RIEMANN PROJECT · NS104 · SEPTEMBER 23, 2026 · VERSION 1
The near-zero pattern is present.
The proposed explanation is not proved.
The archived finite trial has tiny transform values at all 64 tested zeta ordinates. Both precision runs also find opposite signs at the endpoints of each interval γj ± 0.01. This supports Claude’s stated prediction for that trial. It does not establish the same pattern for the complete ground, identify a null-space mechanism, or imply convergence to Xi.
This is a diagnostic and interpretation audit. It adds no manuscript version, theorem node, larger window or arithmetic estimate. The pre-registration was published in PR57 before the NS104 claim and scan.
What was actually evaluated
The archive contains the exact, normalized, even finite trial used in v1.42–1.43: column 16, with all 4,097 cosine coefficients. It does not contain the complete infinite eigenfunction. We evaluated this unchanged trial at the first 64 zeta ordinates with Arb at 1,024 and 1,280 bits. All 647 paired serialized numerical enclosures overlap; source hashes match.
The energy quantity ρ ≈ 2.45362 × 10−75 is the trial’s Rayleigh budget. We know 0 < μ0 ≤ ρ; we have not measured μ0. The saved complete residual ingredients were reused, not reassembled.
| Zero index | Height | |Transform value| | −F/F′ |
|---|---|---|---|
| 1 | 14.1347 | 1.06412 × 10−40 | −3.42306 × 10−38 |
| 2 | 21.0220 | 1.64476 × 10−40 | −3.03117 × 10−36 |
| 29 | 98.8312 | 1.38694 × 10−42 | 1.05140 × 10−18 |
| 64 | 169.9120 | 9.17068 × 10−41 | 1.30881 × 10−4 |
The slopes shrink sharply, so tiny values do not imply uniformly tiny displacements. At the first ordinate, √ρ/|F′| is approximately 1.59341 × 10−35, about 465 times larger than the linearized shift. The proposed scale is not an observed equality for this trial. Neither quantity determines the complete ground’s actual displacement.
Three corrections matter
1. The two window parameters differ. In the manuscript, λ = 4 means support [−log 4, log 4]. NS103 calls the logarithmic half-width “λ.” Converting its heuristic 2πe2a with a = log 4 gives 32π ≈ 100.531, between the 29th and 30th ordinates. The observed trial pattern continues beyond that point.
NS103 measured four detection windows using finite bases, a grid and a numerical threshold. These are not proved minimum windows or a universal height law. A compactly supported function’s entire transform is not confined to a finite frequency interval. The effective bandwidth count does not identify the minimizer or bound its omitted-zero contribution.
Fixed-window certificates establish genuine positivity and spectral facts. They do not verify a new height of zeta zeros or establish a cofinal theorem. The manuscript now states this scope explicitly. The cutoff-cost theorem applies to its specified scalar majorant; it is not a universal cost barrier for every method.
2. The 9 × 10−33 radius already comes from energy. v1.43 proves a complete evaluator inequality with Cℓ ≈ 79,920.06157 and uses the derivative floor d = 0.0016:
|z1(4) − γ1| ≤ √(Cℓρ)/d ≈ 8.75208 × 10−33.
This is an upper bound whose interval includes zero. The energy-to-proximity connection is already in the proof; the radius is not a measured discrepancy. Replacing Cℓ by 1 needs another argument. Off-line zero pairs can contribute signed cross products, so one cannot simply discard an unknown remainder of the zero sum.
The existing ordinary derivative-transfer allowance is about 2.08757 × 10−4, already larger than the trial slope at γ2. Those bounds do not transfer the additional trial roots to the complete ground. This limits the available estimate, not the possible behavior of the ground itself.
3. A uniform negative floor need not resolve tiny positive energies. The v1.36 target is qa[f] ≥ −C‖f‖² with one finite C for all required windows and tests. An absolute remainder bound independent of the window may suffice even when it exceeds a tiny positive eigenvalue. Certifying each eigenvalue’s sign asks more. The complete parity, source, prime, pole, exterior and cross-term conditions remain mandatory.
Controls and what remains open
The Davenport–Heilbronn implementation sanity check was replayed. There is no matched archived ground, Rayleigh certificate or projection estimate for that analogue, so the proposed ground-proximity prediction is still untested there. The sanity example does not screen this different prediction. No generic positivity inference is admitted through that mismatch.
Wall check: Same open gap. Closest results: NS43/46 and v1.40–1.43. The finite prediction was a distinct diagnostic; CCM-LIMIT and WEIL-FLOOR remain frozen and unproved. The 64-point scan is complete. A larger scan alone would not supply the missing arithmetic input.
The next useful direction is a specific inequality retaining the exact prime coefficients together with the archimedean term and bounding their signed remainder by a common constant. No such new inequality is supplied here. It must be written and screened before a new research row is claimed; an equivalent floor formula is insufficient.
Review scope: main d922641, all 104 registered conclusions and 14 continuation groups, affected NS103 interpretation and archived v1.40–1.43 dependencies. Local author review, not an independent audit or replay of every historical proof. Both missing-original evidence groups remain OPEN. G2 and RH remain open. Full data, limits and reproduction instructions: evidence/diag_ns104_ground_mechanism/.