02. Arithmetic generators and boundary conditions
STATUS: RIEMANN HYPOTHESIS OPEN
Recorded 2026-09-29. Date of the recorded report; individual start and finish times are not reconstructed.
Starting idea
Could a generator reconstructed from primes and Gamma factors carry boundary conditions that force the zero spectrum onto the critical line?
What was tested
The comparison covered de Bruijn–Newman evolution, an inverse spectral m-function and modular-surface scattering. In each case the question was whether the actual arithmetic object, including multiplicities, was preserved.
What survived
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The scattering coefficient gives a precise map from nontrivial zeta zeros to scattering poles, with a no-cancellation check.
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The prime-side analytic germ, the heat-time convention and the distinction between a resonance and an L² eigenvalue were fixed.
What failed or remains unproved
Self-adjointness of the surface Laplacian does not make its scattering resonances real. Requiring the actual inverse spectral function to be Herglotz returns to a condition equivalent to RH. The arithmetic definition of the heat family does not establish its needed threshold at time zero.
Why the work moved on
The next comparison asked what finite-field Frobenius has that these number-field constructions lack: a positive pairing on the same space that carries the zeros.
Current scope
Exact correspondences retained; no independent confinement mechanism obtained.
“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.