnected d.quotientDomainᶜ) |
| (hbad : ∀ p ∈ d.physicalDomainᶜ, π p ∈ d.quotientDomainᶜ) |
| (Flags : Set Sphere) (hFlags : Flags.Countable) (hFlagsB : Flags ⊆ d.quotientDomainᶜ) |
| (f : Sphere → ℝ) {M : ℝ} (hM : ∀ z ∈ d.quotientDomain, |f z| ≤ M) |
| (hf : ContinuousOn f d.quotientDomain) |
| (hv : ContDiffOn ℝ 1 (fun z : ℂ => f (π (z : Sphere))) |
| (((↑) : ℂ → Sphere) ⁻¹' d.physicalDomain)) |
| (henergy : (∫⁻ z in ((↑) : ℂ → Sphere) ⁻¹' d.physicalDomain, |
| ENNReal.ofReal ‖fderiv ℝ (fun z : ℂ => f (π (z : Sphere))) z‖ ^ 2) < ∞) |
| {K : Set ℂ} (hK : IsCompact K) |
| (hzero : ∀ z : ℂ, (z : Sphere) ∈ d.physicalDomain → z ∉ K → f (π (z : Sphere)) = 0) |
| (hinfty : OnePoint.infty ∈ d.physicalDomain → f (π OnePoint.infty) = 0) |
| (P : ProbabilityMeasure (BasePathCode d)) |
| (ρ : Sphere → ℝ≥0∞) (hρ : Measurable ρ) |
| (hocc : meanCodeOccupation d P = sphericalArea.withDensity ρ) |
| (hρ2 : (∫⁻ y, ρ y ^ 2 ∂sphericalArea) < ∞) |
| (i : ℕ) (hm : (support (d.levelWeights i)).Countable) |
| (hweight : countingSquareNorm (fun x : support (d.levelWeights i) => |
| d.levelWeights i x.val) < ∞) |
| (hvisits : countingSquareNorm (fun x : support (d.levelWeights i) => |
| meanVisitCount (codePathLaw d P) x.val) < ∞) |
| (hband : ∀ z ∈ d.quotientDomainᶜ, ENNReal.ofReal |
| (limsup f (𝓝[d.quotientDomain] z) - liminf f (𝓝[d.quotientDomain] z)) ≤ |
| d.levelWeights i z) |
| (hmFree : ∀ z ∈ d.quotientDomainᶜ \ Flags, d.levelWeights i z = 0) |
| (hcover : ∀ x ∈ support (d.levelWeights i), ∀ᶠ n in atTop, x ∈ d.flagDeletions n) |
| (hdeletions : ∀ n, (d.flagDeletions n : Set Sphere) ⊆ Flags) |
| (hH : ∀ j, IsClosed (d.levels j)) |
| (hP : OrderedLevelNullCompatibility d.levels (d.quotientDomainᶜ \ Flags)) |
| (hunique : ∀ x ∈ d.quotientDomainᶜ \ Flags, ∀ j a b, |
| (a, x) ∈ d.levels j → (b, x) ∈ d.levels j → a = b) |
| (ω : TupleType → ℝ≥0∞) [SFinite (tupleReference ω)] |
| (hτ : meanCodeTuple d P ≪ tupleReference ω) |
| (hlevel : ∀ θ x, (θ, x) ∈ d.levels i ↔ (x, θ) ∈ circleBandGraph d.quotientDomain f) : |
| (∫⁻ θ : UnitAddCircle, ∫⁻ Γ : BasePathCode d, |
| pathImageCount {x | (θ, x) ∈ d.levels i} Γ.val.1 ∂(P : Measure _)) < ∞ := by |
| have hfree := ae_code_real_free_level_incidence_null hsard hsch d hG hU hdense hdust |
| π hπ hclosed hsurj hr he hTD hbad Flags hFlags hFlagsB f hM hf hv henergy P ρ hρ hocc hρ2 |
| i hm hweight hvisits hband hmFree hcover hdeletions hH hP hunique ω hτ hlevel |
| have hfinite : ∀ᵐ γ ∂(codePathLaw d P : Measure C(PathTime, Sphere)), |
| physicalDomainLength d.domainEquiv d.basePoint γ < ∞ := by |
| change ∀ᵐ γ ∂(P : Measure (BasePathCode d)).map (codePath d), _ |
| apply (ae_map_iff (measurable_codePath d).aemeasurable |
| (measurableSet_finitePhysicalDomainLength d.domainEquiv hU d.basePoint)).mpr |
| exact Eventually.of_forall fun Γ => Γ.property.1.1 |
| have hdom := mean_real_domain_circle_count_finite hcoarea hG d.domainEquiv hU d.basePoint π hπ he |
