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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 | 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
import OAI.Analysis.CircleDomains.Selection.CodeRealMeanIntersections
import OAI.Analysis.CircleDomains.Probability.LevelLawQuantifiers
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 code_real_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)
(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) :
∀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 hmean := mean_code_real_level_intersections_finite hcoarea hsard hsch d
hG hU hdense hdust π hπ hclosed hsurj hr he hTD hbad Flags hFlags hFlagsB f hM hf hv henergy
hK hzero hinfty P ρ hρ hocc hρ2 i hm hweight hvisits hband hmFree hcover hdeletions
hH hP hunique ω hτ hlevel
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 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 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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