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Claim. Under the distinct-factor reading of Problem 786, call $A\subseteq\{1,\ldots,N\}$ admissible when two finite subsets of $A$ whose products agree always have the same number of elements. Shisheng Li's full proof claim, submitted to the site's proof-claim tab on 28 September 2026 and declared as produced with GPT-6 Astra and GPT-5.6 Sol (OpenAI) and Claude (Anthropic), asserts two theorems. For the finite question: there is an absolute $\eta>0$ such that every admissible $A\subseteq\{1,\ldots,N\}$ has $\lvert A\rvert<(1-\eta)N$ for all large $N$, so no admissible set has size $(1-o(1))N$. For the infinite question: an admissible $A\subset\mathbb N$ has $\sum_{a\in A,\,a\le N}1/a\le\tfrac12\log N+O((\log\log N)^2)$, so its logarithmic density, and with it its natural density where that exists, is at most $1/2$, and no admissible set has density above $1-\epsilon$ for $\epsilon<1/2$. Both answers are no. The route the summary gives for the finite bound: when some $a\in A$ factors as a product of $t$ ratios $q_j$, each of which is the quotient of at least $2t$ pairwise disjoint pairs $(x,q_jx)$ of members of $A$, the identity $a\prod_jx_j=\prod_jq_jx_j$ equates a product of $t+1$ members with a product of $t$ members, which admissibility forbids, and some $a\in A$ admits such a factorization with $t=O(\log^2N)$; the Turán-Kubilius inequality shows that the primes most of whose multiples are missing from $A$ have a small sum of reciprocals, and a prime $p$ with many surviving multiples is written as a ratio of smaller quantities through a randomized pairing of $pz$ with $mu$, where $m$ is a smooth integer near $pk$ and $k$ the largest small-prime divisor of $z$. The logarithmic bound comes from a parity argument on divisor boxes. The claim's notes say the constant $\eta$ is tiny and not optimized, where Tao's construction on the problem page suggests the true deficit is $0.1715\ldots$. The distinct-factor reading is the problem page's Statement (precise), so the claim, if it stands, answers it in full. Because the distinct-factor property is the weaker condition on $A$, a negative answer under it is a negative answer under the repetitions-allowed reading as well.

Submission note. Posted to erdosproblems.com as a proof claim by Shisheng Li (account daizisheng) on 28 September 2026, giving "GPT-6 Astra, GPT-5.6 Sol (OpenAI); Claude (Anthropic)" as the AI used:

We answer both parts negatively (distinct factors). (ii): there is an absolute $\eta>0$ with $|A|<(1-\eta)N$ for every admissible $A\subseteq[1,N]$, $N$ large. If $a\in A$ is a product of $t$ ratios $q_j$, each with $\geq 2t$ disjoint pairs $(x,q_jx)$ in $A$, then $a\prod_j x_j=\prod_j q_jx_j$ is a forbidden identity; we show some $a\in A$ has such a factorisation with $t=O(\log^2 N)$. Primes whose multiples are heavily deleted have small reciprocal sum (Turán–Kubilius). For other primes $p$, pair $pz$ with $mu$, where $z=ku$, $k$ is the largest small-prime divisor of $z$ and $m$ is a random smooth number $\approx pk$; since $m$ is recovered from $mu$, $mu$ is nearly uniform, so $pk/m$ has many pairs and $p=(pk/m)\cdot m/k$ reduces to smaller primes. (i): $\sum_{a\in A}1/a\leq \frac{1}{2}\log N+O((\log\log N)^2)$, by a divisor-box parity argument, so the logarithmic density is at most $1/2$. Both formal-conjectures statements are proved in Lean. Notes: Historical note: Erdős (1980, p. 114) reports that Ruzsa had shown both answers are negative in the distinct-factor setting; no proof has appeared, and we claim no priority. Part (i) was answered earlier by Gessel (natural density $\leq 7/8$); our bound $1/2$ is stronger. The constant $\eta$ is not optimised (it is tiny); Tao's construction suggests the optimal deficit is $0.1715\ldots$. The Lean statements are copied verbatim from formal-conjectures (answer False); only the standard axioms are used.

Postings. The full proof claim on the site's proof-claim tab, submitted 28 September 2026 and declared as produced with GPT-6 Astra, GPT-5.6 Sol and Claude; the manuscript and the Lean folder in the author's GitHub repository, linked here at the repository's last commit before the submission (28 September 2026, 08:11 UTC, by the repository's commit list accessed), where the submission links the default branch. The claim's notes say its Lean statements are copied from the formal-conjectures file for the problem with the answer False and that the kernel reports only the three standard axioms.

Acceptance. None on record. The site's label is OPEN (page last edited 11 April 2026, before the claim; proof-claim tab accessed 2026-10-06), the claim has no comments, and no publication or named review exists. The claim's own note says that Erdős's 1980 survey reports negative answers by Ruzsa in this setting, for which no proof has appeared, and claims no priority; it credits Gessel's partial claim with the earlier answer to the first question and calls its own bound stronger. The formalization is not listed as evidence: it has not been built or audited in this corpus, and no statement-fidelity review exists. The claim stays claimed, and the problem's standing is claimed, disproved, through this page.

Depends on. No page of this wiki.