Diagnostic of the frozen finite trial, not a new complete-ground certificate.
The stated prediction is supported for the archived finite trial: its transform is tiny at all 64 tested zeta ordinates, with opposite signs at each pair of endpoints gamma_j +/- 0.01. This does not identify the cause of the small Rayleigh quotient, establish the pattern for the complete ground, or turn a proximity bound into a measured displacement.
Pre-registration: draft PR57,
commit 0513e84, preceded the NS104 claim at 605bbe7 and the scan.
The unchanged proposal is in PROPOSAL.md. No new version or theorem node.
We evaluate the normalized exact column 16 of the even 4097-row trial in
evidence/v126/g2_simultaneous/simultaneous_even_M4096_s16_normalized_witness.json.gz.
This is the same f used in v1.42/1.43. It is not a saved infinite
eigenfunction. Every cosine coefficient (modes 0 through 4096) is included.
The complete ground xi_4 is accessed in the earlier certificates through
its projection h=P_(xi_4)f and energy estimates.
Arb evaluations at 1024 and 1280 bits agree; serialized intervals overlap for every paired numerical field. The Rayleigh enclosure is inherited from the saved complete residual ingredients at 768 and 896 bits, respectively; those assemblies are not recomputed. There is no new cutoff, eigensolve or omitted finite-coefficient remainder. Dependency hashes are in both reports.
.venv/bin/python evidence/diag_ns104_ground_mechanism/evaluate_trial.py --bits 1024 --output /tmp/ns104-fresh-1024.json
.venv/bin/python evidence/diag_ns104_ground_mechanism/evaluate_trial.py --bits 1280 --output /tmp/ns104-fresh-1280.json
.venv/bin/python evidence/diag_ns104_ground_mechanism/validate_results.py
Use fresh output paths. The diagnostic stops at 64 ordinates as registered.
Write F=widehat(f) and rho=q_W[f], with rho = 2.45361754930714…e-75. The complete eigenvalue satisfies 0 < mu_0 <= rho; rho is not a measured value of mu_0. The table shows display approximations, not additional root enclosures.
| j | gamma_j | abs(F(gamma_j)) | F’(gamma_j) | linearized shift -F/F’ |
|---|---|---|---|---|
| 1 | 14.13472514 | 1.06412e-40 | 3.10869e-3 | -3.42306e-38 |
| 2 | 21.02203964 | 1.64476e-40 | -5.42615e-5 | -3.03117e-36 |
| 3 | 25.01085758 | 2.50187e-41 | 4.88381e-6 | 5.12277e-36 |
| 10 | 49.77383248 | 3.32304e-44 | -6.40098e-13 | 5.19146e-32 |
| 29 | 98.83119422 | 1.38694e-42 | 1.31914e-24 | 1.05140e-18 |
| 32 | 105.44662305 | 6.66414e-42 | -2.65060e-26 | 2.51420e-16 |
| 48 | 139.73620895 | 5.64238e-41 | -8.91357e-33 | 6.33010e-9 |
| 64 | 169.91197648 | 9.17068e-41 | -7.00686e-37 | 1.30881e-4 |
All 64 endpoint pairs have opposite signs. This gives nearby trial roots by continuity, but no uniqueness or sharper root-displacement certificate is supplied. One Newton step is also evaluated; it is not a root enclosure. Tiny values alone do not imply uniformly tiny root displacements: the slopes decrease sharply. The largest linearized displacement is about 1.77081e-4 at j=63.
The largest abs(F(gamma_j))/sqrt(rho) is 0.00332047 (rounded upward), at j=2. The formal two-sided partial sample sum 2 sum_(j<=64) F(gamma_j)^2 / rho is 0.0001018608892…. It is not claimed to be an energy fraction: no complete zero-side representation with a nonnegative omitted remainder is established here. In particular, this number does not prove that the rest of the energy comes only from zeros above a crossover.
At j=1, sqrt(rho)/abs(F’) is 1.59340587e-35, about 465.49 times the linearized displacement 3.42306343e-38. Thus the suggested scale is not an observed equality for this trial. Since mu_0 itself is not determined, this comparison neither proves nor refutes a proposed asymptotic law for the complete ground. The earlier finite N=120 discrepancy is a different object.
