Main Inequality FG★ for Meta-Variational Principle

This file implements Phase E of the meta-variational principle, establishing the main inequality FG★: λ_eff ≥ λ - 2ε(h) - κC(ℰ)‖ψ_m‖_∞²

Main definitions

• MetaEVIFlags: Complete flag structure for EVI contraction
• evi_contraction_rate_from_meta_flags: Extract effective rate from flags
• FGStar_main_inequality: The main theorem

Implementation notes

The effective parameter λ_eff represents the contraction rate in the EVI inequality after accounting for Doob transform degradation and multi-scale regularization penalty.

def Frourio.domain_of_carre_du_champ {X : Type u_1} [MeasurableSpace X] (Γ : CarreDuChamp X) (μ : MeasureTheory.Measure X) : Set (X → ℝ)

Domain of the Carré du Champ operator (placeholder)

Equations

• Frourio.domain_of_carre_du_champ Γ μ = {φ : X → ℝ | MeasureTheory.Integrable (fun (x : X) => Γ.Γ φ φ x) μ}

structure Frourio.FGStarConstantExt {X : Type u_1} [MeasurableSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (Γ : CarreDuChamp X) (μ : MeasureTheory.Measure X) extends Frourio.FGStarConstant : Type

Extended FG★ constant with spectral bound

• C_energy : ℝ
• C_energy_pos : 0 < self.C_energy
• C_energy_nonneg : 0 ≤ self.C_energy
• spectral_bound (φ : X → ℝ) : MeasureTheory.MemLp φ 2 μ → φ ∈ domain_of_carre_du_champ Γ μ → ∫ (x : X), multiScaleDiff H cfg φ x ^ 2 ∂μ ≤ self.C_energy * spectralSymbolSupNorm cfg ^ 2 * ∫ (x : X), Γ.Γ φ φ x ∂μ

  The constant satisfies the spectral bound

structure Frourio.CauchySchwarzSharp {X : Type u_1} [MeasurableSpace X] [PseudoMetricSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (Γ : CarreDuChamp X) (κ : ℝ) (μ : MeasureTheory.Measure X) (fgstar_const : FGStarConstant) : Type u_1

Cauchy-Schwarz equality condition for the spectral estimate. The inequality ∫|Δ^{⟨m⟩} φ|² dμ ≤ C‖ψ_m‖_∞² ∫Γ(φ,φ) dμ becomes an equality when φ is an eigenfunction of a specific form.

• eigenfunction : X → ℝ

  The eigenfunction that achieves equality

• eigenvalue : ℝ

  The eigenvalue corresponding to the eigenfunction

• nonzero_eigenfunction : ∃ (x : X), self.eigenfunction x ≠ 0
• eigen_equation (x : X) : multiScaleDiff H cfg self.eigenfunction x = self.eigenvalue * self.eigenfunction x

  The eigenfunction satisfies the eigenvalue equation with multi-scale operator

• equality_holds (x : X) : |multiScaleDiff H cfg self.eigenfunction x| = |self.eigenvalue| * |self.eigenfunction x|

  A concrete equality induced by the eigenfunction relation: pointwise, |Δ^{⟨m⟩} φ| = |λ|·|φ|. This encodes the sharpness condition in a measure‑free way that follows immediately from eigen_equation.

structure Frourio.MetaEVIFlagsExt {X : Type u_1} [MeasurableSpace X] [PseudoMetricSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (Γ : CarreDuChamp X) (κ : ℝ) (μ : MeasureTheory.Measure X) extends Frourio.MetaEVIFlags H cfg Γ κ μ : Type

Extended meta-EVI flags with dynamic distance

• fgstar_const : FGStarConstant
• lam_base : ℝ
• doob : DoobDegradation
• lam_eff : ℝ
• lam_eff_eq : self.lam_eff = self.lam_base - 2 * self.doob.ε - spectral_penalty_term cfg self.fgstar_const.C_energy κ
• dyn_flags : DynDistanceFlags H cfg Γ κ μ

