Certified Convex Positivity for the Pole-Inclusive Weil Form at the Prime-Free Endpoint

We establish a certified coercive lower bound for the pole-inclusive archimedean Weil quadratic form at the complete prime-free endpoint. For every real test function ψ supported on an interval of length at most log 2 and satisfying the stated logarithmic Fourier-integrability condition, we prove Q₀(ψ) ≥ 1.2 × 10⁻⁴ ||ψ||²₂. The proof converts the quadratic form exactly into a convex functional of the normalized autocorrelation. Positive definiteness then yields a finite dual-certificate criterion. At D = log 2, the criterion is verified by a positive atomic measure with 484 atoms using outward-rounded Arb ball arithmetic. The verification includes an analytic origin gate, a fixed uniform pass over 500,000 cells, certified evaluation of the singular-kernel constants, a rigorous ladder-tail bound, and provable interval comparisons. The record includes the article, the machine-readable certificate, an independent verifier, the execution environment, the verification log, and SHA-256 checksums. The paper also proves the exact Gaussian transfer identity Q_A(μ) = Q₀(φ_A * μ) and places the result in relation to the work of Yoshida, Bombieri, Connes–Consani, Suzuki, and subsequent computational approaches to the Weil functional.