The Riemann Hypothesis: Arithmetic Covariance Bounds and Complete Signed Criteria
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Abstract
We prove a uniform inverse-source estimate for a changing arithmetic Weil operator and derive complete-source covariance asymptotics at every fixed polynomial ridge. A power-saving prime-number estimate extends this control uniformly to a superpolynomial regularization window. The estimate controls an entire coefficient unit ball through arithmetic phase averaging, polar-legal preimages and theta localization, with all residual directions retained. At ridge M^(-b), for fixed b>0, the covariance norm differs from one by at most C_b M^(-1/2) sqrt(log M) exp(48 sqrt(b log M)) for sufficiently large M. The bound is uniform over moving and growing finite genuine operator-null restrictions. A lower bound identifies the cost of the positive-majorant method under every vanishing regularization schedule. A critical-scale characterization relates the Riemann hypothesis to vanishing normalized positive covariance excess. We also give a separate constructive certification theorem: a strictly positive finite matrix enclosure proves a supported lower bound in the original weighted norm, with explicit payment for the full complement and mixed terms. Accepted certificates on an unbounded growing-support sequence are equivalent to the Riemann hypothesis. Under that hypothesis each admitted fixed instance has a recognizable certificate in a computably specified trial family. Complete evaluated instances and quantitative approximation bounds accompany this independent construction.
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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
Uniform inverse-source and complete covariance-norm bounds on the original odd weighted Weil domain, including polynomial ridges and quantified joint regularization windows. Exact signed criteria, ridge-independent auxiliary-majorant cost bounds and fixed-cutoff mixed-Gram divergence are accompanied by constructive supported certificates and complete residual, mixed-term and tail payments.
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.