Localized Weil Positivity Beyond the Prime-7 Threshold: Relative Coupling and Finite-Rank Tail Reduction at $X = 7.01$

We study the bottom $\\lambda_a$ of the spectrum associated with the closed localization $Q_W^a$ of the Weil quadratic form in the Connes–Consani–Suzuki operator-theoretic framework. At $$X = 7.01, \\qquad a = \\frac{1}{2}\\log(7.01),$$ we prove, by a rigorous computer-assisted argument, that $$\\lambda_a > 0.$$ By variational monotonicity, the same conclusion holds for every $$0 < a \\le \\frac{1}{2}\\log(7.01).$$ The result is unconditional and does not imply the Riemann Hypothesis. The endpoint $X = 7.01$ lies beyond the prime-7 threshold: the prime $7$ has entered the finite von Mangoldt contribution to the localized explicit formula. The proof concerns the full closed localized quadratic form, not merely a positive finite matrix compression. The main structural ingredients are: A uniform coercive lower bound on the infinite high-frequency sector; A relative-coupling Feshbach reduction that measures the eliminated covariance against the positive energy of the complementary block; A fixed-order coefficient-Gram representation of the infinite arithmetic tail, giving a finite-rank covariance up to a rigorously controlled remainder; Exact-rational positivity certification of the remaining $8 \\times 8$ critical blocks after anisotropic scaling. The proof therefore separates the genuinely infinite-dimensional analysis from the final finite-dimensional certificate. Version v3.1 is a referee-facing exposition revision of the public v3.0 release. It substantially revises the title, abstract, introduction, proof overview, comparison with the earlier $X = 5.01$ certificate, and discussion in order to clarify the mathematical significance and reduce the entry barrier for readers outside the immediate localized-Weil literature. The main theorem, mathematical proof architecture, certified numerical bounds, theorem-critical verification logic, and computational certificate are unchanged from Version v3.0. The verification archive included with Version v3.1r1 is byte-identical to the archive released with Version v3.0. Verification archive SHA-256: 93546c1171a7a6b3d5c0e042be2ddbc0b108912d2383cb6d1b81c4dc2b0ef356 No theorem-critical source file, certified numerical bound, exact-rational certificate, or verification logic has been altered in Version v3.1r1. The verification package contains the scalar interval computation, high-sector coercivity certificate, relative-coupling and coefficient-Gram tail modules, even/odd eight-mode reductions, exact-rational interval certificates, consistency checks, and reproducibility manifests. Files included Revised manuscript PDF, Version v3.1r1 LaTeX source of the revised manuscript Verification archive, byte-identical to the Version v3.0 release Previous public version Version v3.0: DOI 10.5281/zenodo.22267554 Keywords localized Weil quadratic form; Weil explicit formula; Riemann zeta function; localized positivity; computer-assisted proof; rigorous numerics; computational number theory; Feshbach reduction; Schur complement; relative coupling; coefficient-Gram matrix; finite-rank tail reduction; interval arithmetic; exact rational certification; Connes–Consani; Suzuki