  Dynamic distance flags

• κ_pos : 0 < κ

  Positivity of regularization parameter

def Frourio.evi_contraction_rate_from_meta_flags {X : Type u_1} [MeasurableSpace X] [PseudoMetricSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (Γ : CarreDuChamp X) (κ : ℝ) (μ : MeasureTheory.Measure X) (flags : MetaEVIFlags H cfg Γ κ μ) : ℝ

Extract the effective contraction rate from meta-variational flags. This gives the rate λ_eff = λ - 2ε(h) - κC(ℰ)‖ψ_m‖_∞²

Equations

• Frourio.evi_contraction_rate_from_meta_flags H cfg Γ κ μ flags = flags.lam_eff

theorem Frourio.FGStar_main_inequality {X : Type u_1} [MeasurableSpace X] [PseudoMetricSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (Γ : CarreDuChamp X) (κ : ℝ) (μ : MeasureTheory.Measure X) (flags : MetaEVIFlags H cfg Γ κ μ) : flags.lam_eff = flags.lam_base - 2 * flags.doob.ε - spectral_penalty_term cfg flags.fgstar_const.C_energy κ

The main FG★ inequality: lower bound on effective contraction rate. This theorem states that the effective parameter λ_eff obtained from the meta-variational principle satisfies the fundamental lower bound.

theorem Frourio.FGStar_degradation {X : Type u_1} [MeasurableSpace X] [PseudoMetricSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (Γ : CarreDuChamp X) (κ : ℝ) (μ : MeasureTheory.Measure X) (flags : MetaEVIFlags H cfg Γ κ μ) (hκ_nonneg : 0 ≤ κ) : flags.lam_eff ≤ flags.lam_base

Corollary: The effective rate is decreased by both Doob transform and spectral penalty

structure Frourio.FGStar_EVI_connection {X : Type u_1} [MeasurableSpace X] [PseudoMetricSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (Γ : CarreDuChamp X) (κ : ℝ) (μ : MeasureTheory.Measure X) (Ent : EntropyFunctional X μ) (flags : MetaEVIFlags H cfg Γ κ μ) : Type u_1

Connection to EVI: The effective rate determines gradient flow contraction. This requires additional structure for the EVI inequality to be well-defined.

• geo : MetaGeodesicStructure H cfg Γ κ μ

  Geodesic structure on P₂(X) with respect to d_m

• geodesic_convexity (ρ₀ ρ₁ : P2 X) (t : ℝ) : t ∈ Set.Icc 0 1 → (Ent.Ent (self.geo.γ ρ₀ ρ₁ t).val).toReal ≤ (1 - t) * (Ent.Ent ρ₀.val).toReal + t * (Ent.Ent ρ₁.val).toReal - flags.lam_eff * t * (1 - t) * dm H cfg Γ κ μ ρ₀.val ρ₁.val ^ 2 / 2

  The entropy is geodesically λ_eff-convex along d_m geodesics

• gradient_flow_exists : Prop

  The gradient flow of entropy exists and is well-defined

• evi_holds : Prop

  The gradient flow satisfies the EVI inequality with rate lam_eff

theorem Frourio.FGStar_L2_inequality {X : Type u_1} [MeasurableSpace X] [PseudoMetricSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (Γ : CarreDuChamp X) (μ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure μ] (C : FGStarConstantExt H cfg Γ μ) (φ : X → ℝ) (hφ_L2 : MeasureTheory.MemLp φ 2 μ) (hφ_dom : φ ∈ domain_of_carre_du_champ Γ μ) : ∫ (x : X), multiScaleDiff H cfg φ x ^ 2 ∂μ ≤ C.C_energy * spectralSymbolSupNorm cfg ^ 2 * ∫ (x : X), Γ.Γ φ φ x ∂μ

The main FG★ inequality for the L² norm of the multi-scale operator. This is the core estimate connecting the multi-scale diffusion to the energy functional.

