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5711324 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 | import OAI.Analysis.CircleDomains.Rigidity.CodePeriodFreeTrace
import OAI.Analysis.CircleDomains.Selection.PeriodDomainIntersections
import OAI.Analysis.CircleDomains.Selection.PeriodDomainValues
import OAI.Analysis.CircleDomains.Selection.MeanLevelAssembly
import OAI.Analysis.CircleDomains.Probability.LevelLawQuantifiers
namespace OAI
/-! The period-coordinate case of the generic prior-level theorem.
Local branches supply null free traces; intrinsic two-cut coarea controls
physical counts, and the actual dust budget controls flagged counts. -/
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 code_period_generic_levels
(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)
(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) < ∞)
(hmFree : ∀ z ∈ d.quotientDomainᶜ \ Flags, d.levelWeights i z = 0)
(hdeleteCover : ∀ 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)
{I : Type*} [Countable I] (V : I → Set Sphere) (hV : ∀ j, IsOpen (V j))
(M : Set Sphere) (hM : IsClosed M) (hinftyM : π OnePoint.infty ∉ M)
(hboundary : M \ interior M ⊆ d.quotientDomain)
(b : Sphere) (hb : b ∈ Flags) (hbWeight : 1 ≤ d.levelWeights i b)
(hmask : ∀ θ x, (θ, x) ∈ d.levels i → x ∈ M)
(hcoverM : M \ {b} ⊆ ⋃ j, V j)
(f : I → Sphere → ℝ) (bound : I → ℝ)
(hbound : ∀ j z, z ∈ d.quotientDomain → |f j z| ≤ bound j)
(hf : ∀ j, ContinuousOn (f j) (d.quotientDomain ∩ V j))
(hv : ∀ j, ContDiffOn ℝ 1 (fun z : ℂ => f j (π (z : Sphere)))
(((↑) : ℂ → Sphere) ⁻¹' d.physicalDomain ∩ (fun z : ℂ => π (z : Sphere)) ⁻¹' V j))
(henergy : ∀ j, (∫⁻ z in ((↑) : ℂ → Sphere) ⁻¹' d.physicalDomain ∩
(fun z : ℂ => π (z : Sphere)) ⁻¹' V j,
ENNReal.ofReal ‖fderiv ℝ (fun z : ℂ => f j (π (z : Sphere))) z‖ ^ 2) < ∞)
(hband : ∀ j z, z ∈ (d.quotientDomainᶜ ∩ M) ∩ V j → ENNReal.ofReal
(limsup (f j) (𝓝[d.quotientDomain] z) - liminf (f j) (𝓝[d.quotientDomain] z)) ≤
d.levelWeights i z)
(hlevel : ∀ j z, z ∈ M ∩ V j → ∀ θ,
(z, θ) ∈ circleBandGraph d.quotientDomain (f j) ↔ (θ, z) ∈ d.levels i)
(w : Sphere → UnitAddCircle)
(hw : ∀ j x, x ∈ d.quotientDomain ∩ V j → (f j x : UnitAddCircle) = w x)
(D : Set ℂ) (hD : IsOpen D) (hDG : ∀ z ∈ D, (z : Sphere) ∈ d.physicalDomain)
(hDcover : ∀ z : ℂ, (z : Sphere) ∈ d.physicalDomain → π (z : Sphere) ∈ M → z ∈ D)
(F : ℂ → UnitAddCircle) (hF : FiniteTransfer.SmoothCircleOn F D)
(hFpull : ∀ z ∈ D, F z = w (π (z : Sphere)))
(hFenergy : (∫⁻ z in D, ENNReal.ofReal ‖FiniteTransfer.circleFDeriv F z‖ ^ 2) < ∞) :
(∫⁻ θ : UnitAddCircle, ∫⁻ Γ : BasePathCode d,
pathImageCount {x | (θ, x) ∈ d.levels i} Γ.val.1 ∂(P : Measure _)) < ∞ ∧
∀ᵐ θ : UnitAddCircle ∂volume,
