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Claim. Let $P,Q$ be finite sets of primes with $\bigl(\sum_{p\in P}1/p\bigr)\bigl(\sum_{q\in Q}1/q\bigr)=1$, the question of Problem 307. The manuscript On the equation $n''=n$ and a problem of Erdős and Barbeau on products of prime-reciprocal sums identifies such a pair with a two-cycle of the arithmetic derivative $n\mapsto n'$ that is not a fixed point and proves necessary conditions: its Lemma 4.2 (p. 6 of the pinned PDF), titled after Rosen, gives $|P\cup Q|\ge59$, since the reciprocal sum of $P\cup Q$ exceeds $2$ and the first $58$ primes fall short of that, the observation Julian Rosen made in a MathOverflow comment of 15 January 2019; its Theorem 4.3 (p. 6) adds $\prod_{P\cup Q}p\ge\Pi_{59}\approx8.77\times10^{112}$ and $\min(\prod P,\prod Q)\ge2.09\times10^{56}$; and its Proposition 4.5 (p. 7) excludes $|P\cup Q|=59$ by an exact-integer verification over the $49{,}961$ admissible supports of $59$ primes, so that

$$|P\cup Q|\ge60 \qquad\text{and}\qquad \min\Bigl(\prod_{p\in P}p,\ \prod_{q\in Q}q\Bigr)>3.50\times10^{57}.$$

Covers. Necessary conditions on a solution, refuting every solution with at most $59$ primes or with products below the stated bounds: a partial no to the problem's existence question, which is why the claim value is disproved. Not covered: whether a solution exists, which the manuscript itself leaves open. The manuscript presents Kovič's refereed 2012 conditions (their claim page) as the prior literature and shows that the proof of Kovič's Proposition 18 is invalid (§14, p. 63). The manuscript's own contributions are the product bounds and the step from $59$ to $60$: the disjointness it credits to Robert Israel and the bound $59$ to Rosen, both from the MathOverflow comments of January 2019 recorded on the problem page, where the $59$ is also checked; the step to $60$ rests on the manuscript's finite verification, of which no rerun is recorded.

Standing. Claimed. The work was first posted on 13 June 2026 as the repository's version 1.0 (paper, Lean formalization and computations; the release of 14 June 2026 is linked above), and the exclusion of $59$-prime supports came with version 1.1.0 of 2 July 2026; version 1.74.0 of 2 September 2026 was published on Zenodo on 3 September 2026 as a software record (a zip of the repository, not a PDF), and later versions continue; the manuscript PDF in the repository at its pinned commit of 6 September 2026, the preprint link above, is version 1.75.0, also dated 2 September 2026, 136 pages, and is the version cited on this page. The author declares in the acknowledgments (p. 62) substantial assistance from Anthropic's Claude with the derivations, the computations, the Lean formalization and the drafting, and takes responsibility for the statements; the claimant is the author. The manuscript is unrefereed, has no arXiv version and no independent review, and the site's page does not cite it; the site's commentary states the bound $60$ without attribution, and the manuscript says that it claims the proof, not the statement. The author states that Lean files in the repository prove the two barriers, the $60$ through a native_decide search; the formal-conjectures statement file for the problem points to them from two variants marked solved, which are statement files and not a formalization of this claim. This corpus has not built or audited the repository, so no formalized evidence is listed; the page records the manuscript's statements (pp. 1--3, 6--7 and 62 of the pinned PDF) and not its proofs. The site labels the problem VERIFIABLE and its proof-claim tab is empty.