Multi-Scale Difference Operator for Meta-Variational Principle

This file defines the m-point multi-scale difference operator Δ^{⟨m⟩} and spectral symbols for the meta-variational principle framework.

Main definitions

• MultiScaleConfig: Configuration for multi-scale parameters (α, τ)
• multiScaleDiff: The m-point multi-scale difference operator
• spectralSymbol: The spectral symbol ψ_m(λ)
• SpectralBound: Flags for spectral bounds

Implementation notes

We use PNat for m to ensure m ≥ 1 at the type level, and separate positive reals for scale parameters τ to maintain mathematical validity.

structure Frourio.MultiScaleConfig (m : ℕ+) : Type

Configuration for m-point multi-scale parameters. Contains weights α and scales τ with the constraint that weights sum to zero.

• α : Fin ↑m → ℝ

  Weights for each scale, must sum to zero

• τ : Fin ↑m → ℝ

  Time scales, must be positive

• hτ_pos (i : Fin ↑m) : 0 < self.τ i

  Positivity constraint for scales

• hα_sum : ∑ i : Fin ↑m, self.α i = 0

  Zero-sum constraint for weights

• hα_bound (i : Fin ↑m) : |self.α i| ≤ 1

  Weights are bounded (for technical reasons)

structure Frourio.HeatSemigroup (X : Type u_1) : Type u_1

Abstract heat semigroup structure for multi-scale analysis

• H : ℝ → (X → ℝ) → X → ℝ

  The semigroup operator H_t

• isSemigroup (s t : ℝ) (φ : X → ℝ) : self.H s (self.H t φ) = self.H (s + t) φ

  Semigroup property: H_s ∘ H_t = H_{s+t}

• isIdentity (φ : X → ℝ) : self.H 0 φ = φ

  Identity at t=0: H_0 = I

• linear_in_function (t a b : ℝ) (φ ψ : X → ℝ) : (self.H t fun (x : X) => a * φ x + b * ψ x) = fun (x : X) => a * self.H t φ x + b * self.H t ψ x

  Linearity in function argument

• preserves_constants (t c : ℝ) : (self.H t fun (x : X) => c) = fun (x : X) => c

  Preserves constant functions

• measurable_in_function (t : ℝ) (φ : X → ℝ) [MeasurableSpace X] : Measurable φ → Measurable fun (x : X) => self.H t φ x

  Measurability preservation: under any measurable structure on X, if φ is measurable then so is H_t φ. We quantify the instance explicitly to avoid requiring [MeasurableSpace X] at the structure level.

• l2_continuous (t : ℝ) (φ : X → ℝ) [MeasurableSpace X] (μ : MeasureTheory.Measure X) : MeasureTheory.MemLp φ 2 μ → MeasureTheory.MemLp (fun (x : X) => self.H t φ x) 2 μ

  L² continuity: H_t preserves L² functions

• l2_contractive (t : ℝ) (φ : X → ℝ) [MeasurableSpace X] (μ : MeasureTheory.Measure X) : 0 ≤ t → MeasureTheory.MemLp φ 2 μ → MeasureTheory.MemLp (fun (x : X) => self.H t φ x) 2 μ

  L² contractivity: H_t is a contraction on L²

noncomputable def Frourio.multiScaleDiff {X : Type u_1} {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (φ : X → ℝ) : X → ℝ

The m-point multi-scale difference operator Δ^{⟨m⟩}{α,τ}. Definition: Δ^{⟨m⟩} φ := ∑ α_i H{τ_i} φ - (∑ α_i)I φ. Since ∑ α_i = 0 (zero-sum constraint), the second term vanishes, giving us Δ^{⟨m⟩} φ = ∑ α_i H_{τ_i} φ

Equations

• Frourio.multiScaleDiff H cfg φ x = ∑ i : Fin ↑m, cfg.α i * H.H (cfg.τ i) φ x

theorem Frourio.multiScaleDiff_linear {X : Type u_1} {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (a b : ℝ) (φ ψ : X → ℝ) : (multiScaleDiff H cfg fun (x : X) => a * φ x + b * ψ x) = fun (x : X) => a * multiScaleDiff H cfg φ x + b * multiScaleDiff H cfg ψ x

