RH: Complete Signed Certificates and Canonical Null Synthesis
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Abstract
We give quantitative signed certificates retaining every arithmetic shift and mixed term of a weighted Weil form, and construct exact null corrections in its original norm. For each fixed ε>0, the specified inverse-receiving construction admits a trial cutoff m_M≤M^(1+ε) for all sufficiently large M. In the guarded input format, construction and verification have polynomial bit cost in the numeric cutoff M. The polar constraint is retained. Under RH, the admitted fixed instances have recognizable strict certificates. We also prove that the closed span of the canonical derivative generators equals the full-multiplicity Mellin zero-jet class in the original reciprocal-theta norm. An equivalent continuous-area norm and uniform bounds on zero-free exterior arms yield strong inward approximation. Contour moments give a computable orthogonal projection and optimal exact lifts of any fixed controllable finite head. At exact unit row moment and profile tolerance η, the logarithm of the least genuine-null physical norm has order η^(-4/(8j+13)) for a fixed row of highest index j. Simultaneous lifts have physical squared-norm budget exp(D_J e^(2L)), with D_J=exp(C(J+1)) and L≥C(J+1) for fixed auxiliary parameters, and allow prescribed orders J=o(log log M). A complete evaluated certificate and paid localization give the whole-domain lower bound −3×10^(-8) in the original physical norm. The covariance estimates and critical signed criteria connect these results to arithmetic Weil positivity.
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Edition files
- main.pdf454.0 KiB · pdf
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b1495149eed8aa47b83ff9618e16792891fed60428702d189a8105115d8f170c
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, 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 67-page main paper contains the principal certificate, quantitative receiving proofs, canonical synthesis/projection and sharp interpolation analysis. The indexed 155-page technical supplement supplies complete covariance proofs, alternative implementations, relative mixed-return estimates, actual inverse-energy approximation and physical cancellation costs. 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.