The Riemann Hypothesis through Arithmetic Weil Positivity
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Abstract
We give a finite-verifiable criterion for the Riemann hypothesis through arithmetic Weil positivity on its original odd weighted domain. RH is equivalent to the existence of strictly positive finite certificates along an unbounded sequence of explicit growing supports. Each certificate retains every active prime power, the polar constraint, all mixed terms, and the entire receiving complement. Its entries use finite function integrals and a finite constraint-Gram inverse. Under RH, every admitted cutoff has a strict certificate from a computable trial family. A complete physical Green-tail estimate, a positive dyadic theta-ridge floor, and a stable finite-sine core establish this conditional completeness with explicit residual and rounding tolerances. A quantitative receiving-return enclosure gives an explicit sufficient finite trial-size bound. The core has subpolynomial dimension; response and potential representations use logarithmically many cells and features up to a log-log factor. We also prove a complete arithmetic covariance estimate of order M^(-1/2+epsilon) at every fixed polynomial ridge, uniformly over finite genuine-null restrictions. Exact signed identities, physical lift estimates, and complete complementary bounds connect that quantitative formulation to the same Weil positivity endpoint. All realization-specific supporting proofs are included; an indexed technical supplement provides alternative exact-response methods and reproducible numerical applications.
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- main.pdf288.0 KiB · pdf
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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
- 2026-10-28
- Reader reuse
- none
Scope
Finite-verifiable cofinal characterization of RH through arithmetic Weil positivity, complete arithmetic covariance, and quantitative full receiving-return enclosures. Original-domain supporting proofs are included. The cofinal strict finite certificates are the stated equivalent positivity condition.
Disclosures
OpenAI tools assisted with literature comparison, proof development, exposition, and verification code. The author assumes responsibility for all content.
Reproduction
The scientific package contains the 38-page main paper, 54-page indexed technical supplement, editable sources, reproducibility code and source-bound numerical data: 97 direct files and 18 nested numerical-array members. All 64 prior formal statements are preserved mathematically across the main paper and supplement. Finite certificate, constrained inverse, arithmetic commutator and cubic receiving-return proofs are included. Code controls, clean typesetting and inherited large-array checks retain the scopes stated in the package.