11. The global remainder beyond local asymptotic orders
STATUS: RIEMANN HYPOTHESIS OPEN
Recorded 2026-09-30. Date of the recorded report; individual start and finish times are not reconstructed.
Starting idea
If every coefficient in a local asymptotic expansion vanishes, where does the Möbius sum remain? The work used a fixed Gaussian readout to separate local expansion data from global singularities.
What was tested
For R(u) = Σ μ(n)e^{−(u−log n)²}, absolute convergence gives the transform √π e^{s²/4}/ζ(s) for Re s > 1. Contour shifts were checked with explicit height grouping and multiplicities.
What survived
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Finite left shifts yield a controlled grouped residue description. This is not an arbitrarily reordered or absolutely convergent sum over individual zeros.
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For this fixed readout, forward subexponential growth of A(u) = e^{−u/2}R(u) is equivalent to RH. Finite-jet models demonstrate that finite local data can miss global poles.
What failed or remains unproved
An infinite shift to the left and the proposed unrestricted trivial-zero series fail their convergence requirements. A full analytic germ does determine its connected meromorphic continuation; it cannot be changed while keeping every Taylor coefficient. Vanishing Selberg–Delange coefficients are not vanishing Taylor coefficients of 1/ζ.
Why the work moved on
The program paused new constructions for an inventory of actual arithmetic estimates and the precise pointwise gap.
Current scope
Exact transform and finite-shift statements retained; the target growth estimate remains equivalent to RH.
“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.