NS-43 · NS-44 · NS-45 / SEPTEMBER 22, 2026 / VERSION 1
Two comparison nodes close. The cofinal estimate is still missing.
The direct uniform lower floor remains the priority. No defensible numerical probability of proving G2 or RH follows from the current record. This ranking uses the strength and independence of the missing estimate.
The substantive v1.52 draft is local on codex/ns43-route-selection, based on PR #20. It is not merged, tagged or pushed. No new window or tail metric was introduced. G2 and RH remain open.
What changed
| Existing comparison | Fresh result | Scope |
|---|---|---|
| λ=5, Q₁₆ W Q₁₆ ≥ 10⁻⁸ Q₁₆ D Q₁₆ | q/D ≈ 6.4235244×10⁻¹⁸ even; 1.1127923×10⁻¹⁷ odd. | Both are below 10⁻⁸. Both Weil energies are positive. Only this lower comparison is false. |
| λ=5, N=25 tested dyadic norm comparison | Exact comparison Rayleigh quotients 341/250 = 1.364 and 2093/1500 ≈ 1.39533. | Both exceed 1. No admissible smaller block metric or positive row weighting fixes this partition's criterion. |
Both use newly frozen exact dyadics and fresh Arb interval gates at 320 and 448 bits. Independent signed-index assemblies and reviewer replays pass. The block result needs only the first three existing blocks. Its proof handles the metric-existence prerequisite explicitly; it does not assume unverified full-block positivity.
The new certificates replace missing proof inputs for these two comparison questions. They do not recover the original v1.27/v1.28 witnesses or validate the historical four-block tables. Both original NS-1 recovery groups stay OPEN.
NS-44 certificate report · NS-45 certificate report
Where to spend the next effort
- Complete signed lower floor. Seek an arithmetic estimate valid on the whole even source complement and the whole odd sector, with one finite error independent of the window. This is the weakest sufficient cofinal target already proved in the repository. CAE and unrestricted exact adaptive concentration are equivalent ways to state it; neither is a new estimate.
- CCM relative selection. A concrete sufficient input is a complete second-eigenvalue separator above the explicit cutoff source's Rayleigh value, large enough relative to its residual to control the entire Fourier transform. An alternative is Rayleigh excess divided by the true complete gap, together with evaluator control or weighted tightness. A finite-compression gap does not provide this input. Global-ground use also requires odd-sector ordering.
- Arithmetic structure in joint concentration. Estimate the signed operator jointly, with the actual source removed. Even exact scalar concentration extrema for every set can miss its lower edge: two exact commuting countermodels have identical scalar data but signed bottoms −1/5 and +1/10. This refutes a general shortcut, not the actual arithmetic concentration route.
- QG plus packets, as an obstruction investigation. This could rule out bounded level complexity. It supplies no positivity mechanism, and packet feasibility remains separate. NS-40's literature assessment remains in PR #19.
Additional shortcuts now delimited
Deleting prime channels costs too much for this Gaussian weight. The new proof covers every fixed prime-power channel and omitted channels uniformly below the strict endpoint, including a bounded retained count or a cutoff o(exp(2a)). Exact even source-admissible and odd tests retain all cross terms. A fixed loss forces an unbounded ordinary-norm error. The conclusion requires unchanged other signed terms; coefficient-one, compensated, endpoint-approaching or different-weight estimates remain open.
A proposed small test family is not necessarily a weaker input. For an explicit smooth compact probe, bounded translated prime discrepancy would already exclude every off-line zeta zero. The proof uses the absolutely convergent von Mangoldt logarithmic-derivative identity, followed by analytic continuation and an explicit transform with only imaginary zeros. No such discrepancy bound is proved. For arbitrary probes, off-line zeros could still be masked at common transform zeros.
Absolute residuals do not select a ground profile. An exact abstract family has simple even nonnegative grounds with only real transform zeros, arbitrarily fast residual decay, ordinary resolvent collapse and tight ground support, yet its profiles alternate. The family is explicitly not the support-consistent arithmetic Weil family. A separate theorem rules out a nonzero single-profile limit only under its specified coarse-scale, nonnegative full-form liminf hypotheses. Finer arithmetic selection remains open.
The map closes only these named comparison or inference shortcuts. CCM step (b) remains the sole gold node. Circle remains open with the requested note: “measurement lane; closure claims withdrawn after NS-31; geometry and exact identities retained”.
Review and verification
Assisting agents produced and independently reviewed the capacity, concentration and CCM proofs, and the two fresh certificates. Astra RH1 reviewed the capacity argument and integrated comparison statements; Astra RH2 reviewed the bump-strength implication. Review corrected the CCM parity scope, made the generalized concentration form-domain hypothesis explicit, and corrected the map's transform-masking wording. The final versions have no outstanding mathematical or provenance findings in those review scopes.
Actual build: 265 pages; 0 undefined references; 0 duplicate references; 0 overfull boxes. Python diagnostics: 0 errors and 0 warnings. Earlier v1.50 and v1.51 manifests each verify 6/6. The new version manifest and external artifact index accompany this report.
Self-contained manuscript · New propositions and complete proofs · Review record · Updated research map
No new cofinal signed estimate was obtained. A failure of these sufficient comparisons is not a negative Weil direction. Existing PRs #18–20 remain open; this local draft changes neither their reviewed commits nor the public main branch.