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Claim. With $c(N)$ the largest size of a set $A\subseteq\{1,\ldots,N\}$ carrying signs $\delta:A\to\{-1,1\}$ whose signed reciprocals sum to zero while no nonempty proper subset of $A$ sums to zero, as Problem 319 defines it, the claim asserts

$$c(N)=N-o(N),$$

that is, for every $\varepsilon>0$ and all large $N$ a minimal signed zero-sum relation exists on all but at most $\varepsilon N$ of the integers $1,\ldots,N$. Its method, as the claimant's summary describes it: a fixed finite block of denominators carrying the sign $+1$ is chosen together with a squarefree modulus $Q$ so that the block's reciprocal sum has a denominator prime to $Q$ while the reciprocal sum of every nonempty proper subfamily of the block does not; the sign $-1$ goes on almost all integers of an interval $(aN,N]$ that are prime to $Q$ and have only small prime-power factors, and the bounded gap between their reciprocal sum and the block's is filled by further denominators below $aN$, also prime to $Q$, produced by a completion lemma that at each step removes the largest prime-power factor of the running denominator through a subset-sum theorem of Conlon and coauthors on residues and, when no prime power above a fixed bound is left, by a fixed reservoir of denominators taken from Croot's theorem. Minimality follows because a zero-sum subfamily must, by the arithmetic of $Q$, contain the whole block or none of it, and the signs then force it to contain the whole negative side or none of it.

Submission note. Posted to erdosproblems.com as a proof claim by popular_12345 (account popular_12345) on 16 July 2026, giving "OpenAI GPT-5.6" as the AI used:

Claim: This proof claims c(N)=N-o(N). Main idea: A fixed positive modular atom P is built with a squarefree Q so that the reciprocal sum of P has denominator coprime to Q, while no nonempty proper subfamily does. For the negative side, take almost all Q-coprime, powersmooth denominators in (aN,N]. Their bounded reciprocal deficit is completed below aN by a Q-avoiding Egyptian-fraction lemma. At each step that lemma removes the largest exact prime-power component of the residual denominator using the modular-inverse subset-sum theorem of Conlon et al. Once only bounded prime powers remain, a fixed reservoir obtained from Croot's theorem represents the terminal residual. All negative denominators are coprime to Q. Thus any zero-sum subfamily uses either none or all of the modular atom; positivity then forces none or all of the negative side. This gives a primitive relation on (1-o(1))N integers. Notes: This is an unrefereed candidate proof of the density-one asymptotic, not an exact evaluation of c(N) or of N-c(N). The accompanying Lean development verifies Proposition 4.1, the Q-avoiding completion lemma, relative to two explicit proposition-valued interfaces corresponding to published results of Conlon et al. and Croot. It does not yet formalize Proposition 3.1, the powersmooth-density input, the dense-side asymptotics, the primitivity argument, or the global theorem.

Covers. The lower bound $c(N)\ge N-o(N)$, which would lift the Croot-based $(1-\frac1e+o(1))N$ recorded on the problem page to density one; with the trivial bound $c(N)\le N$ it gives $c(N)\sim N$. The order of $c(N)$, which is $N$, already follows from the two recorded bounds, the lower one pending on Adenwalla's claim page. It does not evaluate $c(N)$ exactly or estimate $N-c(N)$, which the claimant's notes name as outside the claim; the claimant filed it as a partial claim. The claim value is proved because the result proves a bound, $c(N)\ge N-o(N)$, and determines neither $c(N)$ nor $N-c(N)$.

Depends on. Croot's Main Theorem, for the reservoir of denominators; the claim also uses a subset-sum theorem of Conlon and coauthors, which the manuscript cites.

Provenance. The claim was submitted to the site's proof-claim tab on 16 July 2026 by the pseudonymous account popular_12345 and declares the AI system OpenAI GPT-5.6. The repository g-01234/erdos-319-density-one, whose single commit of 16 July 2026 is linked above, holds the manuscript, a candidate proof by its own title, and a Lean directory Erdos319. By the claimant's notes the Lean development proves the completion lemma (the manuscript's Proposition 4.1) relative to two assumed interfaces standing for the published results of Conlon and coauthors and of Croot, and does not formalize the density input (Proposition 3.1), the asymptotics of the negative side, the minimality argument or the main theorem; the claimant calls the manuscript unrefereed. The development has not been built or audited by this corpus, and one resting on assumed interfaces gives no formalized evidence in any case.

Standing. Claimed. The site's label is OPEN and its commentary does not mention the claim (2026-10-07); the tab carries its standing notice that listing a claim implies no examination. The one comment under the claim, of 17 July 2026, reports a brief reading that found nothing wrong and a separate consultation of GPT-5.6 Pro which, by the commenter's own timestamps, preceded the claim and arrived at the same protected-atom idea but completed the construction differently; it is not a review. No refereed or arXiv version and no independent review were found on 2026-09-18. The claim is pending and, as a partial claim, derives no standing for the problem.