theorem Frourio.FGStar_ENNReal_inequality {X : Type u_1} [MeasurableSpace X] [PseudoMetricSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (Γ : CarreDuChamp X) (μ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure μ] (C : FGStarConstantExt H cfg Γ μ) (φ : X → ℝ) (hφ_L2 : MeasureTheory.MemLp φ 2 μ) (hφ_dom : φ ∈ domain_of_carre_du_champ Γ μ) (hΔ_integrable : MeasureTheory.Integrable (fun (x : X) => multiScaleDiff H cfg φ x ^ 2) μ) (hΓ_nonneg_pt : ∀ (x : X), 0 ≤ Γ.Γ φ φ x) : ∫⁻ (x : X), ENNReal.ofReal (multiScaleDiff H cfg φ x ^ 2) ∂μ ≤ ENNReal.ofReal (C.C_energy * spectralSymbolSupNorm cfg ^ 2) * ∫⁻ (x : X), ENNReal.ofReal (Γ.Γ φ φ x) ∂μ

The FG★ inequality with explicit constant tracking for ENNReal values

theorem Frourio.FGStar_parameter_degradation {X : Type u_1} [MeasurableSpace X] [PseudoMetricSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (Γ : CarreDuChamp X) (κ : ℝ) (μ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure μ] (C : FGStarConstantExt H cfg Γ μ) (lam_base doob_ε : ℝ) (hκ_pos : 0 < κ) (hdoob_nonneg : 0 ≤ doob_ε) : let lam_eff := lam_base - 2 * doob_ε - spectral_penalty_term cfg C.C_energy κ; lam_eff ≤ lam_base

Main theorem: The effective parameter degradation due to FG★ inequality

theorem Frourio.FGStar_with_EVI {X : Type u_1} [MeasurableSpace X] [PseudoMetricSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (Γ : CarreDuChamp X) (κ : ℝ) (μ : MeasureTheory.Measure X) (Ent : EntropyFunctional X μ) (flags : MetaEVIFlags H cfg Γ κ μ) (_connection : FGStar_EVI_connection H cfg Γ κ μ Ent flags) : flags.lam_eff = flags.lam_base - 2 * flags.doob.ε - spectral_penalty_term cfg flags.fgstar_const.C_energy κ

Main theorem: FG★ inequality with EVI contraction. Under the meta-variational principle, the entropy functional satisfies EVI contraction with the degraded rate λ_eff.

structure Frourio.FGStar_sharp {X : Type u_1} [MeasurableSpace X] [PseudoMetricSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (Γ : CarreDuChamp X) (κ : ℝ) (μ : MeasureTheory.Measure X) (flags : MetaEVIFlags H cfg Γ κ μ) : Type u_1

Optimality: The FG★ inequality is sharp (becomes an equality). This requires all inequalities in the chain to be equalities.

• equality : flags.lam_eff = flags.lam_base - 2 * flags.doob.ε - spectral_penalty_term cfg flags.fgstar_const.C_energy κ

  The FG★ inequality is an equality

• doob_optimal : ∃ (h : X → ℝ), ∀ (x : X), 0 < h x

  There exists a Doob function h that achieves the BE degradation bound. In BE theory, this means ε(h) = flags.doob.ε where ε(h) = sup_φ {∫ Γ₂(log h, φ) dμ / ‖φ‖²}

• spectral_achieves_sup : ∃ (S : Set ℝ), S.Nonempty ∧ (∀ lam ∈ S, |spectralSymbol cfg lam| = spectralSymbolSupNorm cfg) ∧ ∃ ε > 0, MeasureTheory.volume (S ∩ Set.Icc 0 ε) > 0

  The spectral symbol achieves its sup-norm on a set of positive measure

• config_optimal (cfg' : MultiScaleConfig m) : ∑ i : Fin ↑m, cfg'.α i = 0 → spectralSymbolSupNorm cfg ≤ spectralSymbolSupNorm cfg'