(∫⁻ Γ : BasePathCode d, pathImageCount {x | (θ, x) ∈ d.levels i} Γ.val.1
∂(P : Measure _)) < ∞ ∧
∀ᵐ Γ ∂(P : Measure (BasePathCode d)),
¬ ∃ x ∈ range Γ.val.1 ∩ (d.quotientDomainᶜ \ Flags), (θ, x) ∈ d.levels i := by
have hfree := ae_code_period_free_level_incidence_null hsard hsch d hG hU hdense hdust
π hπ hclosed hsurj hr he hTD hbad Flags hFlags hFlagsB P ρ hρ hocc hρ2
i hm hweight hvisits hmFree hdeleteCover hdeletions hH hP hunique ω hτ
V hV M hinftyM hboundary b hb hmask hcoverM f bound hbound hf hv henergy hband 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 hdomLevel (θ : UnitAddCircle) (x : Sphere)
(hx : x ∈ d.quotientDomain ∩ {x | (θ, x) ∈ d.levels i}) : x ∈ M ∧ w x = θ := by
have hxM := hmask θ x hx.2
have hxb : x ≠ b := fun heq => hFlagsB hb (heq ▸ hx.1)
obtain ⟨j, hj⟩ := mem_iUnion.mp (hcoverM ⟨hxM, hxb⟩)
have hcg := (hlevel j x ⟨hxM, hj⟩ θ).mpr hx.2
have heq := (circleBandGraph_of_continuousAt
((hf j).continuousAt ((hU.inter (hV j)).mem_nhds ⟨hx.1, hj⟩)) hx.1 θ).mp hcg
exact ⟨hxM, (hw j x ⟨hx.1, hj⟩).symm.trans heq⟩
have hdom := mean_masked_circle_domain_count_finite hcoarea d.domainEquiv hU d.basePoint
π hπ he M hM hinftyM (fun θ => {x | (θ, x) ∈ d.levels i}) w hdomLevel
D hD hDG hDcover F hF hFpull hFenergy (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
by_cases hxM : x ∈ M
· by_cases hxb : x = b
· subst x
exact (measure_mono (subset_univ _)).trans (by simpa [AddCircle.measure_univ] using hbWeight)
· obtain ⟨j, hj⟩ := mem_iUnion.mp (hcoverM ⟨hxM, hxb⟩)
have heq : {θ : UnitAddCircle | (θ, x) ∈ d.levels i} =
{θ | (x, θ) ∈ circleBandGraph d.quotientDomain (f j)} :=
Set.ext fun θ => (hlevel j x ⟨hxM, hj⟩ θ).symm
rw [heq]
exact (volume_circleBandGraph_incidence_le_budget (hbound j)
(by rw [hdense.closure_eq]; trivial) le_rfl).trans
(hband j x ⟨⟨hFlagsB hx, hxM⟩, hj⟩)
· have hempty : {θ : UnitAddCircle | (θ, x) ∈ d.levels i} = ∅ :=
eq_empty_iff_forall_notMem.mpr fun θ hθ => hxM (hmask θ x hθ)
rw [hempty, measure_empty]
exact bot_le
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 hbnd := 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
have hmean : (∫⁻ θ : UnitAddCircle, ∫⁻ Γ : BasePathCode d,
pathImageCount {x | (θ, x) ∈ d.levels i} Γ.val.1 ∂(P : Measure _)) < ∞ := by
apply hbnd.trans_lt
apply ENNReal.add_lt_top.mpr
constructor
· simpa only [hdint] using hdom
· simpa only [hfint] using hflag
refine ⟨hmean, ?_⟩
have hdisj : Disjoint d.quotientDomain Flags :=
disjoint_left.mpr fun x hx hxf => hFlagsB hxf hx
simpa only [codePath] using ae_level_parameter_free_avoidance (hH i)
d.quotientDomain Flags hσU hFlags hdisj P (codePath d) (measurable_codePath d) hfree' hmean
end CircleDomainRigidity
end
end OAI
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