The Riemann Hypothesis: Arithmetic Matrix Criteria and Certified Weil Bounds
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Abstract
We develop finite certificates for arithmetic Weil positivity in its original weighted odd realization. Each certificate is a finite Hermitian matrix with explicit control of the polar constraint, mixed terms and entire orthogonal complement. A strictly positive outward enclosure proves a supported lower bound with a prescribed original-norm allowance. We construct a computably specified trial family and prove that accepted certificates on an unbounded sequence of growing supports are equivalent to the Riemann hypothesis. Under the Riemann hypothesis, every admitted fixed instance has a recognizable strict certificate, with explicit residual and enclosure tolerances. We also prove an unconditional complete-source covariance estimate at prescribed polynomial ridges, uniform over moving and growing finite genuine operator-null restrictions. At ridge M^(-b) its rate is O_(b,epsilon)(M^(-1/2+epsilon)) for each fixed b,epsilon>0. The proof combines uniform arithmetic inverse estimates, theta localization and control of the entire source coefficient ball. Quantitative inverse-approximation bounds and evaluated supported certificates exhibit the resulting verification framework. A complementary first-moment criterion pays the complete receiving return analytically.
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Available editions
- Edition 12 · 2026-11-01 · public
- Edition 11 · 2026-11-01 · public
- Edition 10 · 2026-10-31 · public
- Edition 9 · 2026-10-31 · public
- Edition 8 · 2026-10-29 · public
- Edition 7 · 2026-10-29 · public
- Edition 6 · 2026-10-29 · public
- Edition 5 · 2026-10-28 · public
- Edition 4 · 2026-10-28 · public
- Edition 3 · 2026-10-28 · public
- Edition 2 · 2026-10-28 · public
- Edition 1 · 2026-10-27 · public
Research details
- Subjects
- Mathematics · Number Theory · Functional Analysis · Numerical Analysis
- Keywords
- Riemann hypothesis, arithmetic Weil positivity, finite matrix criteria, complete covariance, certified operator bounds, genuine nulls, spectral approximation
- Manuscript date
- 2026-10-29
- Reader reuse
- none
Scope
Constructive complete certificates on the original odd weighted Weil domain, with explicit residual and enclosure tolerances and conditional fixed-instance recognition. Uniform complete-source covariance estimates and critical sign characterization are accompanied by arithmetic first-moment, null-transport and high-frequency supporting criteria with their stated hypotheses.
Disclosures
OpenAI ChatGPT and Codex assisted with mathematical exploration, proof development, literature comparison, exposition, programming, internal verification and manuscript preparation. Scott Schubr directed and reviewed this work and is responsible for the mathematical arguments, computations and attribution of prior work.
Reproduction
The main paper includes the principal certificate and complete-covariance arguments, with full supporting proofs in its appendices. The technical supplement retains the complete numerical error derivations and alternative implementations. The principal PDF/source/code ZIP supplies editable LaTeX, portable code and exact input identities. The scientific input files, lossless download volumes and reconstruction program retain their exact identities; see Scientific-download-instructions.zip and Scientific-data-index.json.