def Frourio.RH : Prop

RH predicate (abstract): all nontrivial zeros lie on Re s = 1/2. At this phase, we keep it as a single Prop to be connected with the ζ API later.

Equations

• Frourio.RH = True

structure Frourio.Prepared (σ : ℝ) (f : ℕ → ↥(Hσ σ)) : Type

Preparedness conditions for a golden-lattice test sequence. This bundles the assumptions coming from plan2: frame bounds, Γ-収束, and Gaussian width control. Each field is a Prop placeholder to keep the API light.

• frame : Prop
• gamma : Prop
• width : Prop

def Frourio.FW_criterion [ZetaLineAPI] (σ : ℝ) : Prop

Frourio–Weil criterion at height σ: for every prepared golden test sequence, each element has nonnegative ζ-quadratic energy, and if it is zero then the Mellin trace vanishes at ζ zeros with the prescribed multiplicity.

Equations

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

def Frourio.disc_consistency [ZetaLineAPI] (_σ : ℝ) (_F : GoldenTestSeq _σ) : Prop

Auxiliary: discrete–continuous consistency of Qζ along prepared golden sequences.

Equations

• Frourio.disc_consistency _σ _F = True

def Frourio.kernel_multiplicity_bridge [ZetaLineAPI] (_σ : ℝ) (_F : GoldenTestSeq _σ) : Prop

Auxiliary: kernel–multiplicity bridge specialized to elements of a prepared sequence.

Equations

• Frourio.kernel_multiplicity_bridge _σ _F = True

def Frourio.off_critical_contradiction : Prop

Auxiliary: contradiction derived from an off-critical zero using the prepared toolkit.

Equations

• Frourio.off_critical_contradiction = True

def Frourio.concentrates_at [ZetaLineAPI] (σ : ℝ) (F : GoldenTestSeq σ) (τ₀ : ℝ) : Prop

Concentration at τ₀ along a golden test sequence: the Mellin trace mass outside any fixed neighborhood of τ₀ goes to zero.

Equations

• Frourio.concentrates_at σ F τ₀ = ∀ ε > 0, ∃ (N : ℕ), ∀ n ≥ N, ∫ (τ : ℝ) in {τ : ℝ | |τ - τ₀| > ε}, ‖Frourio.LogPull σ (F.f n) τ‖ ^ 2 < ε

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

Standard golden test sequence with δ n = 1/(n+1)

• f : ℕ → ↥(Hσ σ)
• δ : ℕ → ℝ
• hδ_pos (n : ℕ) : 0 < self.δ n
• hδ_lim : Filter.Tendsto self.δ Filter.atTop (nhds 0)
• hδ_bound (n : ℕ) : self.δ n ≤ 1 / (↑n + 1)
• gaussian_form (_n : ℕ) : ∃ (_τ₀ : ℝ) (w : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)), ‖w‖ = 1
• variational_property (n : ℕ) (y : ↥(Hσ σ)) : Qζσ σ (self.f n) ≤ Qζσ σ y + self.δ n
• δ_standard (n : ℕ) : self.δ n = 1 / (↑n + 1)

  The width parameter has the standard form

def Frourio.exists_golden_peak [ZetaLineAPI] (σ : ℝ) : Prop

Auxiliary: existence of a golden-lattice peak sequence concentrating at a target zero.

Equations

• Frourio.exists_golden_peak σ = ∀ (τ₀ : ℝ), ∃ (F : Frourio.GoldenTestSeq σ), Frourio.concentrates_at σ F τ₀

theorem Frourio.limiting_energy_nonneg [ZetaLineAPI] (σ : ℝ) (f : ↥(Hσ σ)) : 0 ≤ limiting_energy σ f

The limiting_energy is non-negative for elements in our construction

def Frourio.gamma_converges_to (σ : ℝ) (E_n : ℕ → ↥(Hσ σ) → ℝ) (E : ↥(Hσ σ) → ℝ) : Prop

Predicate for Γ-convergence of a sequence of functionals to a limit functional. For golden sequences, we track convergence of the energy functionals.

Equations

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

theorem Frourio.disc_consistency_proof [ZetaLineAPI] (σ : ℝ) (F : GoldenTestSeq σ) : disc_consistency σ F

Phase 3.2: Discrete–continuous consistency along prepared golden sequences (statement-level). With current placeholders for bounds and Γ-収束, we record the result as a direct proof of the Prop scaffold.

theorem Frourio.FW_implies_RH [ZetaLineAPI] (σ : ℝ) : FW_criterion σ → RH

(ii) ⇒ (i): From the Frourio–Weil criterion at height σ, conclude RH. At this phase, RH is an abstract predicate and the bridge lemmas are recorded as propositional placeholders to be instantiated in later phases.

theorem Frourio.RH_implies_FW [ZetaLineAPI] (σ : ℝ) : RH → FW_criterion σ

(i) ⇒ (ii): Assuming RH, every prepared golden test sequence satisfies the Frourio–Weil criterion. In this phase, nonnegativity comes from Qζσ_pos and the vanishing implication comes from the multiplicity bridge placeholder.