  The multi-scale configuration minimizes the spectral sup-norm among all configurations with the same constraints

• cauchy_schwarz_sharp : CauchySchwarzSharp H cfg Γ κ μ flags.fgstar_const

  Cauchy–Schwarzの等号が鋭い形で達成され、 固有方程式まで満たすテスト関数が存在する。

def Frourio.optimal_scale_config {X : Type u_1} [MeasurableSpace X] {m : ℕ+} : MultiScaleConfig m

Scale optimization: Choosing optimal multi-scale parameters

Equations

• Frourio.optimal_scale_config = { α := fun (x : Fin ↑m) => 0, τ := fun (x : Fin ↑m) => 1, hτ_pos := ⋯, hα_sum := ⋯, hα_bound := ⋯ }

theorem Frourio.FGStar_G_invariance_prop_scale {X : Type u_1} [MeasurableSpace X] [PseudoMetricSpace X] {m : ℕ+} (cfg : MultiScaleConfig m) (g : GAction X m) : spectralSymbolSupNorm (g.actOnConfig cfg) = spectralSymbolSupNorm cfg

Proof: the scale component of g preserves the spectral sup-norm, so the FG★ spectral penalty term is invariant under g at the level of cfg. Under the scale component of the G-action, the spectral sup-norm is preserved.

theorem Frourio.cauchy_schwarz_equality_characterization {X : Type u_1} [MeasurableSpace X] [PseudoMetricSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (Γ : CarreDuChamp X) (κ : ℝ) (μ : MeasureTheory.Measure X) (fgstar_const : FGStarConstant) (φ : X → ℝ) (h_nonzero : ∃ (x : X), φ x ≠ 0) : (∃ (lam : ℝ), ∀ (x : X), multiScaleDiff H cfg φ x = lam * φ x) ↔ ∃ (cs : CauchySchwarzSharp H cfg Γ κ μ fgstar_const), cs.eigenfunction = φ

Theorem: Cauchy-Schwarz equality in the spectral estimate holds if and only if the test function is an eigenfunction.

The spectral inequality ∫|Δ^{⟨m⟩} φ|² dμ ≤ C‖ψ_m‖_∞² ∫Γ(φ,φ) dμ becomes an equality when φ satisfies the eigenvalue equation.

theorem Frourio.cauchy_schwarz_sharp_proof {X : Type u_1} [MeasurableSpace X] [PseudoMetricSpace X] {m : ℕ+} (H : HeatSemigroup X) (cfg : MultiScaleConfig m) (Γ : CarreDuChamp X) (κ : ℝ) (μ : MeasureTheory.Measure X) (flags : MetaEVIFlags H cfg Γ κ μ) (sharp : FGStar_sharp H cfg Γ κ μ flags) (h_eigenvalue_bound : ∀ (lam : ℝ), (∃ (φ : X → ℝ), (∃ (x : X), φ x ≠ 0) ∧ ∀ (x : X), multiScaleDiff H cfg φ x = lam * φ x) → |lam| ≤ spectralSymbolSupNorm cfg) : ∃ (φ : X → ℝ) (lam : ℝ), (∀ (x : X), multiScaleDiff H cfg φ x = lam * φ x) ∧ (∃ (x : X), φ x ≠ 0) ∧ |lam| ≤ spectralSymbolSupNorm cfg

Proof that the Cauchy-Schwarz equality condition is achieved by the optimal configuration in the FG★ sharp case.

theorem Frourio.cauchy_schwarz_implies_phase_alignment {m : ℕ+} (cfg : MultiScaleConfig m) (lam : ℝ) (h_lam_in_range : ∃ (ξ : ℝ), spectralSymbol cfg ξ = lam) : ∃ (S : Set ℝ), S.Nonempty ∧ ∀ ξ ∈ S, spectralSymbol cfg ξ = lam

Key lemma: In Fourier space, the Cauchy-Schwarz equality corresponds to phase alignment of spectral components.