| f hf hv henergy hK hzero hinfty (codePathLaw d P) hfinite ρ hρ |
| ((meanPhysicalOccupation_codePathLaw d hU P).trans hocc) hρ2 |
| have hvol : ∀ x ∈ Flags, volume {θ : UnitAddCircle | (θ, x) ∈ d.levels i} ≤ d.levelWeights i x := by |
| intro x hx |
| have heq : {θ : UnitAddCircle | (θ, x) ∈ d.levels i} = |
| {θ : UnitAddCircle | (x, θ) ∈ circleBandGraph d.quotientDomain f} := |
| Set.ext fun θ => hlevel θ x |
| rw [heq] |
| exact (volume_circleBandGraph_incidence_le_budget hM |
| (by rw [hdense.closure_eq]; exact mem_univ x) le_rfl).trans (hband x (hFlagsB hx)) |
| have hflag := mean_flag_intersection_count_finite_of_support (hH i) Flags hFlags |
| (d.levelWeights i) hvol (codePathLaw d P) hm hweight hvisits |
| have hσU : IsSigmaCompact d.quotientDomain := by |
| let : LocallyCompactSpace d.quotientDomain := hU.locallyCompactSpace |
| exact isSigmaCompact_iff_sigmaCompactSpace.mpr inferInstance |
| have hfree' : ∀ᵐ Γ ∂(P : Measure (BasePathCode d)), ∀ᵐ θ : UnitAddCircle ∂volume, |
| ¬ ∃ x ∈ range (codePath d Γ) ∩ (d.quotientDomainᶜ \ Flags), (θ, x) ∈ d.levels i := by |
| filter_upwards [hfree] with Γ hΓ |
| exact ae_iff.mpr (by simpa only [not_not, codePath] using hΓ) |
| have hb := mean_parameter_level_count_le_of_no_free_hits (hH i) |
| d.quotientDomain Flags hσU hFlags P (codePath d) (measurable_codePath d) hfree' |
| have hDmeas := measurable_sigmaCompact_restricted_relation_path_count hσU |
| ((hH i).preimage continuous_swap) |
| have hFmeas := measurable_flag_intersection_count (hH i) Flags hFlags |
| have hdint (θ : UnitAddCircle) : |
| (∫⁻ Γ : BasePathCode d, pathImageCount (d.quotientDomain ∩ {x | (θ, x) ∈ d.levels i}) |
| (codePath d Γ) ∂(P : Measure _)) = |
| ∫⁻ γ, pathImageCount (d.quotientDomain ∩ {x | (θ, x) ∈ d.levels i}) γ |
| ∂(codePathLaw d P : Measure _) := by |
| have hmeas : Measurable (fun γ : C(PathTime, Sphere) => |
| pathImageCount (d.quotientDomain ∩ {x | (θ, x) ∈ d.levels i}) γ) := |
| hDmeas.comp (measurable_id.prodMk measurable_const) |
| rw [codePathLaw, ProbabilityMeasure.toMeasure_map] |
| exact (lintegral_map hmeas (measurable_codePath d)).symm |
| have hfint (θ : UnitAddCircle) : |
| (∫⁻ Γ : BasePathCode d, pathImageCount (Flags ∩ {x | (θ, x) ∈ d.levels i}) |
| (codePath d Γ) ∂(P : Measure _)) = |
| ∫⁻ γ, pathImageCount (Flags ∩ {x | (θ, x) ∈ d.levels i}) γ |
| ∂(codePathLaw d P : Measure _) := by |
| have hmeas : Measurable (fun γ : C(PathTime, Sphere) => |
| pathImageCount (Flags ∩ {x | (θ, x) ∈ d.levels i}) γ) := |
| hFmeas.comp (measurable_id.prodMk measurable_const) |
| rw [codePathLaw, ProbabilityMeasure.toMeasure_map] |
| exact (lintegral_map hmeas (measurable_codePath d)).symm |
| apply hb.trans_lt |
| apply ENNReal.add_lt_top.mpr |
| constructor |
| · simp_rw [hdint] |
| simpa only [hlevel] using hdom |
| · simpa only [hfint] using hflag |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
