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LeanPool.Superdiffusion.Algsuperdiff.Section4.Provider.ExcessDecay.ProviderEpsFree

ProviderEpsFree #

Ported from scottnarmstrong/Superdiffusion at 7f20476385d4af418f110303085ad0740c9eba46. Module imports, support namespace and local option directives adapted for Lean Pool.

The ε-free provider, unconditional #

What is proved #

One public theorem, harmonic_approximation_provider_epsFree, whose statement is the frozen block verbatim (up to the single (fun omega' => ↦ (fun _omega' => rename that the zero-warning gate forces, the same rename the proved siblings carry), and whose proof has no hypothesis: no hres, no hepsPin, no hmean, no hgen, no hbdry.

The composition, slot by slot #

  RebaseGeneralClauseRegated.exists_generalClauseReal_frontierEmpty_regated   (the interior/
                                                                      general side)
        │
        ├─ RebaseJoinRegated  ──────────────────────────── the two clauses, one constant
        │
  ProviderBoundary.exists_boundaryClauseReal   (the boundary side)
        │      ├─ AssemblyBoundaryClause.hbdryOfScalarCoarse
        │      ├─ StitchDisplayLegs.exists_scalarLeg_le_displayLegs_of_epsPin
        │      │       └─ ResiduePlugin.exists_flushResidue_le_displayLegs_of_epsPin
        │      │       └─ ResidueBudget.exists_scalarControl_of_boundaryBranch_…
        │      ├─ GoodEventCaps.ae_errorRepresentative_le_of_mem_goodEventAt  (the pin)
        │      ├─ ReindexSlot.exists_inv_sigmaBar_add_three_le      (σ_{n+3} → σ_n)
        │      └─ MeanControlWindowCube.overhang_of_boundaryBranch  (the gate)
        │
  RebaseSealDistance.exists_harmonicApproximation_provider_of_boundaryClauseReal
        │
        ▼
  the frozen block

C is chosen last, after every d-only constant of the chain, so the closing fit is sound.

