RESEARCH RECORD · SEPTEMBER 21, 2026 · V1.41
The CCM benchmark matches.
The NS-14 basis needed correction.
NS-2, NS-3 and NS-4b are complete. NS-1 remains OPEN. The new computation stays at the existing λ = 3 window and CCM’s N = 120 cutoff. G2 and RH remain open.
What had to change
Exact identity. The matrix uses Fourier modes on [0,L]. Moving its cosine coefficients into the centered interval [−L/2,L/2] requires vn → (−1)nvn. The original NS-14 script omitted this factor and therefore evaluated a different function. Its sinc-lattice conclusion is withdrawn; the original files are preserved.
At a retained lattice point ωk = 2πk/L, the correct transform equals √(L/2)(−1)kvk. A small nonzero coefficient does not make that value zero. Low-mode mass concentration cannot establish the locations of transform zeros.
The numerical lock
Certified computation, finite compression only. Fresh Arb assembly and verified eigenpairs at 768 and 1024 bits give identical proposed decimal root centers. Each accepted bracket has radius 10−110, with opposite interval signs and a derivative excluding zero. All eight discrepancies match the published rounding in CCM §6, Figure 1.
| Zeta index | Reproduced discrepancy | Published |
|---|---|---|
| 1 | 1.582329697193127e-34 | 1.6e-34 |
| 2 | 2.060550289594801e-31 | 2.1e-31 |
| 3 | 1.460904856368295e-29 | 1.5e-29 |
| 4 | 8.293686860378121e-27 | 8.3e-27 |
| 5 | 1.279455965166440e-25 | 1.3e-25 |
| 6 | 1.174389453606719e-23 | 1.2e-23 |
| 7 | 7.528486586592898e-22 | 7.5e-22 |
| 8 | 6.565874264666424e-21 | 6.6e-21 |
Open input. These are unique local roots near the first eight zeta ordinates. The computation does not enumerate all transform roots, rerun NS-14’s N = 256 experiment, or transfer the root intervals to the complete ground. Complete-ground transfer and cofinal convergence remain unproved.
Assigned tasks
NS-1 — OPEN: all four available local ZIPs, available repository refs and saved project records were checked. No original later script or witness was recovered. GitHub exposes only the already restored v1.26 release. The previous v1.34 library-download failure is documented; the current tools do not expose that download connector. An accessible original v1.28–v1.34 archive or extracted evidence with provenance is still needed. The v1.31 report alone does not close the gap.
NS-2 — complete: exact coordinate dictionary, eight published discrepancies and two-precision interval gates are archived. The required modification is to the diagnostic’s reconstruction, not to the physical Weil form or the existing λ = 4 certificate.
NS-3 — complete: the a = log λ translation table, Zhu and Groskin comparisons, scoped cutoff-cost citation and CCM μλ paragraph are integrated. The older empirical “roughly 20×” comparison is no longer presented as a normalization check.
NS-4b — complete: the existing ten-route table in the v1.39 adversarial review, §§6–7, meets the task. Its exact location is now linked from the manuscript; the signed concentration mechanism is explicitly outside the two no-go classes.
Verification and scope
The full manuscript builds to 215 pages, with 0 undefined or duplicate references and 0 overfull boxes. Pyright reports zero errors or warnings for the new certificate script. The earlier v1.40 evidence is retained unchanged. This pass has not received a new independent human or agent audit.
W4 ≥ 0 and its certified simple even ground remain established within the existing record. G2 and RH remain open. No new window or positivity-transfer claim is made.