09. Prime-logarithm phases: the arithmetic comma track
STATUS: RIEMANN HYPOTHESIS OPEN
Recorded 2026-09-30. Date of the recorded report; individual start and finish times are not reconstructed.
Starting idea
Prime logarithms generate a torus flow. Could nonresonance, long-time phase statistics or pretentious distance control the distinguished phase at which the Möbius sum is read?
What was tested
Exact finite Fourier identities, Mellin/Perron reconstruction, time-averaged sinc factors, high moments and near returns were checked. The cost of removing smoothing was kept explicit.
What survived
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The phase-to-arithmetic reconstruction is exact within its stated cutoff and convergence domains.
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Synthetic comparisons show that identical Haar phase statistics can coexist with very different values at the distinguished arithmetic phase. Quantitative Halász input provides logarithmic savings at its stated scale.
What failed or remains unproved
Equidistribution and root-mean-square time control do not give the required pointwise square-root estimate. Hardy-space boundary evaluation can be unbounded. A normalization by the number of sign configurations is not cancellation of the actual signed sum.
Why the work moved on
The next track retained the actual prime-product constraint as a Boolean weighted halfspace rather than replacing it by a phase distribution.
Current scope
Exact bridges and their losses retained; phase statistics alone did not provide the missing estimate.
“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.