ReindexGeneralClause

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

The general clause in real form, at frontier-empty windows

This is the real-form general clause at frontier-empty windows, re-cut at the frozen slot: the composed chain of ReindexComposed (good event, flux index and σ index all at n+3, binder n + 3 ≤ m) carried onto the frozen right-hand side by the three proved moves, in one absorption:

• the σ gap-3 conversion σ_{n+3}^{-1} ≤ 4 σ_n^{-1} (ReindexSlot.exists_inv_sigmaBar_add_three_le; the same constant 4 the proved two-scale conversion pays — the gap is free);
• the two window moves to W' = (z + □_{n+3}) ∩ □_m, at the volume ratio 81^d, i.e. K = 2·9^d on the L² leg and 9^d on the force leg (RecutAtoms, unchanged);
• the three ∇h legs, which are nonnegative summands of the frozen right-hand side and are simply added (the comparison solution carries a constant boundary datum, so no ∇h term is produced anywhere in the chain).

Every s-power, every 3-power, the in-bracket 3^n |(∇h)_W| companion at the n+2 window (a deliberate asymmetry of the frozen statement), and the interior gate are byte-untouched.

The MemLp side conditions

Exactly the re-cut general clause's: u.toFun almost-everywhere strongly measurable on the normalized W, and the two window quantities finite so that the ℝ≥0∞ bounds descend to ℝ. Both come from the anchor's own binders. No new hypothesis is introduced.

References

• ABK26, l.harmonic.approximation.good.scales; the frozen block of Algsuperdiff/Frozen/Section4/HarmonicApproximation.lean.

1. The re-cut absorption, on abstract reals

theorem Algsuperdiff.Section4.Provider.ExcessDecay.general_absorb_window {Cf C K A A' B B' T3 T4 : ℝ} (hK : 0 ≤ K) (hAA' : A ≤ K * A') (hA' : 0 ≤ A') (hBB' : B ≤ 4 * K * B') (hB' : 0 ≤ B') (hT3 : 0 ≤ T3) (hT4 : 0 ≤ T4) (hCf : 0 ≤ Cf) (h1 : K * Cf ≤ C) (h4 : 4 * K * Cf ≤ C) : Cf * (A + B) ≤ C * (A' + B' + T3 + T4)

The re-cut absorption at the anchor window: the composed two-summand bound is absorbed into the four-summand frozen right-hand side, with the window factor K paid on both surviving legs and the σ gap-3 factor 4 on the force leg.

theorem Algsuperdiff.Section4.Provider.ExcessDecay.prefactor_window_move {a X Y K : ℝ} (ha : 0 ≤ a) (hXY : X ≤ K * Y) : a * X ≤ K * (a * Y)

Absorbing a window factor through a nonnegative prefactor.

theorem Algsuperdiff.Section4.Provider.ExcessDecay.sigma_window_move {p q q' r Y Y' K : ℝ} (hp : 0 ≤ p) (hq : 0 ≤ q) (hr : 0 ≤ r) (hK : 0 ≤ K) (hY' : 0 ≤ Y') (hqq' : q ≤ 4 * q') (hYY' : Y ≤ K * Y') : p * q * r * Y ≤ 4 * K * (p * q' * r * Y')

Absorbing the σ-index factor and the window factor through the force leg's three nonnegative prefactors.

2. The general clause in real form, gated at the interior regime

theorem Algsuperdiff.Section4.Provider.ExcessDecay.exists_generalClauseReal_frontierEmpty_with_slack (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 * (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) → 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) → (fun (y' : HCPolySupport.Vec d) => z + y') '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 2)) ∩ frontier (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 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))) → ∀ (gradientBesovLeg gradientL2Leg : ℝ), 0 ≤ gradientBesovLeg → 0 ≤ gradientL2Leg → ∀ (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 ≤ 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)))).toReal + 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‖) + 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).toReal + gradientBesovLeg + gradientL2Leg)

The frontier-empty general clause with arbitrary nonnegative gradient slack.

The sigma conversion and two window moves are proved once; the two display interfaces specialize the supplied slack terms to their respective weights.

theorem Algsuperdiff.Section4.Provider.ExcessDecay.exists_generalClauseReal_frontierEmpty (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 * (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) → 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) → (fun (y' : HCPolySupport.Vec d) => z + y') '' HCPolySupport.openCubeSet (HCPolySupport.originCube d (n + 2)) ∩ frontier (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 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) → (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 ≤ 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)))).toReal + 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‖) + 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).toReal + s.rpow (-(9 / 2)) * 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).toReal + s.rpow (-(9 / 2)) * 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)))).toReal)

The general clause with the unweighted boundary-gradient slack terms.