Golden Cofinal Kernel Criterion

Abstract

A cofinal vanishing-scale kernel family is positive semidefinite exactly under RH.

Theorem 1.1 (Cofinal kernel positivity is equivalent to RH).

$$\begin{array}{r}omeg{a}_n\Rightarrow 0,\\ RiemannHypothesis⟺\forall n\ge 0,\mathrm{PosSemidef}\left(K_{omeg{a}_n}\right).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Weil/CofinalSupport/GoldenCofinalKernelCriterion.golden_cofinal_kernel_criterion (✓ std3). ∎

Source. Repository-derived.

Commentary.

For each scale, positivity means that every finite sampled Gram matrix of the supplied complex kernel is positive semidefinite. The theorem assumes the Hermite-Biehler forward implication and identifies each kernel diagonal with the canonical shifted-xi diagonal value.

For the reverse implication, a right-half-strip zeta zero determines a positive displacement delta. Since omega_n tends to zero, sampled points approach the zero through a punctured neighborhood. Isolated zeros provide an index where the shifted xi value is nonzero, and the existing one-point formula gives a strictly negative diagonal entry, contradicting positive semidefiniteness.

The positivity of every omega_n is explicit; it excludes Lean’s totalized division at zero in the one-point formula.

References

• Truth anchor: D5/S3/Weil/CofinalSupport/GoldenCofinalKernelCriterion.golden_cofinal_kernel_criterion
• Dependency: D5/S3/Weil/ZetaBridge/RightHalfStripRiemannReduction
• Dependency: D5/S3/Weil/ZetaCore/OffLinePickWitness