17. Hierarchical selection in fixed derivative families
STATUS: RIEMANN HYPOTHESIS OPEN
Recorded 2026-09-30. Date of the recorded report; individual start and finish times are not reconstructed.
Starting idea
After the leading boundary energy becomes degenerate, do successive terms select k within F_m = span{k,k″,…,k⁽²ᵐ⁾}? The order of operations is the actual projection P_N R_a.
What was tested
The work derived fixed-m support asymptotics, controlled their Gram matrices, compared Fourier resolution with the boundary scale, and separately audited prime rank-one perturbations and Sonin filtrations.
What survived
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For every fixed finite m, the support-restricted smallest eigenvalue is eventually simple and its normalized minimizer tends to k/‖k‖. The recorded leading term is a(m!)²Y^(7/2−2m)e^(−2Y)/(√π‖k‖²), with Y = πe^(2a).
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The same restricted selection survives P_N R_a under liminf N/(aY) > 4/π². This is a sufficient resolution condition; necessity or a sharp threshold was not proved.
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The prime-event derivative has a rank-one description, but finite updates need not be positive. The Sonin/Gamma dictionary is exact in its own spaces and does not intertwine the current energy with a replacement selector.
What failed or remains unproved
Constants and errors are not uniform in growing m. No estimate controls the complementary directions of the full finite matrix. Evenness/simplicity of the relevant full ground states and G* remain missing. Restricted selection is not full ground capture.
Why the work moved on
A separately dated update asked only whether the union of these derivative families is dense in even L². It did not start the later full-ground steps.
Current scope
Fixed finite-m hierarchy established within its stated domain; full ground, growing m and RH remain open.
“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.