The manuscript’s lambda=4 is multiplicative: support is [-log 4, log 4], and L=2 log 4. NS103 uses the same letter for the logarithmic half-width, support [-lambda,lambda]. On converting its heuristic 2 pi exp(2a) to a=log 4, the value is 32 pi = 100.5309649…, between gamma_29 and gamma_30, not 2 pi exp(8). The trial pattern continues past that value through gamma_64; a hard cutoff reading is not supported.
NS103’s four detection windows are first detected points on its chosen finite basis/grid and threshold. They are not proved minimum support sizes or a universal detection law. Compact support in physical space does not give compact support of the entire Fourier transform in frequency. An effective bandwidth count is not the dimension of the full Paley-Wiener space, which is infinite dimensional. Counting finite zero constraints alone does not identify a minimizer or bound the residual tail. No sharp zero-height interpretation of fixed-window positivity is proved. The relation of the explicit-formula conventions must be checked on the same tests before transferring a numerical threshold; calling one object “semilocal” does not by itself settle that comparison.
The fixed-window results establish positivity, spectral structure and ground-transform zeros. They do not verify a new height of zeta zeros or establish cofinal convergence. The cutoff-cost theorem is for its specified scalar majorant, not a universal lower bound on every possible method.
v1.43 proves abs(ell(v))^2 <= C_ell q_W[v] using the complete Schur comparison, with C_ell = 79920.06156776…. For h=P_(xi_4)f and the certified derivative lower bound d=0.0016, it obtains
abs(z_1(4)-gamma_1) <= sqrt(C_ell rho)/d = 8.75208157...e-33.
Thus the connection between the energy upper budget and the proximity upper budget was already part of the certificate. The radius is not an observed displacement; its interval includes zero. Replacing C_ell by 1 requires an additional argument. Dropping the rest of a zero sum is not justified without its nonnegativity: off-line pairs contribute signed cross products. Global nonnegativity of the W4 form does not make each zero contribution nonnegative. No RH assumption is used in the Schur argument.
The existing coarse transfer bound is norm(f-h) <= sqrt(rho/1e-67), about 1.5664e-4. Its transform and derivative allowances are about 2.60823e-4 and 2.08757e-4. The latter already exceeds abs(F’(gamma_2))=5.42615e-5. Consequently those displayed bounds do not transfer this scan’s later trial roots to the complete ground. This is a limitation of the available bounds, not proof that the complete pattern fails or cannot be certified.
No Davenport-Heilbronn complete ground, matched Rayleigh bound or projection
certificate is present. Its proposed analogous proximity remains untested.
The unchanged example_screen.py run in control-sanity.json checks the
control implementation, not this ground prediction. The actual prediction
is marked not-applicable to that unmatched archive; no positivity implication
is being admitted through this exception. NS100/101 have the same mismatch.
v1.36 asks for q_a[f] >= -C norm(f)^2 with one finite C on the complete admissible domains cofinally. It does not ask for an error smaller than each positive eigenvalue. A uniform absolute remainder bound can suffice even when it exceeds a tiny positive energy. Resolving that energy’s sign is a stronger task. No finite list of numerical windows proves a cofinal bound, but that is different from saying the floor must numerically resolve all near-null directions. NS46’s obstruction to a positive spectral gap does not obstruct this weaker negative-floor target. The full parity, source, prime, pole, exterior and cross-term obligations remain in place.
Reviewed main d92264127a6bf40cbfb85899e1db676576457546, including all
104 conclusion nodes, the 14 continuation groups, current checkpoints,
board, evidence ledger and changed NS103 claims. Node objects are unchanged
from the prior reviewed main a47d8a1; the affected continuation summary is
corrected forward. This is a conclusions/dependency and interpretation
review, not a replay of all historical proofs. Both missing-original groups
remain OPEN. Local author review only; no independent reviewer is claimed.
Wall check: Distinct test of the stated finite-trial prediction; Same open gap for CCM-LIMIT and WEIL-FLOOR. Closest: NS43/46, v1.40–1.43, NS103. The observation supports a near-zero pattern but does not prove a null-space mechanism, convergence, an energy-tail law or a new arithmetic estimate. No frozen family is thawed. Stop at the registered diagnostic; any complete-ground extension needs its own explicit transfer estimate and control assessment before another row is claimed.