Basic linearity property of the multi-scale difference operator

theorem Frourio.multiScaleDiff_const_zero {X : Type u_1} {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (c : ℝ) : (multiScaleDiff H cfg fun (x : X) => c) = fun (x : X) => 0

Constants are annihilated by the multi-scale difference operator

theorem Frourio.multiScaleDiff_zero {X : Type u_1} {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) : (multiScaleDiff H cfg fun (x : X) => 0) = fun (x : X) => 0

The zero function is mapped to zero by the multi-scale difference operator

theorem Frourio.heatSemigroup_measurable {X : Type u_1} [MeasurableSpace X] (H : HeatSemigroup X) (t : ℝ) (φ : X → ℝ) (hφ : Measurable φ) : Measurable fun (x : X) => H.H t φ x

HeatSemigroup preserves measurability

theorem Frourio.heatSemigroup_l2_preserves {X : Type u_1} [MeasurableSpace X] (H : HeatSemigroup X) (t : ℝ) (μ : MeasureTheory.Measure X) (φ : X → ℝ) (hφ : MeasureTheory.MemLp φ 2 μ) : MeasureTheory.MemLp (fun (x : X) => H.H t φ x) 2 μ

HeatSemigroup preserves L² functions

theorem Frourio.heatSemigroup_l2_contraction {X : Type u_1} [MeasurableSpace X] (H : HeatSemigroup X) (t : ℝ) (ht : 0 ≤ t) (μ : MeasureTheory.Measure X) (φ : X → ℝ) (hφ : MeasureTheory.MemLp φ 2 μ) : MeasureTheory.MemLp (fun (x : X) => H.H t φ x) 2 μ

HeatSemigroup is L² contractive for non-negative time

theorem Frourio.multiScaleDiff_measurable {X : Type u_1} [MeasurableSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (φ : X → ℝ) (hφ : Measurable φ) : Measurable (multiScaleDiff H cfg φ)

Measurability property for multiScaleDiff

theorem Frourio.multiScaleDiff_square_integrable {X : Type u_1} [MeasurableSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (μ : MeasureTheory.Measure X) (φ : X → ℝ) (hφ : MeasureTheory.MemLp φ 2 μ) : MeasureTheory.MemLp (multiScaleDiff H cfg φ) 2 μ

Integrability property for multiScaleDiff squared

noncomputable def Frourio.spectralSymbol {m : ℕ+} (cfg : MultiScaleConfig m) (lam : ℝ) : ℝ

The spectral symbol ψ_m(λ) = ∑ α_i (exp(-τ_i λ) - 1) for λ ≥ 0

Equations

• Frourio.spectralSymbol cfg lam = ∑ i : Fin ↑m, cfg.α i * (Real.exp (-cfg.τ i * lam) - 1)

theorem Frourio.spectralSymbol_at_zero {m : ℕ+} (cfg : MultiScaleConfig m) : spectralSymbol cfg 0 = 0

The spectral symbol at λ = 0 vanishes

theorem Frourio.spectralSymbol_basic_bound {m : ℕ+} (cfg : MultiScaleConfig m) (lam : ℝ) (hlam : 0 ≤ lam) : |spectralSymbol cfg lam| ≤ ∑ i : Fin ↑m, |cfg.α i|

Basic bound: |ψ_m(λ)| ≤ ∑|α_i| for all λ ≥ 0

theorem Frourio.exp_diff_le {a b : ℝ} : |Real.exp a - Real.exp b| ≤ max (Real.exp a) (Real.exp b) * |a - b|

theorem Frourio.exp_diff_le_of_nonpos {a b : ℝ} (ha : a ≤ 0) (hb : b ≤ 0) : |Real.exp a - Real.exp b| ≤ |a - b|

