RH PROJECT · TEST AND REASSESSMENT · 2026-09-29 · V1

A failed boundary test, and a better rule for choosing the next task

The extra attempt failed. More significantly, the reviews identify why we keep arriving at these stops: we have been testing convenient properties before establishing what their success would prove about the actual arithmetic problem.

Final wall check: Same open gap — KERNEL-API. The boundary ordering was a distinct auxiliary test, rejected at admission. It supplies no new arithmetic estimate, proved wall, theorem node or thaw. The classification is narrower than a new obstruction to exact Euler methods.

What the last attempt tested

PR #63 showed that prime-chain bounds lose essential coefficient information. This attempt kept the exact prime-2 and prime-3 Euler denominators. With T a translation on the whole line, they are

D_p=(1+1/p)I−(T_{log p}+T_{−log p})/√p.

These factors are positive and commute on the whole line. Restricting them to a finite window introduces a mixed exterior term K. A nonpositive K would let their symmetrized compressed product dominate the positive full-line product. The proposed shortcut was that this error had the needed sign.

Using four narrow packets in the window (−2,2), the direct overlap calculation instead gives boundary Rayleigh values approximately +0.2041241452 and −0.2041241452, in both even and odd parity. The 50- and 80-digit runs agree on every stored matrix and energy value. Both one-sided orderings are therefore rejected for this attempt.

These are floating-point diagnostic evaluations of a bounded boundary operator. They are not interval certificates, negative Weil energies or first-zero nullvectors. An indefinite boundary error does not make the compressed product indefinite. The mathematical reviewer found no material sign, normalization or parity error. We stopped without a larger scan or additional certification.

What the three reviews changed

Mathematical review: the packet calculation is consistent, but the same sign pattern occurs for generic translations and positive weights. The exact Euler parameters appear without doing essential arithmetic work. Retaining the original numbers is not the same as exploiting their relations.

Process review: the control gate successfully catches failures and limits claims. Its weakness is candidate selection. The exemption for admission audits has allowed a stream of tests whose success would still leave an unspecified bridge to the complete signed Weil form. Publication, extra precision and algebra checks cannot substitute for that bridge.

Route triage: no new RH proposal is admitted on the present record. The useful next action is a targeted independent reconstruction of NS57’s first-zero control, because recent route decisions inherit it. Version 1.36 already has an archived independent review; this recommendation is not to repeat every upstream proof. Reopen an upstream dependency only if the NS57 audit identifies a specific issue.

My assessment

My mistake was repeatedly choosing stronger properties that would be convenient if true, rather than first showing how they would control the actual missing signed term. The latest admission record openly said that even success would leave the complete-Weil bridge missing. That should have reduced its priority before computation.

The project has gained reproducible examples, scoped exclusions and clearer statements. Those are useful records, but they do not accumulate into an arithmetic lower bound. The mathematical burden remains the unproved estimate, not the number of representations or completed tests.

For this last attempt, the target connection was absent; arithmetic use was partial (exact parameters retained, generic overlap geometry used); decision value was limited to rejecting this ordering. This is a retrospective assessment of the actual work, not a numerical score or a probability of success.

The rule now applied before another calculation

Show the implication. Write what the proposed estimate would prove about one named open input, with domains, parity, source restrictions, rates and all additional unproved terms. If the bridge is missing, make that bridge the task.
Show where arithmetic matters. Identify the step using the original data. Check cheap generic counterexamples on paper first. A supporting lemma may be generic, provided its conditional transfer and separate arithmetic hypotheses are explicit; it need not individually defeat every false analogue.
Show the decision. State what changes after success and after failure. A failed screen does not automatically license an adjacent variation with the same missing connection.

These requirements are added to AGENTS.md and the existing proposal template, including pre-claim candidate screens. Evidence recovery, interpretation checks and audits of existing proofs remain valid work with their own stated question. They should be reported as validation, rather than as a new RH route.

The next bounded task

Independently reconstruct the NS57 chain: approximation by the total near-radical family, detection of a bounded perturbation’s negative direction, preservation of small-window positivity, continuity and attainment of the first zero, and the parity/domain qualifications. Track each inherited premise to its existing source and review.

Success: a separately checked dependency that supports or precisely narrows our control decisions. Failure: identify the invalid implication and correct the affected downstream conclusions. Either outcome has direct value because the project is already relying on that argument. A clean audit would not itself establish RH, admit a new API route or thaw the frozen groups. This task is recommended here, not claimed as completed.

After that, the next research proposal must bring a concrete estimate or a precisely matched theorem to one selected arithmetic term. No replacement positivity screen is proposed in this report.

Reviewed public base: ccebc6524cf8adf5e83143ad00daa55b190737bd. All 104 existing node conclusions and 14 continuation inputs were reviewed for scope; historical proofs and certificates were not replayed. The unchanged manuscript builds to 358 pages with zero undefined or duplicate references. Both historical original-evidence gaps remain open.

Test and replay record · Mathematical review · Process review · Route triage