Step 4: Bridge towards kernel-dimension = zero multiplicity (statements).

We set up propositional interfaces that connect vanishing of the ζ-kernel quadratic form to vanishing at the ζ zeros with a specified multiplicity. Proofs are deferred to the subsequent chapter.

def Frourio.RH_pred : Prop

RH predicate (placeholder). Will encapsulate the Riemann Hypothesis.

Equations

• Frourio.RH_pred = True

noncomputable def Frourio.Mult (_τ0 : ℝ) : ℕ

Abstract multiplicity of a zero at τ₀ on the critical line (placeholder).

Equations

• Frourio.Mult _τ0 = 0

def Frourio.VanishAtZeros : ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume) → (ℝ → ℕ) → Prop

Predicate: the L² trace g = Uσ f vanishes at the ζ zeros with the specified multiplicities (design-level placeholder).

Equations

• Frourio.VanishAtZeros x✝¹ x✝ = True

theorem Frourio.zero_enforces_vanishing [ZetaLineAPI] (σ : ℝ) (f : ↥(Hσ σ)) : Qζσ σ f = 0 → VanishAtZeros (MeasureTheory.MemLp.toLp (LogPull σ f) ⋯) Mult

Statement: If Qζσ σ f = 0, then the Mellin transform vanishes at the ζ zeros with multiplicities recorded by Mult. This is the intended endpoint of the bridge; a full proof will rely on golden-lattice sampling, Γ-convergence, and kernel characterizations from previous chapters.