RH: Complete Signed Certificates and Canonical Null Synthesis
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Abstract
We give quantitative complete signed certificates and describe exact null corrections for a weighted realization of Weil's form. For every fixed epsilon>0, the specified inverse-receiving construction has a sufficient trial cutoff m_M<=M^(1+epsilon) for all sufficiently large M, retaining the polar constraint, all arithmetic shifts and the complete mixed return. In its guarded input format, construction and verification have polynomial bit cost in the numeric cutoff M. Under RH, the admitted fixed instances have recognizable strict certificates. We also identify the closed canonical derivative carrier with the entire full-multiplicity Mellin zero-jet class in the original reciprocal-theta norm. An equivalent continuous-area norm and uniform estimates on zero-free exterior arms prove strong inward synthesis. Contour moments then give a computable orthogonal projection and optimal exact lifts of any fixed controllable finite head. At exact unit row moment and profile tolerance eta, the logarithm of the least genuine-null physical norm has order eta^(-4/(8j+13)) for a fixed row of highest index j; simultaneous lifts admit linear-order exponential budgets and prescribed orders J=o(log log M). A complete evaluated certificate and paid localization give the whole-domain lower bound -3x10^(-8) in the original physical norm. The covariance estimates and critical signed criteria connect these constructive and geometric results to arithmetic Weil positivity.
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Edition files
- main.pdf494.8 KiB · pdf
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49ef6547a2de02371533a5d262497f1ed015d2adf338b2c5ab986bb5723f2103 - M25-data-volume-11.zip62364.1 KiB · package
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5d861866f5811cff786e0dbc1d1bfdbd67cadcb8a2e2ad04a372cf0e9802f9cc - M25-data-volume-12.zip62079.1 KiB · package
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ae549252eea253b7521cda407d7c8ffcde49b47063054bbde1deafd0dd193c5e - M25-data-volume-13.zip61631.6 KiB · package
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6f9d64414a9110f19ca20d35a71e2330c5f50d455918d4f47ee62ce40520f25c - M25-data-volume-16.zip61162.9 KiB · package
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15e08962d609301cf16e1ae058e61d3e50bba435376659bb605f35ef6503990f - M24-data-volume-03.zip85456.1 KiB · package
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240584294a00479bd6a92d1fc1fd8dc20ee6dd41b93e6d1bbafec0c140d1f10a - Ridge-continuation-data-volume-01.zip135202.3 KiB · package
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b1495149eed8aa47b83ff9618e16792891fed60428702d189a8105115d8f170c
Available editions
- 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, Weil positivity, complete signed certificates, canonical null synthesis, weighted interpolation, arithmetic covariance
- Manuscript date
- 2026-10-29
- Reader reuse
- none
Scope
Complete quantitative signed receiving certificates in the original odd polar weighted Weil realization; actual reciprocal-theta-norm full-multiplicity canonical carrier synthesis, computable projection and optimal fixed controllable-head lifts; sharp genuine-null interpolation and linear-order lift budgets; original-domain M25 lower bound -3e-8; complete covariance estimates and critical signed criteria.
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 75-page main paper contains the principal certificate, quantitative receiving proofs, canonical synthesis/projection and sharp interpolation analysis. The indexed 140-page technical supplement supplies complete covariance proofs and alternative implementations. The source/code ZIP contains editable LaTeX, portable code and exact input identities. M25 lossless download volumes and M25-download-instructions.zip supply the separate complete numerical inputs. Earlier M20/M24 and ridge-continuation downloads remain available with their exact identities. Saved-array replay is distinct from fresh moment acquisition and from the analytic original-domain return. The named scalar localization-fee checker and its exact rational output are included in the source package; the weighted/signed verification records are regenerated by its directed verifiers.