RESEARCH RECORD · SEPTEMBER 21, 2026 · REPORT VERSION 2
The saved CCM certificate passes again.
The saved v1.41 candidate was recovered and its eight local root checks replayed at 768 and 1024 bits. Current main remains v1.40. Both NS-1 evidence groups remain OPEN. G2 and RH remain open. No new window was computed.
What changed since the handoff
The fetched main commit 289961d already contains the later NS-2 diagnostic, NS-4b coverage analysis, NS-3/6 draft inserts, and a retraction of the sinc-lattice and line-symbol conclusions. The previous session’s isolated worktree additionally retained commit a6b1d8f, with the finite-compression interval certificate and manuscript positioning edits. These are two different states; neither has been overwritten or silently substituted for the other.
What had to change
Exact identity. Moving the unshifted cosine coefficients on [0,L] to [−L/2,L/2] requires vn → (−1)nvn. The original NS-14 reconstruction omitted that phase and evaluated a different function. At a retained lattice point ωk = 2πk/L, the correct transform is √(L/2)(−1)kvk. A small coefficient does not force a zero.
The obstruction was in the diagnostic’s reconstruction and inference, not a certified failure of the Weil form or of CCM’s convergence proposal. The corrected benchmark uses CCM’s N = 120, rather than the old NS-14 N = 256 experiment. No claim that it locates the complete ground’s zeros follows.
Replayed result and its precise limit
Certified computation, finite compression only. At λ = 3, N = 120, K = 128, the original assembler and verified eigenpairs pass all eight endpoint-sign and nonzero-derivative root gates at both precisions. Decimal centers agree with the original archived reports, and discrepancy intervals overlap across both precisions. The first discrepancy is approximately 1.582329697193127 × 10−34, matching CCM §6, Figure 1 at its printed precision.
Exact limitation. A → cA + sI for c > 0 preserves the ground eigenspace and its transform zeros. This numerical benchmark alone cannot detect a common positive scale factor or an identity shift. The manuscript now says so explicitly; full normalization depends separately on the coefficient identities. Its CCM infimum is also written μCCMλ to distinguish it from the inverse-approximation parameter.
Open inputs. Complete-ground zero transfer and cofinal convergence to Xi remain unproved. The interval certificate does not enumerate every transform zero. No G2, RH, or positivity-transfer claim is made.
Assigned work
| Task | Current status |
|---|---|
| NS-1 | OPEN. The prior search record covers four local archives, available refs and saved project records; no later original script or witness was recovered. The v1.31 report remains only a partial recovery. An accessible original v1.28–v1.34 archive or extracted original evidence with provenance is needed. |
| NS-2 | Main contains the broader midpoint diagnostic. The saved local candidate adds eight interval root certificates, freshly replayed here. These have different scopes and both should be retained. |
| NS-3 | Citations, half-width translation and the CCM infimum paragraph are integrated in the local candidate. Main separately contains draft inserts. Integration remains pending. |
| NS-4b | Main contains the detailed coverage table: the probe theorem covers one of ten closures; the protected-orbit theorem covers none of the ten as recorded. Signed concentration lies outside both. The saved candidate already points to the earlier v1.39 coverage review. |
Build and integration status
The updated local candidate builds with the repository’s actual build script to 215 pages, with 0 undefined or duplicate references and 0 overfull boxes. The certificate script was unchanged. The two fresh replays agreed with the archived reports; v1.40’s manifest also verified without missing or mismatched files.
The shared-repository rule requires coordination of the integration target. That question remains pending. No branch was cut, no shared-checkout branch was switched, and nothing was merged or pushed. Integration must preserve the newer main diagnostics and retractions, resolve the task-board and research-map differences, and avoid restoring the obsolete KMS paragraph still present in the older candidate. This report does not declare the combined manuscript ready.
Replay record · Build record · NS-1 availability search · Preserved report version 1
W4 ≥ 0 and its simple even ground remain the fixed-window results of v1.40. G2 and RH remain open.