13. Arithmetic restoring force
STATUS: RIEMANN HYPOTHESIS OPEN
Recorded 2026-09-30. Date of the recorded report; individual start and finish times are not reconstructed.
Starting idea
Could the actual theta kernel impose a force that confines zeros, without simply assuming a global sign condition equivalent to RH?
What was tested
The track derived off-axis cosine/hyperbolic-sine equations, examined logarithmic derivatives and fixed the de Bruijn–Newman heat sign and time convention. It separated real-axis repulsion from confinement of complex branches.
What survived
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Known forward strip contraction and the simple-zero evolution law remain valid in their stated domains. Multiple collisions require separate treatment.
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Specific sign information outside a region does not become a sign everywhere. Synthetic heat examples clarify why individual complex zeros need not obey a universal radial inward law.
What failed or remains unproved
The required global restoring sign returns to an RH-equivalent criterion. Positivity and evenness of the theta kernel do not supply it. A proposed total-positivity property of the actual kernel fails; this is distinct from the Fourier real-zero question.
Why the work moved on
The next comparison asked whether existing local real-zero theorems could be connected to the actual global approximation family.
Current scope
Correct evolution laws retained; the time-zero confinement regime remains unproved.
“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.