05. Continuous scale flow and prime return maps
STATUS: RIEMANN HYPOTHESIS OPEN
Recorded 2026-09-29. Date of the recorded report; individual start and finish times are not reconstructed.
Starting idea
Prime periodic orbits have lengths log p. The question was whether their actual arithmetic return maps could explain the real part 1/2 of every zero.
What was tested
The audit separated a mapping torus, its fiber monodromy, local p-adic transverse action and a genuine smooth Poincaré return map. It also separated local orbit modes from the global zero representation.
What survived
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The known orbit C_p = ℝ₊×/p^ℤ has length log p. In the chosen lift, the deck convention and the positive-time inverse return convention must be distinguished.
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The p-adic normal action can have valued growth exponent +1 in the stated orientation. Thus a literal claim of no transverse growth is false, without identifying that exponent with a zeta-zero character.
What failed or remains unproved
Local Haar unitarity does not recover a discrete zeta divisor: the local model has extra modes. A faithful common Hilbert metric on the global arithmetic quotient remains missing. Symplecticity, indefinite pairings and time reversal do not replace it.
Why the work moved on
The next track asked for the weaker requirement of two-sided subexponential growth of the return operator at p = 2.
Current scope
Known arithmetic geometry clarified; no new purity theorem or spectral realization claimed.
“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.