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 regularized signed certificates. An explicit factorial bound in the original norm controls their growing physical lift cost, while small-set inverse estimates retain the full frequency exterior. The critical covariance excess o(M^(-1/2)) yields vanishing original-norm allowances and hence RH, with complete signed certificates formulated in the original source metric.
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- Edition 12 · 2026-11-01 · public
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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: an exact critical complete-covariance characterization in the original odd weighted realization, a uniform complete covariance bound at every fixed polynomial ridge, and constructive genuine-null certificates with physical lift and exterior budgets. Under the stated critical covariance condition, the Riemann hypothesis holds.
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 54-page manuscript, editable LaTeX sources, portable reader programs, saved covariance and inverse-action data, and the supporting scientific inputs. Ten short verification programs pass in normal and optimized modes, including replay of 598,418 saved covariance entries. A clean source rebuild reproduces all 54 pages of manuscript text. The unchanged large source-generation datasets retain their existing verification scope; full regeneration was not repeated for this strengthening.