AssemblyComposed

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

theorem Algsuperdiff.Section4.Provider.ExcessDecay.exists_boundaryClauseComposed_honest (d : ℕ) [NeZero d] : ∃ (C : ℝ) (Cfin : ℝ), 0 < C ∧ 0 < Cfin ∧ ∀ (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 * (1 / 2) → ∀ (hs : 0 < s) (L m n : ℤ), m ≤ L → n + 3 ≤ m → ∀ (x z : HCPolySupport.Vec d), x ∈ 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), omega ∈ Frozen.Section4.goodEventAt M (Support.cgEllipLowerConstant d) (n + 3) z ⟨s / 8, ⋯⟩ (1 / 2) → ∀ (u hdat : HCPolySupport.H1Function (HCPolySupport.openCubeSet (HCPolySupport.originCube d m))) (g : HCPolySupport.Vec d → HCPolySupport.Vec d) (S : ℝ), Support.IsDirichletSolutionOn (Section3.Cutoff.coefficientCutoff M.nu L omega).toCoeffField (HCPolySupport.originCube d m) u hdat 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 hdat.grad) 2 (Support.normalizedGagliardoMeasureOn (HCPolySupport.openCubeSet (HCPolySupport.originCube d m))) → |HCPolySupport.volumeAverage ((fun (y : HCPolySupport.Vec d) => wellPlacedCentre x m (n + 2) + y) '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 2))) fun (y : HCPolySupport.Vec d) => u.toFun y - hdat.toFun y| ≤ S → ∀ (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) → (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)))).toReal ≤ Cfin * (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)))).toReal + s.rpow (-7) * (↑(Section3.Annealed.sigmaBar M (n + 3)))⁻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).toReal + 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 hdat.grad).toReal + s.rpow (-6) * Real.rpow 3 ↑n * (MeasureTheory.eLpNorm hdat.grad 2 (Support.normalizedVolumeMeasureOn ((fun (y' : HCPolySupport.Vec d) => z + y') '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 3)) ∩ HCPolySupport.openCubeSet (HCPolySupport.originCube d m)))).toReal + s.rpow (-4) * Support.fluxCorrectedErrorRepresentative M L (n + 3) ⟨s / 8, ⋯⟩ (Section3.Cutoff.translateCutoffSample z omega) * S)

The unweighted boundary clause follows by capping the representative error.