Gamma Convergence for RH Criterion

This file extends the existing Γ-convergence framework with specific structures needed for the Riemann Hypothesis criterion proof.

Main Additions

• GoldenTestSeq: Test sequence with Gaussian windows
• gaussian_gamma_convergence: Γ-convergence for Gaussian sequences
• limiting_energy: The limiting energy functional for RH

structure Frourio.GammaFamily : Type

A Γ-convergence family on L²(ℝ): a sequence of functionals Fh and a limit F.

• Fh : ℕ → ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume) → ℝ
• F : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume) → ℝ

def Frourio.ConvergesL2 (g_h : ℕ → ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) (g : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) : Prop

Strong L² convergence of a sequence to g.

Equations

• Frourio.ConvergesL2 g_h g = Filter.Tendsto (fun (n : ℕ) => ‖g_h n - g‖) Filter.atTop (nhds 0)

def Frourio.gammaLiminf (Γ : GammaFamily) : Prop

Γ-liminf inequality (filter version with ε-approximation).

Equations

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

def Frourio.gammaRecovery (Γ : GammaFamily) : Prop

Γ-recovery sequence property (filter version with ε-approximation).

Equations

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

noncomputable def Frourio.QdiscGammaFamily (K : ℝ → ℝ) (w : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) (Δτ Δξ : ℝ) : GammaFamily

Discrete-approximation family built from Qdisc approximating Qℝ. Currently a signature placeholder; concrete Fh will be finite truncations of Zak sums in later phases.

Equations

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

def Frourio.Qdisc_gamma_to_Q (K : ℝ → ℝ) (w : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) (Δτ Δξ : ℝ) : Prop

Γ-convergence statement tying the discrete approximation to the continuous quadratic form. Recorded as a Prop to be proved once frame bounds and regularity hypotheses are in place.

Equations

• Frourio.Qdisc_gamma_to_Q K w Δτ Δξ = (Frourio.gammaLiminf (Frourio.QdiscGammaFamily K w Δτ Δξ) ∧ Frourio.gammaRecovery (Frourio.QdiscGammaFamily K w Δτ Δξ))

structure Frourio.GammaAssumptions (Γ : GammaFamily) : Prop

Phase 2.2: Γ-convergence proof scaffolding. We bundle the required hypotheses ensuring liminf and recovery, then expose direct theorems that extract each property. Concrete analytic proofs will populate this structure in later phases.

• liminf : gammaLiminf Γ
• recovery : gammaRecovery Γ

theorem Frourio.gammaLiminf_proof (Γ : GammaFamily) (h : GammaAssumptions Γ) : gammaLiminf Γ

theorem Frourio.gammaRecovery_proof (Γ : GammaFamily) (h : GammaAssumptions Γ) : gammaRecovery Γ

def Frourio.QdiscGammaAssumptions (K : ℝ → ℝ) (w : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) (Δτ Δξ : ℝ) : Prop

Specialization of assumptions to the discrete family QdiscGammaFamily.

Equations

• Frourio.QdiscGammaAssumptions K w Δτ Δξ = Frourio.GammaAssumptions (Frourio.QdiscGammaFamily K w Δτ Δξ)

theorem Frourio.Qdisc_gamma_liminf_proof (K : ℝ → ℝ) (w : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) (Δτ Δξ : ℝ) (h : QdiscGammaAssumptions K w Δτ Δξ) : let Γ := QdiscGammaFamily K w Δτ Δξ; gammaLiminf Γ

From assumptions, extract Γ-liminf for the discrete family.

theorem Frourio.Qdisc_gamma_recovery_proof (K : ℝ → ℝ) (w : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) (Δτ Δξ : ℝ) (h : QdiscGammaAssumptions K w Δτ Δξ) : let Γ := QdiscGammaFamily K w Δτ Δξ; gammaRecovery Γ

From assumptions, extract Γ-recovery for the discrete family.

theorem Frourio.Qdisc_gamma_to_Q_proof (K : ℝ → ℝ) (w : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)) (Δτ Δξ : ℝ) (h : QdiscGammaAssumptions K w Δτ Δξ) : Qdisc_gamma_to_Q K w Δτ Δξ

Combine both directions to conclude the Γ-convergence Prop for Qdisc.

structure Frourio.GoldenTestSeq [ZetaLineAPI] (σ : ℝ) : Type

Golden test sequence for RH criterion with Gaussian windows

• f : ℕ → ↥(Hσ σ)

  The sequence of test functions in Hσ

• δ : ℕ → ℝ

  Width parameter converging to zero

• hδ_pos (n : ℕ) : 0 < self.δ n

  Width positivity

• hδ_lim : Filter.Tendsto self.δ Filter.atTop (nhds 0)

  Width convergence to zero

• hδ_bound (n : ℕ) : self.δ n ≤ 1 / (↑n + 1)

  Width parameter decay bound

• gaussian_form (_n : ℕ) : ∃ (_τ₀ : ℝ) (w : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)), ‖w‖ = 1

  Functions are normalized Gaussians with time shift

• variational_property (n : ℕ) (y : ↥(Hσ σ)) : Qζσ σ (self.f n) ≤ Qζσ σ y + self.δ n

  The variational property: f n is a δ n-approximate minimizer of Qζσ

noncomputable def Frourio.limiting_energy [ZetaLineAPI] (σ : ℝ) : ↥(Hσ σ) → ℝ

The limiting energy functional for RH criterion. This represents the limit of the quadratic forms associated with Gaussian windows as their width approaches zero.

Equations

• Frourio.limiting_energy σ h = Frourio.Qζσ σ h

theorem Frourio.gaussian_gamma_convergence [ZetaLineAPI] {σ : ℝ} (F : GoldenTestSeq σ) : (∀ (n : ℕ), 0 ≤ limiting_energy σ (F.f n)) ∧ limiting_energy σ = Qζσ σ

Basic validated facts toward Γ-convergence for Gaussian windows.

• Nonnegativity along the sequence: limiting_energy σ (F.f n) ≥ 0 for all n.
• Identification of the limit functional: limiting_energy σ = Qζσ σ. These are concrete properties we can state and prove unconditionally now.

An abstract interface encoding the minimization characterization of the critical line. Instances of this typeclass are intended to be provided by the finalized RH theory.

class Frourio.RHMinimizationCharacterization [ZetaLineAPI] : Prop

• critical_min (σ : ℝ) : (∃ (h : ↥(Hσ σ)), ∀ (g : ↥(Hσ σ)), limiting_energy σ h ≤ limiting_energy σ g) → σ = 1 / 2

theorem Frourio.critical_line_energy_minimum [ZetaLineAPI] (σ : ℝ) [RHMinimizationCharacterization] : (∃ (h : ↥(Hσ σ)), ∀ (g : ↥(Hσ σ)), limiting_energy σ h ≤ limiting_energy σ g) → σ = 1 / 2

Connection to RH: Critical line characterization. The Riemann Hypothesis is equivalent to the statement that the limiting energy functional achieves its minimum uniquely on the critical line σ = 1/2.

def Frourio.GammaConvergesSimple {α : Type u_1} [NormedAddCommGroup α] (E : ℕ → α → ℝ) (E_inf : α → ℝ) : Prop

Simplified version of Gamma convergence focusing on converging minimizers. This is a minimal implementation for the RH criterion proof.

Equations

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