03. The missing Frobenius mechanism
STATUS: RIEMANN HYPOTHESIS OPEN
Recorded 2026-09-29. Date of the recorded report; individual start and finish times are not reconstructed.
Starting idea
Finite-field proofs supply a model for asking where arithmetic origin, trace, duality and purity enter. The aim was to identify the missing theorem over the integers, not to transfer a finite-field analogy by name.
What was tested
The audit compared the curve/Jacobian polarization argument, broader cohomological purity, Deninger proposals and the existing adelic quotient that represents zeta zeros.
What survived
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For the finite-field abelian-variety argument, positivity of the Rosati pairing and the relation π†π = q act on the same representation.
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The known adelic test quotient represents the relevant zeros with multiplicities. Its topology and trace statements were distinguished from a positive Hilbert realization.
What failed or remains unproved
Formal duality and positive integer point counts alone do not establish purity. A natural positive ambient Hilbert norm collapses the intended quotient: its null subspace is dense in that completion. No positive pairing with the required faithful arithmetic descent was supplied.
Why the work moved on
The work split into an arithmetic-polarization audit and a parallel examination of continuous scaling and prime return maps.
Current scope
Missing positivity/purity isolated. The historical main state stops here; later tracks have separate states.
“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
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.