References #

source
theorem Algsuperdiff.Section4.Provider.ExcessDecay.harmonic_approximation_provider_epsFree (d : ℕ) :
∃ (C : ℝ), 0 < C ∧ ∀ (M : Section3.ABKModel d), ∀ s ∈ Set.Icc (64 * M.gamma) 1, M.gamma ≤ C⁻¹ * Section3.Disorder.cstar M ^ 10 → M.gamma * |Real.log M.gamma| ^ 2 ≤ (s / 8).rpow (3 / 2) * Section3.Disorder.cstar M ^ 2 * (C⁻¹ * s ^ 4) → ∀ (hs : 0 < s) (L m n : ℤ), m ≤ L → n + 3 ≤ m → ∀ (x z : HCPolySupport.Vec d), x ∈ HCPolySupport.openCubeSet (HCPolySupport.originCube d m) → z ∈ HCPolySupport.openCubeSet (HCPolySupport.originCube d m) → (fun (y : HCPolySupport.Vec d) => x + y) '' HCPolySupport.openCubeSet (HCPolySupport.originCube d n) ⊆ (fun (y : HCPolySupport.Vec d) => z + y) '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 1)) ∩ HCPolySupport.openCubeSet (HCPolySupport.originCube d m) → ∀ᵐ (omega : Section3.Cutoff.CutoffSample d) ∂↑(Section3.Cutoff.cutoffSampleLaw M), ∀ (u h : HCPolySupport.H1Function (HCPolySupport.openCubeSet (HCPolySupport.originCube d m))) (g : HCPolySupport.Vec d → HCPolySupport.Vec d), Support.IsDirichletSolutionOn (Section3.Cutoff.coefficientCutoff M.nu L omega).toCoeffField (HCPolySupport.originCube d m) u h g → MeasureTheory.MemLp g 2 (Support.normalizedVolumeMeasureOn (HCPolySupport.openCubeSet (HCPolySupport.originCube d m))) → MeasureTheory.MemLp (HCPolySupport.Gagliardo.gagliardoKernel s 2 g) 2 (Support.normalizedGagliardoMeasureOn (HCPolySupport.openCubeSet (HCPolySupport.originCube d m))) → MeasureTheory.MemLp (HCPolySupport.Gagliardo.gagliardoKernel s 2 h.grad) 2 (Support.normalizedGagliardoMeasureOn (HCPolySupport.openCubeSet (HCPolySupport.originCube d m))) → ∀ (v : HCPolySupport.H1Function ((fun (y : HCPolySupport.Vec d) => x + y) '' HCPolySupport.openCubeSet (HCPolySupport.originCube d n))) (w : HCPolySupport.H10Function ((fun (y : HCPolySupport.Vec d) => x + y) '' HCPolySupport.openCubeSet (HCPolySupport.originCube d n))), Support.IsWeaklyHarmonicOn ((fun (y : HCPolySupport.Vec d) => x + y) '' HCPolySupport.openCubeSet (HCPolySupport.originCube d n)) v → (∀ (y : HCPolySupport.Vec d), v.toFun y = u.toFun y - w.toFun y) → (∀ (y : HCPolySupport.Vec d), v.grad y = u.grad y - w.grad y) → (Frozen.Section4.goodEventAt M (Support.cgEllipLowerConstant d) (n + 3) z ⟨s / 8, ⋯⟩ (C⁻¹ * s ^ 4)).indicator (fun (_omega' : Section3.Cutoff.CutoffSample d) => MeasureTheory.eLpNorm (fun (y : HCPolySupport.Vec d) => u.toFun y - v.toFun y) 2 (Support.normalizedVolumeMeasureOn ((fun (y : HCPolySupport.Vec d) => x + y) '' HCPolySupport.openCubeSet (HCPolySupport.originCube d n)))) omega ≤ ENNReal.ofReal (C * s.rpow (-4) * Support.fluxCorrectedErrorRepresentative M L (n + 3) ⟨s / 8, ⋯⟩ (Section3.Cutoff.translateCutoffSample z omega)) * (MeasureTheory.eLpNorm (fun (y : HCPolySupport.Vec d) => u.toFun y - HCPolySupport.volumeAverage ((fun (y' : HCPolySupport.Vec d) => z + y') '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 3)) ∩ HCPolySupport.openCubeSet (HCPolySupport.originCube d m)) u.toFun) 2 (Support.normalizedVolumeMeasureOn ((fun (y' : HCPolySupport.Vec d) => z + y') '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 3)) ∩ HCPolySupport.openCubeSet (HCPolySupport.originCube d m))) + ENNReal.ofReal (s.rpow (-(3 / 2)) * Real.rpow 3 ↑n * ‖HCPolySupport.volumeAverageVec ((fun (y' : HCPolySupport.Vec d) => z + y') '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 2)) ∩ HCPolySupport.openCubeSet (HCPolySupport.originCube d m)) h.grad‖)) + ENNReal.ofReal (C * s.rpow (-7) * (↑(Section3.Annealed.sigmaBar M n))⁻¹ * Real.rpow 3 ((1 + s) * ↑n)) * Support.normalizedGagliardoESeminormOn ((fun (y' : HCPolySupport.Vec d) => z + y') '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 3)) ∩ HCPolySupport.openCubeSet (HCPolySupport.originCube d m)) s g + ENNReal.ofReal (C * s.rpow (-6) * Real.rpow 3 ((1 + s) * ↑n)) * Support.normalizedGagliardoESeminormOn ((fun (y' : HCPolySupport.Vec d) => z + y') '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 3)) ∩ HCPolySupport.openCubeSet (HCPolySupport.originCube d m)) s h.grad + ENNReal.ofReal (C * s.rpow (-6) * Real.rpow 3 ↑n) * MeasureTheory.eLpNorm h.grad 2 (Support.normalizedVolumeMeasureOn ((fun (y' : HCPolySupport.Vec d) => z + y') '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 3)) ∩ HCPolySupport.openCubeSet (HCPolySupport.originCube d m))) ∧ ((fun (y' : HCPolySupport.Vec d) => z + y') '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 2)) ∩ frontier (HCPolySupport.openCubeSet (HCPolySupport.originCube d m)) = ∅ → (Frozen.Section4.goodEventAt M (Support.cgEllipLowerConstant d) (n + 2) z ⟨s / 8, ⋯⟩ (1 / 2)).indicator (fun (_omega' : Section3.Cutoff.CutoffSample d) => MeasureTheory.eLpNorm (fun (y : HCPolySupport.Vec d) => u.toFun y - v.toFun y) 2 (Support.normalizedVolumeMeasureOn ((fun (y : HCPolySupport.Vec d) => x + y) '' HCPolySupport.openCubeSet (HCPolySupport.originCube d n)))) omega ≤ ENNReal.ofReal (C * s.rpow (-4) * Support.fluxCorrectedErrorRepresentative M L (n + 2) ⟨s / 8, ⋯⟩ (Section3.Cutoff.translateCutoffSample z omega)) * MeasureTheory.eLpNorm (fun (y : HCPolySupport.Vec d) => u.toFun y - HCPolySupport.volumeAverage ((fun (y' : HCPolySupport.Vec d) => z + y') '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 2)) ∩ HCPolySupport.openCubeSet (HCPolySupport.originCube d m)) u.toFun) 2 (Support.normalizedVolumeMeasureOn ((fun (y' : HCPolySupport.Vec d) => z + y') '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 2)) ∩ HCPolySupport.openCubeSet (HCPolySupport.originCube d m))) + ENNReal.ofReal (C * s.rpow (-(19 / 2)) * (↑(Section3.Annealed.sigmaBar M n))⁻¹ * Real.rpow 3 ((1 + s) * ↑n)) * Support.normalizedGagliardoESeminormOn ((fun (y' : HCPolySupport.Vec d) => z + y') '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 2)) ∩ HCPolySupport.openCubeSet (HCPolySupport.originCube d m)) s g)

l.harmonic.approximation.good.scales, the provider — unconditional.

The frozen block, proved from the proved chain with no caller obligation. The boundary instance of the general clause is ProviderBoundary.exists_boundaryClauseReal; the frontier-empty instance, the ℝ≥0∞ shell, the interior clause and the outer join are RebaseSealDistance's.