Diagnostic of the frozen finite trial, not a certificate and not a complete-ground statement.
Arb evaluation (1024 bits, replayed at 1280 bits) of the unchanged archived
normalized even trial, column 16, modes 0-4096, the same object NS104 and
v1.42/1.43 use, through the v1.42 loader. No new eigensolve, cutoff, window,
zero-tail bound, G2 or RH claim. Pre-registered in draft PR #59 before the row
claim; the unchanged proposal is PROPOSAL.md.
NS104 evaluated the trial transform F only at the zeta ordinates and 0.01 to
either side. This scan samples F on the grid t = 0.05, 0.10, …, 400 and asks
where its mass sits and where its real sign changes are. PROPOSAL.md stated
three predictions before the run:
scan_trial.py, scan-1024bits.json, scan-1280bits.json)| Envelope. Maximum of | F | per 20-unit band, log10: |
| band | log10 max abs F |
|---|---|
| 0-20 | -0.062 |
| 20-40 | -3.974 |
| 40-60 | -9.549 |
| 60-80 | -15.135 |
| 80-100 | -19.692 |
| 100-120 | -24.251 |
| 120-140 | -28.338 |
| 140-160 | -32.43 |
| 160-180 | -34.897 |
| 180-200 | -36.943 |
| 200-220 | -37.6 |
| 220-240 | -37.769 |
| 240-260 | -37.849 |
| 260-280 | -37.947 |
| 280-300 | -38.03 |
| 300-320 | -38.255 |
| 320-340 | -38.256 |
| 340-360 | -38.484 |
| 360-380 | -38.36 |
| 380-400 | -38.6 |
The decay is close to exponential, 0.20 to 0.28 decades per unit height between 20 and 160, and then flattens. Above 200 the profile sits on a plateau: max 2.51e-38, median 2.07e-39, i.e. 0.51 and 0.042 times sqrt(rho) = 4.95e-38, where rho = 2.4536e-75 is the trial Rayleigh quotient. The interval radii are below 1e-304 everywhere, so the plateau is the trial’s actual transform, not precision noise.
Sign changes versus ordinates. A sign change is counted when both neighbouring grid values exclude zero at working precision.
| T | sign changes of F below T | N(T) | changes within 0.05 of an ordinate |
|---|---|---|---|
| 50.0 | 10 | 10 | 10 |
| 100.0 | 29 | 29 | 29 |
| 100.53 | 29 | 29 | 29 |
| 150.0 | 52 | 52 | 52 |
| 200.0 | 79 | 79 | 73 |
| 300.0 | 126 | 138 | 85 |
| 400.0 | 172 | 202 | 89 |
No sign change below 100 is unmatched. Between 150 and 200, 21 of 27 changes are within 0.05 of an ordinate. Above 200 the counts fall below N(T) and the matching stops: the last matched change is at 396.3 (ordinate 200), but most changes there are not near ordinates.
Replay. 1280 bits with identical parameters (validate_replay.py,
validation.json): all 8000 intervals overlap, midpoints differ by at
most 0.00e+00 absolutely, sign-change heights and all counts are
identical, and the envelope table is identical to three decimals.
The archived trial is a function supported in [-log 4, log 4] whose transform carries essentially all of its mass below height about 150, decays exponentially through that range, vanishes near every zeta ordinate there (29 below the crossover, 52 below 150), and then sits at the level allowed by its energy budget. That is a description of this finite object, consistent with reading the small Rayleigh quotient as “avoid the ordinates where the mass is, spend the budget elsewhere”. It is not a proof of a null-space mechanism, not a statement about the complete ground xi_4 (NS104: no transfer of trial roots beyond gamma_1 under the displayed bounds), not a convergence statement, and it supplies no arithmetic estimate. Nothing here bears on zeros of zeta beyond what v1.40-1.43 already certify.
Wall check: Distinct test of a stated prediction on an archived object; Same open gap for CCM-LIMIT. Closest: NS104, v1.40-1.43, NS103. What changed: the sampled set and the question. No group thawed; no route opened or closed; no version.
cd <repo root>
.venv/bin/python evidence/diag_ns105_trial_spectrum/scan_trial.py --bits 1024 --output /tmp/ns105-1024.json # ~70 s
.venv/bin/python evidence/diag_ns105_trial_spectrum/scan_trial.py --bits 1280 --output /tmp/ns105-1280.json # ~90 s
.venv/bin/python evidence/diag_ns105_trial_spectrum/validate_replay.py
Fresh output paths are required. Depends on evidence/v142/certify_ground_zero.py
and the v1.26 witness archive; hashes are recorded in the scan JSON.