A Certified Lower Bound for the Localized Weil Positivity Frontier Across the Prime-7 Threshol

Abstract

Version v2.0 We prove a computer-assisted full-domain positivity theorem for Suzuki’s localized Weil quadratic form at $$X = \frac{701}{100} = 7.01, \qquada = \frac{1}{2}\log\frac{701}{100}= 0.9736688505232493612\dots.$$ At this endpoint the active von Mangoldt ledger is $\{2,3,4,5,7\}$, so the certified support lies strictly beyond the threshold at which the prime $7$ becomes active. The proof operates in a substantially deeper near-critical regime than the previously certified prime-$5$ endpoint. It combines four main ingredients. First, an 80-shell directed-rational leakage argument establishes the high-sector coercive bound $$A_{QQ} \succeq 0.75 I$$ at Fourier cutoff $N=3600$. Second, the low/high coupling is treated directly through $M=20000$. Instead of pricing the coupling only through a global operator norm, the proof uses relative-coupling inequalities against the order-one low complement. The certified finite-high relative prices satisfy $$\rho_{\mathrm{fin},+} < 0.18,\qquad\rho_{\mathrm{fin},-} < 0.18.$$ Third, the infinite high-frequency tail is expanded through $K=20$ geometric pairs and compressed into parity-specific $40\times 40$ coefficient-Gram matrices. The explicit tail covariance has algebraic rank at most $40$, independently of the number of active prime powers. Directed bounds give $$\rho_{\mathrm{tail},+} < 0.03,\qquad\rho_{\mathrm{tail},-} < 0.05.$$ After relative-coupling elimination and rigorous control of the geometric remainders, the large coordinate complements retain the order-one coercive bounds $$D_{+,\mathrm{refined}} \succeq 0.22 I,\qquadD_{-,\mathrm{refined}} \succeq 0.21 I.$$ Fourth, the remaining near-critical problem is reduced to eight parity modes in each sector. The effective spectrum is strongly multilayered, with the even critical scale reaching approximately $10^{-28}$. Binary64 factorizations are used only as preconditioners; the critical solves are refined with IEEE binary128 residual evaluation. Directed scalar centers are transferred through the actual fixed-$2^{320}$ to decimal to `strtoflt128` path. The infinite-tail uncertainty is then propagated anisotropically through the residualized eight-mode coefficient map before interval absolute values are taken. The final transformed interval families are certified by exact-rational scaled diagonal dominance. The minimum certified scaled margins are $$m_+ > 0.9824187862,\qquadm_- > 0.9998199977.$$ Exact Schur-complement domination, triangular block congruence, and form-core closure then give $$\lambda_{\frac{1}{2}\log(701/100)} > 0.$$ By variational monotonicity, $$\lambda_a > 0\qquad\text{for every }0 < a \le 0.9736688505232493612\dots,$$ providing a certified lower bound for the localized Weil positivity frontier strictly beyond the prime-$7$ threshold. Two reusable structural statements are made explicit in this version. The first is a relative-coupling Feshbach inequality. If $$C_V C_V^* \preceq \rho D,\qquad0 \le \rho < \mu,$$ then the effective critical block is bounded below by a correction with denominator $\mu-\rho$, rather than by a global absolute coupling price. The second is the fixed-order coefficient-Gram rank bound $$\operatorname{rank}(VGV^*) \le 2K,$$ which separates the explicit infinite-tail rank from the Fourier cutoff and from the number of active prime powers. The result is unconditional and does not assume or prove the Riemann Hypothesis. It does not determine the exact localized positivity frontier, and it does not establish a uniform theorem across all prime thresholds. The accompanying verification archive is source-complete and includes the directed scalar backend, the multiscale high-sector certificate, finite relative-coupling modules, parity-specific $K=20$ coefficient-Gram tail certificates, mixed-precision eight-mode reductions, anisotropic exact interval certificates, deterministic finite-high accumulation, fail-closed dependency checks, SHA-256 manifests, source-completeness audits, and reproducibility information. The canonical verification archive has SHA-256 `94bed4086e8d4a8f8b8f4fdf8d5ba852159e13181772f188e8cd1445233f8cfd` and is also archived at DOI 10.5281/zenodo.22066402 The preceding prime-$5$ certificate is archived at DOI 10.5281/zenodo.22062597 Version: v2.0 Author: Lee Byoungwoo