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. A complete supported certificate at M=2^20 gives an evaluated zero-trial example with the entire infinite receiving complement retained.
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- main.pdf296.6 KiB · pdf
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12310e98e11beeccca128f26b1fc1ddbaa52847421c5d71c0db8dc610ab27fc4 - technical_supplement.pdf396.9 KiB · pdf
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554a543544992b02b4557c892a4b553834bcf7365fdcc16db90fcf6653c9544e - arithmetic_weil_positivity_package.zip202359.5 KiB · package
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7465522ca126d5222b2d83687d55f4ba16f87e13daf89469304b4254a5cda885 - worked_example_data.zip76310.2 KiB · package
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81421b27a034e3fb10ce64be1ed6d23078dcfe105d29a0e690a5f97b7043d423
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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. The supported M=2^20 worked example uses its explicitly stated alternative support and complete positive receiving reference. Its theorem and analytic Schur/order proof are in the main paper, with exact computation and verification in Supplement S19. The central cofinal criterion and specified trial-size bounds retain their original hypotheses and schedule.
Disclosures
OpenAI tools assisted with literature comparison, proof development, exposition, and verification code. The author assumes responsibility for all content.
Reproduction
The 40-page main paper and 58-page technical supplement are accompanied by a principal PDF/source/code package with 106 direct members and 18 nested numerical-array members, plus worked_example_data.zip with four lossless numerical inputs and their manifest. The portable code checks all 6,471 reference cells, regenerates the full 108,241-entry weighted constraint Gram, reconstructs the complete signed lower from 2048-bit primitive moments and replays both 328-pivot signs at 4096 bits. The fresh 8-input moment check covers 6,860 entries; complete 393-input moment production retains its source-bound reviewed computation. All central theorem/proof sources remain byte-identical. Scientific identities and verification scopes are stated in the files.