The Riemann Hypothesis: Arithmetic Matrix Criteria and Certified Weil Bounds
Loading original first page…
Abstract
We give constructive finite-verification criteria for the Riemann hypothesis (RH) through arithmetic Weil positivity on its original domain. The central criterion expresses positivity of the infinite-dimensional form through finite matrices with explicit control of its constraints and entire receiving complement. RH is equivalent to strict certificates along an unbounded sequence of growing supports; under RH, computable finite trial families recognize each admitted instance with specified residual and rounding tolerances. A complementary fixed-shift criterion reduces verification to signed arithmetic first moments and a known diagonal at prescribed spectral cutoffs. A first-commutator, relative-energy estimate pays the full infinite tail analytically and gives a sufficient matrix-dimension bound exp(O(sqrt(log M))) on its stated support scale. Complete-response commutator bounds and bounded profile regularization provide alternative verification constructions. An unconditional complete-covariance estimate has order M^(-1/2+epsilon) at prescribed polynomial ridges, uniformly over finite genuine-null restrictions. Two complete shrinking-allowance examples retain every active prime power, constraint and mixed term; the larger-support example reduces the supported allowance sixteenfold. Two arithmetic-only examples and a continuous ridge interval give further evaluated certificates in their specified realizations. Uniform unweighted lower bounds and support-growth estimates connect the complementary constructions to the same original positivity endpoint. Realization-specific proofs, quantitative error bounds, and scientific code and data are supplied for reproduction.
Read the full paper in the original PDF.
Edition files
- main.pdf447.7 KiB · pdf
SHA-256
209d68519d69043214fa5eab50c11f166e01028d3a2b2af9709ec415c760af1d - Technical_supplement.pdf481.6 KiB · pdf
SHA-256
7bb6672cd1c63ec121789532b6bdd81513b3837c3bfb347a8c9f671842c19342 - RH_manuscript_source_and_code.zip203628.6 KiB · package
SHA-256
6d6f66d00c41d061bf83bba6a752b3eacfaeb2104d8af484f644232f63d5f1ba - Editable_LaTeX_sources.zip208.9 KiB · package
SHA-256
502308c6c559159b14b3f5f2f34c8aeac3f8258e3300794f9b8f459b3f75e7c4 - worked_example_data.zip76310.2 KiB · package
SHA-256
81421b27a034e3fb10ce64be1ed6d23078dcfe105d29a0e690a5f97b7043d423 - Arithmetic_only_scientific_inputs.zip37620.2 KiB · package
SHA-256
d6f78bc7a42f4f2bcc7294998ae12c1409619679c6cfcc9bc799a924890bff1d - Arithmetic-shift-two-data-volume-01.zip135250.5 KiB · package
SHA-256
16af3466ecb0d643bcd8c953befea31f1f328ae35381c09e288d05359c5ae063 - Arithmetic-shift-two-data-volume-02.zip135237.3 KiB · package
SHA-256
17c36f72d3d4892ca0bb060614ac3c15f33703e857baf3a6f7ad12a887e82922 - Arithmetic-shift-two-data-volume-03.zip65240.9 KiB · package
SHA-256
9f042f41913438bac1c6b8362e578339e9f5fd1db031953097ec0862f93e3a28 - M24-data-volume-01.zip135237.7 KiB · package
SHA-256
5023b2b6cab11d0f0a42fff6bdb9b38e4008c3445efe0e74a1cd5a5eeca31af6 - M24-data-volume-02.zip135231.7 KiB · package
SHA-256
15e08962d609301cf16e1ae058e61d3e50bba435376659bb605f35ef6503990f - M24-data-volume-03.zip85456.1 KiB · package
SHA-256
240584294a00479bd6a92d1fc1fd8dc20ee6dd41b93e6d1bbafec0c140d1f10a - Ridge-continuation-data-volume-01.zip135202.3 KiB · package
SHA-256
ca93b19f2cd5d4370c0f84f40650e790dde06ee8960505afbb9d79382b7fe925 - Ridge-continuation-data-volume-02.zip24140.7 KiB · package
SHA-256
3938a56eb582dddba934845f66b3c24c16e5e9d5df440cd91e67314a6960e248 - Scientific-download-instructions.zip3.8 KiB · package
SHA-256
310926e53743a89948025c09832089e814a566c62115d498e1eea43c1bf47489 - Scientific-data-index.json6.2 KiB · manifest
SHA-256
b1495149eed8aa47b83ff9618e16792891fed60428702d189a8105115d8f170c
Available editions
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
- 2026-10-28
- Reader reuse
- none
Scope
Constructive finite-verification criteria for the Riemann hypothesis through arithmetic Weil positivity on its original domain, with complete constraint and receiving-complement control. A complementary fixed-shift first-moment criterion pays the infinite tail analytically. Uniform complete covariance, full-response and capped constructions, realization-specific numerical certificates, and three original-profile certificate transfers supply complementary support.
Disclosures
OpenAI ChatGPT (Sol 6.1) assisted with mathematical exploration, proof development, literature comparison, exposition, programming and internal verification. OpenAI Codex assisted with manuscript preparation, source comparison, code adaptation, compilation, rendering and packaging. Scott Schubr directed these processes through workflows developed in his N=1 AI governance research, including source and claim controls, corrective intervention and review of subsequent outputs. The author is responsible for the mathematical arguments, computations and attribution of prior work.
Reproduction
The 70-page main manuscript and 74-page technical supplement are accompanied by a principal PDF/source/code package, an editable LaTeX package, and scientific inputs with exact identities. M20 worked_example_data.zip and Arithmetic_only_scientific_inputs.zip are direct downloads. The three largest archives are provided as eight lossless download volumes: three M24, two Ridge-continuation, and three Arithmetic-shift-two volumes. Download Scientific-download-instructions.zip and Scientific-data-index.json and use the supplied standard-library Python restoration program to verify the downloads and reconstruct M24_scientific_inputs.zip, Ridge_continuation_scientific_inputs.zip and Arithmetic_only_shift2_scientific_inputs.zip exactly. Then extract the original archives according to the principal README. Decimal intervals, exact dyadic arrays, input manifests, code, norms and complete error contracts are preserved.