The Riemann Hypothesis through Arithmetic Weil Positivity
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Abstract
We develop a quantitative covariance formulation of the Riemann hypothesis through arithmetic Weil positivity. In the original odd weighted realization, RH is equivalent to liminf as M tends to infinity of sqrt(M)(gamma_M - 1)_+ = 0, where gamma_M is the complete covariance norm at ridge 1/M after any sequence of finite genuine-null restrictions. We prove that the complete covariance norm differs from one by O_{b,epsilon}(M^(-1/2+epsilon)) for every epsilon > 0 at each prescribed ridge M^(-b), b > 0 fixed, uniformly over all finite genuine-null restrictions. The estimate includes every negative-source frequency and all mixed terms. Its proof combines a complete arithmetic source majorant, theta-tail bounds, the Vinogradov-Korobov prime-number error and a uniform actual-inverse construction. Genuine derivative nulls then provide complete source coordinates and signed Schur certificates. A complete complementary cap bounded by a fixed q < 1, together with source-normalized physical lift cost o(sqrt(M)), suffices for RH. We prove uniform small-set inverse bounds throughout [0, exp(A sqrt(log(M)))] for every fixed A > 0, unconditionally. Factorial physical lift budgets and actual inverse-weighted exterior and null-spectrum bounds quantify the construction. These results retain all mixed terms and give complete signed criteria for vanishing original-norm allowances and the Riemann hypothesis. A second cofinal RH criterion uses explicit M^(o(1))-dimensional Schur matrices retaining the exact infinite receiving inverse, with a proved complete receiving floor and only O(log(M)) positive Green responses. The response reduction preserves the original-norm vanishing-allowance scale. A finite-sine core gives joint Schur error O(sqrt(log(M))/M^2) in physical core coordinates.
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Research details
- Subjects
- Mathematics · Number Theory · Functional Analysis · Numerical Analysis
- Keywords
- Riemann hypothesis, arithmetic Weil positivity, complete covariance, genuine nulls, critical rate, vanishing allowances, certified operator bounds
- Manuscript date
- 2026-10-27
- Reader reuse
- none
Scope
RH through arithmetic Weil positivity: a complete covariance bound and exact critical characterization, together with a resource-quantified cofinal Schur criterion. The declared cores have subpolynomial dimension and a proved full receiving floor; logarithmically many Green responses preserve the original physical allowance scale. A finite-sine core has raw profile Gram condition below 4L_M and complete joint Green-induced Schur error O(sqrt(log(M))/M^2) in physical core coordinates. The cofinal criterion uses the exact full receiving inverse and the stated Schur-positivity condition.
Disclosures
OpenAI tools assisted with literature comparison, proof development, exposition, and verification code. The author assumes responsibility for all content.
Reproduction
The 93-member scientific package contains the 40-page main paper, 41-page technical supplement, complete editable sources, reader/citation files, numerical data and reproducibility code. All proofs for the new cofinal criterion are internal to Appendix B of the main paper. The 49 original formal statements are preserved; seven formal statements develop the new reduction. Directed Green-schedule controls use 320-bit arithmetic; saved-array checks and source regeneration retain their separately stated scopes. Large inherited certificate arrays retain their source-bound verification history.