04. Arithmetic polarization and intersection pairings
STATUS: RIEMANN HYPOTHESIS OPEN
Recorded 2026-09-29. Date of the recorded report; individual start and finish times are not reconstructed.
Starting idea
Could an existing arithmetic intersection or height theorem provide the positive pairing missing from the zero-bearing quotient?
What was tested
The track compared Arakelov heights, explicit-formula “intersection” language, adelic Fourier involutions and product formulas, checking the actual spaces and subtraction of pole terms.
What survived
-
Known positivity theorems retain their stated geometric domains. The height theory of an arithmetic surface is different from a degree-zero quotient on Spec Z.
-
A dense radical obstructs any nonzero closable positive form that is required to vanish there in the specified ambient topology.
What failed or remains unproved
No exact map identified the established height space with the full zero representation. A natural Fourier-star proposal has negative directions even for actual pole-free tests; this defeats that proposed pairing, not RH or the Weil criterion. Calling an explicit-formula term an intersection does not supply an intersection geometry or a Hodge-index theorem.
Why the work moved on
The separate scaling track examined the real arithmetic prime orbits directly, then weakened the desired conclusion to a bound for a single prime return.
Current scope
Existing geometric theorems retained; the required arithmetic polarization was not constructed.
“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.