14. Local-to-global zero confinement
STATUS: RIEMANN HYPOTHESIS OPEN
Recorded 2026-09-30. Date of the recorded report; individual start and finish times are not reconstructed.
Starting idea
Several local or constrained transforms have real zeros. The task was to identify exactly what must converge to transfer that conclusion to Xi.
What was tested
The audit compared Hermite–Mellin results, local-field statements, completed transforms with finite correction factors, semilocal Sonin stability and the finite Weil-ground construction.
What survived
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The hypotheses of the real-zero theorems include their specific vector, positivity, support or orthogonality conditions. A completed transform containing ζ as a factor cannot remove off-line zeta zeros by a holomorphic correction.
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Two families were kept separate: actual finite Weil ground states, whose real-zero theorem needs the stated evenness and simplicity hypothesis ES; and a prolate proxy whose transform converges locally to a scalar multiple of Xi.
What failed or remains unproved
No theorem identifies the normalized true ground state with that proxy. The missing comparison G is convergence of their transforms on compact subsets of |Im z| < 1/2, together with the required ground-state hypotheses. The combined criterion is sufficient; an RH converse for G was not established.
Why the work moved on
The common-parent track asked whether an actual energy identity, residual estimate or spectral gap could provide this comparison.
Current scope
Local theorems and proxy convergence retained; the global bridge remains 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
- Completion report
- Scoped internal audit
- 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.