Refined bound for exponential differences when arguments are non-positive

theorem Frourio.spectralSymbol_lipschitz {m : ℕ+} (cfg : MultiScaleConfig m) : ∃ (L : ℝ), 0 ≤ L ∧ ∀ (lam₁ lam₂ : ℝ), 0 ≤ lam₁ → 0 ≤ lam₂ → |spectralSymbol cfg lam₁ - spectralSymbol cfg lam₂| ≤ L * |lam₁ - lam₂|

The spectral symbol is Lipschitz continuous in λ

theorem Frourio.spectralSymbol_monotone_decreasing {m : ℕ+} (cfg : MultiScaleConfig m) (hα_nonneg : ∀ (i : Fin ↑m), 0 ≤ cfg.α i) (lam₁ lam₂ : ℝ) : 0 ≤ lam₁ → lam₁ ≤ lam₂ → spectralSymbol cfg lam₂ ≤ spectralSymbol cfg lam₁

Monotonicity: ψ_m(λ) is decreasing for λ ≥ 0 when all α_i ≥ 0

structure Frourio.SpectralBound {X : Type u_1} {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) : Type

Flags for spectral bounds and Bochner-type inequalities. These are assumptions at this stage, to be proved later.

• norm_bound : ℝ

  Uniform bound on the spectral symbol

• norm_bound_nonneg : 0 ≤ self.norm_bound

  The bound is non-negative

• spectral_uniform_bound (lam : ℝ) : 0 ≤ lam → |spectralSymbol cfg lam| ≤ self.norm_bound

  The spectral symbol is uniformly bounded

• bochner_inequality : Prop

  Bochner-type inequality relating L² norm to energy (Γ operator)

noncomputable def Frourio.spectralSymbolSupNorm {m : ℕ+} (cfg : MultiScaleConfig m) : ℝ

The sup-norm of the spectral symbol. Defined abstractly.

Equations

• Frourio.spectralSymbolSupNorm cfg = sSup {y : ℝ | ∃ (lam : ℝ), 0 ≤ lam ∧ y = |Frourio.spectralSymbol cfg lam|}

theorem Frourio.spectralSymbolSupNorm_bounded {m : ℕ+} (cfg : MultiScaleConfig m) : spectralSymbolSupNorm cfg ≤ ∑ i : Fin ↑m, |cfg.α i|

The sup-norm is bounded by ∑|α_i|

def Frourio.spectralSymbolLipschitzConstant {m : ℕ+} (cfg : MultiScaleConfig m) : ℝ

Explicit constant tracking: spectral symbol Lipschitz constant

Equations

• Frourio.spectralSymbolLipschitzConstant cfg = ∑ i : Fin ↑m, |cfg.α i| * cfg.τ i

theorem Frourio.spectralSymbolLipschitzConstant_nonneg {m : ℕ+} (cfg : MultiScaleConfig m) : 0 ≤ spectralSymbolLipschitzConstant cfg

The Lipschitz constant is non-negative

theorem Frourio.spectralSymbol_lipschitz_constant_eq {m : ℕ+} (cfg : MultiScaleConfig m) : ∃ (L : ℝ), L = spectralSymbolLipschitzConstant cfg ∧ ∀ (lam₁ lam₂ : ℝ), 0 ≤ lam₁ → 0 ≤ lam₂ → |spectralSymbol cfg lam₁ - spectralSymbol cfg lam₂| ≤ L * |lam₁ - lam₂|

theorem Frourio.spectralSymbolSupNorm_explicit_bound {m : ℕ+} (cfg : MultiScaleConfig m) (C : ℝ) (hC : ∀ (i : Fin ↑m), |cfg.α i| ≤ C) : spectralSymbolSupNorm cfg ≤ ↑↑m * C

Explicit bound when all coefficients satisfy |α_i| ≤ C

theorem Frourio.spectralSymbolSupNorm_optimal_bound {m : ℕ+} (cfg : MultiScaleConfig m) : spectralSymbolSupNorm cfg ≤ ↑↑m

Optimal bound: When cfg.hα_bound gives |α_i| ≤ 1, the sup-norm is at most m

theorem Frourio.spectralSymbol_zero_at_zero {m : ℕ+} (cfg : MultiScaleConfig m) : |spectralSymbol cfg 0| = 0

