16. Rates, history and finite-cutoff splitting
STATUS: RIEMANN HYPOTHESIS OPEN
Recorded 2026-09-30. Date of the recorded report; individual start and finish times are not reconstructed.
Starting idea
Could the way support and Fourier cutoffs increase select the arithmetic kernel from the global radical?
What was tested
The track separated support restriction R_a from periodic Fourier projection P_N, examined prime-power threshold events, and compared diagnostic paths N ≍ a² and N ≍ a³.
What survived
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For the two-dimensional family span{k,k″}, support-only minimization selects k after normalization. An explicit boundary cancellation improves its rate.
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Full projected energies on every fixed derivative family have unconditional upper bounds tending to zero along the two diagnostic paths. A simple ground projector at the same finite endpoint has no history dependence; prime-power events introduce derivative corners, not jumps of the matrix.
What failed or remains unproved
Energy upper bounds are not leading asymptotics or a ground-selection theorem. Beyond two derivatives, the leading nullspace has larger dimension. At this stage neither path-dependent nor path-independent limiting selection had been proved.
Why the work moved on
The next track resolved a hierarchy for each fixed finite derivative family, using higher-order Schur terms and a stronger Fourier-resolution scale.
Current scope
Rate and finite-endpoint facts retained. Later refinements apply to restricted families and do not retrospectively prove a full-ground statement here.
“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.