Strong Goldbach: where a proof argument stops

A candidate conditional 1.3892 refinement and proof audit, with numerical checks and unresolved analytic dependencies. Goldbach is not proved.

v1.1 · 2026-11-12 · Editorial research note

The question

The strong, or binary, Goldbach conjecture says that every even integer greater than two is a sum of two primes. This note summarizes a shared AI-assisted research discussion about a conditional refinement of a partial result and an audit of its proof dependencies. It does not establish Goldbach.

Li and Liu study representations N = p + rq, where p and q are primes, r is one or a prime, and r has a size limit controlled by a parameter a. Their paper states a conditional result with a = 1.4 under a weighted Elliott–Halberstam assumption. The discussion proposes tightening that parameter to 1.3892, so the extra factor would satisfy r ≤ q^0.3892 for sufficiently large even N, conditional on WEH(0.999) and further analytic estimates used in the paper.

The proposed derivation combines changes in several cutoff-dependent terms and reports a positive exact-arithmetic margin. The conversation also describes a separate reproduction of the rational certificate and interval checks of the numerical constants. Those checks address the numerical step; they do not verify the full analytic theorem.

What the audit found

A later audit in the same conversation identified a counting defect in a printed sieve term: a factor already extracted from a candidate integer could still be removed by the sieve. The discussion gives a concrete witness and proposes a corrected term that excludes both extracted factors. It reports a direct argument for the corrected counting inequality and says the correction leaves the numerical budget unchanged.

This is a substantive proposed repair, but the complete derivation from the distribution assumption alone was not independently reconstructed. The original source estimates, corrected counting argument, and claimed range conditions all require specialist scrutiny. Novelty of the 1.3892 parameter was not established.

Why this is not Goldbach

For a = 1.3892, the representation may still have a prime r > 1, making rq composite. Goldbach requires r = 1 for every even integer above two. A result that holds for each a > 1 only beyond a threshold depending on a would not automatically establish the endpoint a = 1.

The conversation also revisits a claimed 2016 proof by Aouessare, El Haddouchi and Essaaidi. Listing candidate prime pairs and estimating cumulative counts does not prove that every individual even integer has one. A per-integer lower bound or another argument excluding every exception is still needed.

Status and provenance

Status: candidate conditional refinement and proof audit, summarized by Zelnick Research as editorial notes. The shared conversation reports checked arithmetic and a proposed combinatorial repair. The complete analytic argument has not been independently verified, and no priority claim is made. This page is not a proof of strong Goldbach or an established new theorem.

The downloadable note is an editorial summary. The original HTML reports, checker, LaTeX, and research bundle linked inside the conversation are not included in this site's release. The cited papers remain the work of their respective authors. No reuse license has been specified for this note.

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References

Proof failure analysis: shared research conversation — Zelnick Research and ChatGPT discussion, 2026
A Rigorous Proof for the Strong Goldbach Conjecture — Abdelilah Aouessare, Abdeslam El Haddouchi and Mohamed Essaaidi, 2016
Theorem (1+1.9) on the Goldbach Conjecture — Jiamin Li and Jianya Liu, 2026

Zelnick Research (editorial notes) (2026). Strong Goldbach: where a proof argument stops. v1.1. Zelnick Research. https://research.zelnick.pw/work/strong-goldbach-proof-audit/?version=v1.1

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