The spectral symbol achieves its sup-norm at λ = 0 when all α_i have the same sign

def Frourio.spectralSymbolDerivativeBound {m : ℕ+} (cfg : MultiScaleConfig m) : ℝ

Precise constant for the derivative bound of spectral symbol

Equations

• Frourio.spectralSymbolDerivativeBound cfg = ∑ i : Fin ↑m, |cfg.α i| * cfg.τ i

theorem Frourio.spectralSymbol_derivative_bound {m : ℕ+} (cfg : MultiScaleConfig m) (lam : ℝ) (hlam : 0 ≤ lam) (hderiv : deriv (spectralSymbol cfg) lam = ∑ i : Fin ↑m, -cfg.α i * cfg.τ i * Real.exp (-cfg.τ i * lam)) : |deriv (spectralSymbol cfg) lam| ≤ spectralSymbolDerivativeBound cfg

The derivative of spectral symbol at any point is bounded

structure Frourio.FGStarConstants {m : ℕ+} (cfg : MultiScaleConfig m) : Type

Precise tracking of the constant C(ℰ) in FG★ inequality

• spectral_const : ℝ

  Constant from spectral bound

• energy_const : ℝ

  Constant from energy functional properties

• combined_const : ℝ

  Combined constant for FG★ inequality

• const_relation : self.combined_const = self.spectral_const * self.energy_const

  Relation between constants

• spectral_const_nonneg : 0 ≤ self.spectral_const

  All constants are non-negative

• energy_const_nonneg : 0 ≤ self.energy_const

noncomputable def Frourio.constructFGStarConstants {m : ℕ+} (cfg : MultiScaleConfig m) : FGStarConstants cfg

Construction of FG★ constants with explicit values

Equations

• One or more equations did not get rendered due to their size.

structure Frourio.OptimalSpectralBound {m : ℕ+} (cfg : MultiScaleConfig m) : Type

Refined bound with optimal constant tracking

• C_opt : ℝ

  The optimal bound constant

• C_opt_nonneg : 0 ≤ self.C_opt

  Non-negativity

• is_sharp : ∃ (lam : ℝ), 0 ≤ lam ∧ |spectralSymbol cfg lam| = self.C_opt

  The bound is sharp (achieved for some λ)

• uniform_bound (lam : ℝ) : 0 ≤ lam → |spectralSymbol cfg lam| ≤ self.C_opt

  The bound holds uniformly

def Frourio.spectralBoundHypothesis {m : ℕ+} (cfg : MultiScaleConfig m) : Prop

Auxiliary hypothesis: the spectral symbol is bounded. This would follow from detailed Fourier analysis.

Equations

• Frourio.spectralBoundHypothesis cfg = ∀ (lam : ℝ), |Frourio.spectralSymbol cfg lam| ≤ 1

theorem Frourio.le_spectralSymbolSupNorm {m : ℕ+} (cfg : MultiScaleConfig m) (lam : ℝ) (hlam : 0 ≤ lam) : |spectralSymbol cfg lam| ≤ spectralSymbolSupNorm cfg

Under the hypothesis that the spectral symbol is bounded, it is bounded by the sup-norm.

structure Frourio.SpectralSupNormBound {m : ℕ+} (cfg : MultiScaleConfig m) : Type

Alternative formulation: explicit sup-norm bound flag

• bound : ℝ

  The sup-norm bound value

• bound_nonneg : 0 ≤ self.bound

  Non-negativity

• is_bound : spectralSymbolSupNorm cfg ≤ self.bound

  The actual bound

structure Frourio.SpectralPenalty {X : Type u_1} [MeasurableSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) : Type

Spectral penalty evaluation for the multi-scale difference operator. This encodes the key estimate: ∫|Δ^{⟨m⟩} φ|² dμ ≤ C(ℰ)‖ψ_m‖_∞² ∫Γ(φ,φ) dμ

• C_dirichlet : ℝ

  The constant C(ℰ) depending on the Dirichlet form

• C_nonneg : 0 ≤ self.C_dirichlet

  Non-negativity of the constant