A Certified Lower Bound for the Localized Weil Positivity Frontier Across the Prime-5 Threshold: Multiscale Coercivity, Correlation-Preserving Tail Certification, and Five-Mode Feshbach Reduction

We prove a computer-assisted full-domain positivity theorem for Suzuki’s localized Weil quadratic form at X = 501/100 = 5.01, a = 1/2 log 501/100 = 0.8057179575483867153. At this endpoint the active von Mangoldt ledger is \2,3,4,5\, so the certified support lies strictly beyond the threshold at which the prime $5$ becomes active. The proof combines three main ingredients. First, a 64-shell directed-rational leakage argument establishes the high-sector coercive bound AQQ 0.75 I at Fourier cutoff $N=800$. Second, the low/high coupling is treated directly through $M=5000$, while the infinite tail is expanded through $K=12$ geometric pairs and compressed into a 24× 24 coefficient Gram matrix. This retains the correlations that are lost under channel-by-channel majorization. Third, the remaining near-critical spectral problem is reduced to five parity modes above order-one complements and certified by exact-rational Sylvester tests, residual-verified solves, and directed uncertainty transfer. The resulting parity comparison bounds are S+,true 5.010416381129903 × 10⁻¹⁹ I, S-,true 6.626561472715318 × 10⁻¹⁶ I. Exact Schur-complement domination, triangular block congruence, and form-core closure then give λ1/2log(501/100) > 0. By variational monotonicity, λₐ > 0 for every 0 < a ≤ 0.8057179575483867153, providing a certified lower bound for the localized positivity frontier. The result is unconditional and does not assume or prove the Riemann Hypothesis. It also does not determine the exact localized positivity frontier. The accompanying verification archive is source-complete and includes directed scalar enclosures, the multiscale high-sector certificate, even and odd $K=12$ coefficient-Gram modules, five-mode Feshbach certificates, fail-closed dependency checks, deterministic serialized proof data, SHA-256 manifests, and reproducibility information. The final verification package is designed so that theorem-level claims are accepted only through deterministic source-based checks rather than unverified floating-point output. Version: v1.0 Author: Lee Byoungwoo