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.
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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
- No additional permission granted
Scope
RH through arithmetic Weil positivity: a complete covariance estimate, an exact covariance characterization of RH, and complete signed certificates with physical lift estimates. Corollary 7.5 proves that a fixed complete complementary gap together with the stated lift bound yields vanishing original-norm allowances and hence the Riemann hypothesis. Unconditional expanded-collar estimates and actual exterior and null-spectrum bounds quantify the construction.
Disclosures
AI assistants supported derivation, literature comparison, manuscript preparation and independent checks. The author is responsible for the mathematical statements and presentation.
Reproduction
The 90-member publication package contains the complete 58-page manuscript, editable LaTeX sources, portable reader programs, saved covariance and inverse-action data, and supporting scientific inputs. The unchanged short verification programs retain their normal and optimized-mode checks, including replay of 598,418 saved covariance entries. A clean source rebuild reproduces all 58 pages of manuscript text. The large source-generation datasets retain their recorded verification scope.