01. Initial Weil program: finite positivity and its limits
STATUS: RIEMANN HYPOTHESIS OPEN
Recorded 2026-09-29. Date of the cycles 1–23 handoff; individual cycle dates are not available.
Starting idea
The opening program asked whether finite certificates, kernel identities and continuity could establish positivity of the Weil quadratic form on every compact support. Weil positivity is a criterion for RH; proving an equivalent statement still requires an independent argument.
What was tested
Twenty-three recorded cycles checked finite forms, fixed support bounds, operator dictionaries, arithmetic identities and selected external proof claims. Tests repeatedly separated a finite matrix from its infinite tail and from the coupling between the two.
What survived
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A computer-assisted bound was recorded for the full complex form domain on support [−1/2, 1/2]: Q_W(f) ≥ 9 × 10⁻⁸ ‖f‖², with an interval-certified head and analytic tail and coupling bounds.
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Five other fixed-window joint-symbol certificates concern high frequencies only. Conditional finite-to-infinite lemmas and explicit counterexamples to insufficient block, symmetry and kernel arguments remain reusable.
What failed or remains unproved
No positivity argument for all supports survived. Checking coordinate blocks does not control cross terms. A positive kernel or a symmetry of the zero set does not force every zero onto the symmetry line. A gap in an audited proof claim is recorded at the disputed step, rather than treated as a disproof of every result in that paper.
Why the work moved on
The 23-cycle handoff led to a separately authorized study of arithmetic generators and their boundary conditions.
Current scope
Completed exploratory program; all-support Weil positivity and RH remain open.
“Proved” here refers to the stated mathematical claim in the linked research record. Internal AI-agent checks are scoped checks, not external peer review. Numerical and formal checks have separate limits described in Reproducibility. No priority or novelty claim is made.
Canonical records
- Detailed source report
- Scoped internal audit 1
- Scoped internal audit 2
- Scoped internal audit 3
- Scoped internal audit 4
- Historical state record
- Recorded validation
The source archive preserves the original Japanese research prose. These links point to publication copies, not third-party paper downloads. The source map explains historical state files and references distinguish publication status from verification.