The Riemann Hypothesis: Arithmetic Matrix Criteria and Certified Weil Bounds
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Abstract
We construct finite-verification criteria for the Riemann hypothesis (RH) through arithmetic Weil positivity on its original odd domain. The central criterion uses finite matrices with explicit payment for the polar constraint, mixed terms and entire receiving complement. Strict outward certificates on an unbounded sequence of growing supports are equivalent to RH; under RH, computable trial families recognize every admitted instance with prescribed residual and rounding tolerances. Complete-response and relative-energy estimates provide alternative constructions, including an arithmetic-only first-moment criterion with a subexponential sufficient matrix-dimension bound on its stated support scale. With a supplied fixed zero-free half-plane, a complementary high-frequency covariance criterion admits exponential precision governed explicitly by that half-plane. Exact null transport, constrained inverses and Gevrey estimates pay its return to the original domain; an independent classical-input criterion follows from the same complete source analysis. An unconditional complete-covariance estimate of order M^(-1/2+epsilon) is uniform over finite genuine-null restrictions at prescribed polynomial ridges. Evaluated moving-window certificates include the original-norm allowance 2^(-24). Arithmetic-only certificates and continuous-ridge control retain their stated norms, active prime powers, constraints and mixed terms. The paper includes realization-specific proofs and quantitative error budgets, with scientific code and data for reproduction.
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Available editions
Research details
- Subjects
- Mathematics · Number Theory · Functional Analysis · Numerical Analysis
- Keywords
- Riemann hypothesis, arithmetic Weil positivity, finite matrix criteria, complete covariance, certified operator bounds, genuine nulls, spectral approximation
- Manuscript date
- 2026-10-29
- Reader reuse
- none
Scope
Finite-verifiable arithmetic RH criteria on the original odd domain, with complete constraint and infinite-receiver payment. Full-source null transport and actual constrained inverses support additional high-frequency criteria, including a supplied-fixed-strip criterion and an independent classical-input criterion. Original unconditional covariance estimates and complete evaluated certificates retain their stated norms and hypotheses.
Disclosures
OpenAI ChatGPT (Sol 6.1) assisted with mathematical exploration, proof development, literature comparison, exposition, programming and internal verification. OpenAI Codex assisted with manuscript preparation, source comparison, code adaptation, compilation, rendering and packaging. Scott Schubr directed these processes through workflows developed in his N=1 AI governance research, including source and claim controls, corrective intervention and review of subsequent outputs. The author is responsible for the mathematical arguments, computations and attribution of prior work.
Reproduction
The 74-page main paper and 80-page indexed technical supplement include the complete central and high-frequency proofs. The principal PDF/source/code ZIP supplies editable LaTeX, portable code and exact input identities. Earlier M20 details, full fixed-support ridge derivations and the alternative growing-floor criterion are indexed in the supplement. M20 worked_example_data.zip and Arithmetic_only_scientific_inputs.zip are direct downloads. Eight lossless volumes supply the three largest archives. Use Scientific-download-instructions.zip and Scientific-data-index.json to verify and reconstruct the M24, ridge-continuation and arithmetic-shift-two scientific archives. The complete input values, norm contracts, rounding budgets and reproduction instructions are retained.