HarmonicApproximation

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

Harmonic approximation at good scales — [ABK] l.harmonic.approximation.good.scales

On a good scale, a Dirichlet solution on a triadic cube is approximated in L^2 by a harmonic replacement on the child cube, with an error controlled by the flux-corrected coarse-graining error together with the oscillation of the solution, the fractional force seminorm and the boundary gradient data. Two clauses are stated: a general one valid up to the boundary, and a sharper frontier-empty specialization carrying no gradient-of-datum legs.

Proved; reduces to the standard axioms.

theorem Algsuperdiff.Frozen.Section4.harmonic_approximation_good_scales (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), Section4.Support.IsDirichletSolutionOn (Section3.Cutoff.coefficientCutoff M.nu L omega).toCoeffField (HCPolySupport.originCube d m) u h g → MeasureTheory.MemLp g 2 (Section4.Support.normalizedVolumeMeasureOn (HCPolySupport.openCubeSet (HCPolySupport.originCube d m))) → MeasureTheory.MemLp (HCPolySupport.Gagliardo.gagliardoKernel s 2 g) 2 (Section4.Support.normalizedGagliardoMeasureOn (HCPolySupport.openCubeSet (HCPolySupport.originCube d m))) → MeasureTheory.MemLp (HCPolySupport.Gagliardo.gagliardoKernel s 2 h.grad) 2 (Section4.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))), Section4.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) → (goodEventAt M (Section4.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 (Section4.Support.normalizedVolumeMeasureOn ((fun (y : HCPolySupport.Vec d) => x + y) '' HCPolySupport.openCubeSet (HCPolySupport.originCube d n)))) omega ≤ ENNReal.ofReal (C * s.rpow (-4) * Section4.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 (Section4.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)) * Section4.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)) * Section4.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) * Section4.Support.fluxCorrectedErrorRepresentative M L (n + 3) ⟨s / 8, ⋯⟩ (Section3.Cutoff.translateCutoffSample z omega) * 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‖) ∧ ((fun (y' : HCPolySupport.Vec d) => z + y') '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 2)) ∩ frontier (HCPolySupport.openCubeSet (HCPolySupport.originCube d m)) = ∅ → (goodEventAt M (Section4.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 (Section4.Support.normalizedVolumeMeasureOn ((fun (y : HCPolySupport.Vec d) => x + y) '' HCPolySupport.openCubeSet (HCPolySupport.originCube d n)))) omega ≤ ENNReal.ofReal (C * s.rpow (-4) * Section4.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 (Section4.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)) * Section4.Support.normalizedGagliardoESeminormOn ((fun (y' : HCPolySupport.Vec d) => z + y') '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 2)) ∩ HCPolySupport.openCubeSet (HCPolySupport.originCube d m)) s g)