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

Abstract

Version v3.0 We prove a computer-assisted full-domain positivity theorem for Suzuki’s localized Weil quadratic form at $$X = \frac{701}{100} = 7.01, \qquad a = \frac{1}{2} \log \frac{701}{100} = 0.9736688505232493612\dots.$$ The main result is $$\lambda_{\frac{1}{2}\log(701/100)} > 0.$$ By variational monotonicity, $$\lambda_a > 0 \qquad \text{for every } 0 < a \le 0.9736688505232493612\dots.$$ The theorem is unconditional. It is a localized positivity statement and does not assume or prove the Riemann Hypothesis. At $X = 7.01$, the active von Mangoldt ledger is $$\mathcal Q(X) = \{2,3,4,5,7\}.$$ The proof combines high-frequency coercivity, relative-coupling elimination, finite-rank compression of the infinite tail, and a rigorous eight-dimensional comparison in each parity sector. First, an 80-shell directed-rational argument establishes the high-sector lower bound $$A_{QQ} \succeq 0.75 I$$ at Fourier cutoff $N = 3600$. The closed high-sector restricted form therefore defines a positive self-adjoint operator with bounded inverse, $$\Vert{}A_{QQ}^{-1}\Vert{}_{\mathrm{op}} \le \frac{4}{3}.$$ Second, the finite low/high coupling is treated through $M = 20000$ using relative rather than purely absolute coupling estimates. The certified finite-high 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. Its explicit part is compressed into parity-specific coefficient-Gram matrices of algebraic rank at most $$2K = 40,$$ independently of the Fourier cutoff and of the number of active prime powers. Writing the tail coupling columns as $$c_\ell^\pm = \mathcal V_\pm a_\ell^\pm + r_\ell^\pm,$$ the complete covariance decomposition is $$\sum_{\ell>M} c_\ell^\pm (c_\ell^\pm)^* = \mathcal V_\pm G_\pm \mathcal V_\pm^* + E_{\mathrm{rem},\pm},$$ where $E_{\mathrm{rem},\pm}$ contains both mixed main–remainder terms and the pure remainder covariance. These terms are rigorously bounded before their Schur-transported contribution is absorbed into the final non-Gram reserve. The certified tail relative prices are $$\rho_{\mathrm{tail},+} < 0.03, \qquad \rho_{\mathrm{tail},-} < 0.05.$$ After finite and tail relative-coupling elimination, the large coordinate complements retain the coercive bounds $$D_{+,\mathrm{refined}} \succeq 0.22 I, \qquad D_{-,\mathrm{refined}} \succeq 0.21 I.$$ Fourth, the remaining near-critical problem is reduced to eight modes in each parity sector. The number eight is a certificate-design choice used to isolate the near-null spectral geometry; no intrinsic or optimal significance is claimed for this dimension. The effective critical spectrum is strongly anisotropic. Binary64 factorizations are used only as preconditioners, while theorem-critical residual evaluation and interval transport use higher-precision arithmetic and exact-rational certification. The final transformed interval families are verified by exact-rational scaled diagonal dominance, with certified margins $$m_+ > 0.9824187862, \qquad m_- > 0.9998199977.$$ These are certification margins in rationally congruent and anisotropically scaled coordinates; they are not lower eigenvalue bounds for the original comparison matrices. Exact Schur-complement domination, triangular block congruence, and form-core closure then lift the finite-dimensional certified positivity to the full closed quadratic-form domain and yield $$\lambda_{\frac{1}{2}\log(701/100)} > 0.$$ Two structural ingredients of the proof are stated separately. The first is a relative-coupling Feshbach inequality. If $$C_V C_V^* \preceq \rho D, \qquad 0 \le \rho < \mu,$$ then the effective critical block admits a lower bound whose correction is governed by $\mu - \rho$, rather than by a global absolute coupling price. The second is the fixed-order coefficient-Gram rank bound $$\operatorname{rank}(\mathcal V G \mathcal V^*) \le 2K.$$ This separates the algebraic rank of the explicit infinite-tail correction from the Fourier cutoff and from the arithmetic size of the active prime-power ledger. Version v3.0 is a substantial exposition and proof-traceability revision of the public manuscript. The mathematical endpoint theorem and its certified constants are unchanged. The principal changes are: a self-contained derivation bridge from the localized Weil form to the periodic Hermitian matrix; an explicit Fourier-multiplier derivation for the high-frequency analysis; clearer explanation of the relation between the closed form, periodic matrix representation, and Fourier representation; explicit accounting of all geometric tail mixed terms; dimensionally explicit restriction of the tail coefficient map to the coordinate-complement rows; an operator-theoretic explanation of the closed high-sector operator and its bounded inverse; a claim-to-certificate map linking theorem-critical assertions to the corresponding verification modules; a first-referee readability revision of terminology, notation, proof architecture, and full-domain closure. The accompanying verification archive is the journal-facing v3.0 release of the computational supplement. The theorem-critical verifier sources and certified numerical constants are unchanged; the release layer has been cleaned to provide self-contained analytic-closure documentation, corrected source-completeness metadata, regenerated SHA-256 manifests, and consistent v3.0-facing release information. The verification archive contains the directed scalar backend, multiscale high-sector certificates, finite relative-coupling modules, parity-specific $K=20$ coefficient-Gram tail certificates, geometric-remainder and non-Gram error budgets, mixed-precision eight-mode reductions, anisotropic exact interval certificates, deterministic finite-high accumulation, dependency checks, SHA-256 manifests, and reproducibility information. The v3.0 verification ZIP has SHA-256 93546c1171a7a6b3d5c0e042be2ddbc0b108912d2383cb6d1b81c4dc2b0ef356. This Zenodo release contains both the v3.0 manuscript PDF and the corresponding v3.0 verification archive. Version DOI: 10.5281/zenodo.22267554 Version: v3.0 Author: Lee Byoungwoo