A proposed global positivity route was retired after hostile testing

ABSTRACT — ALL READERS

This milestone records the retirement of a stronger positivity-based sufficient route within the Riemann Hypothesis programme. The route required a project-defined spectral quantity to remain non-negative throughout its intended domain. A high-precision interior negative example was found and reproduced through two materially different internal representations. A separate exact countermodel then showed that the motivating local real-zero intuition did not, by itself, force the stronger nonlocal positivity property. The project therefore retired this route while retaining the exact identities developed along the way. The detailed construction, numerical coordinates, formulas and countermodel parameters are withheld from the public version. This does not disprove RH.

PLAIN-LANGUAGE ACCOUNT — GENERAL READER

The programme had identified a mathematically attractive condition which, if true everywhere required, would have supplied a sufficient route towards RH. Because the condition was global, a single reliable negative value would be enough to defeat it.

The project therefore searched actively for failure rather than accumulating favourable samples. It found an interior negative region. The sign was reproduced using two substantially different internal formulations and survived precision, truncation and nearby-parameter checks. The calculation is high-precision numerical evidence rather than an externally certified theorem.

The investigation also produced an exact model showing why the original intuition was too strong: local real-zero structure need not force the stronger nonlocal positivity condition being tested. That exact structural result is distinct from the numerical Riemann calculation.

WHAT CHANGED

The project stopped trying to prove the unchanged global positivity statement. This preserved a useful piece of negative knowledge and redirected effort towards other positivity and passivity frameworks rather than continuing to optimise a route contradicted by its own evidence.

WHAT IS DELIBERATELY WITHHELD

The public version does not disclose the project quantities' exact definitions, Fourier bridge, arithmetic representations, counterexample coordinates or values, exact countermodel, close-zero factorisation, auxiliary integral identities, numerical search design, proof-critical dependencies, code or data.

SCOPE AND NON-CLAIMS

The retired condition was stronger than RH and only sufficient, not equivalent. Its failure therefore does not disprove the Riemann Hypothesis and does not invalidate positivity approaches in general. No external novelty claim is made.

STRATEGIC DISCLOSURE REVISION

On 19 September 2026 the earlier public technical version was withdrawn from live presentation after a project-wide strategic-disclosure audit. Its exact bytes are preserved privately for provenance. This replacement records the research decision and evidential boundary without continuing to publish reusable machinery. .