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Claim. The answer to Problem 652 is yes, with $\alpha_k=\Omega(k^{1/4})$. Tony Feng and twenty-three coauthors, Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems, arXiv:2601.22401, first posted 29 January 2026, present in Section 2.1 a solution the authors attribute to the research agent Aletheia, built upon Gemini Deep Think, listed in the paper as one of its two autonomous resolutions, with the remark that it is an immediate reduction to the literature. Fix $k$ and, for large $n$, a set of $n$ points with $R(x_k)<\alpha_k n^{1/2}$. The $k$ points $x_1,\ldots,x_k$ each span fewer than $\alpha_k n^{1/2}$ distinct distances, so the distance circles centered at them number fewer than $k\alpha_k n^{1/2}$, while every other point lies on one circle of each center and so contributes $k$ incidences. The Pach–Sharir incidence bound for curves with the circles' intersection pattern, with the exponents $(3/5,4/5)$, gives $(n-k)k\ll n^{3/5}(k\alpha_k n^{1/2})^{4/5}+n+k\alpha_k n^{1/2}$; dividing by $n$ and letting $n\to\infty$ leaves $k\ll\alpha_k^{4/5}k^{4/5}+1$, that is $\alpha_k\gg k^{1/4}$. The authors' Remark 2.1 says that the agent's output had the right argument with wrong exponents, cited from a paper they could not locate, and that they corrected the exponents and a limiting step in the written solution. The source card is feng_2026_semi_autonomous_mathematics_discovery_gemini_case and its digest records the statement.

Standing. The preprint is unrefereed, and the only review recorded is the authors' own expert evaluation. The site's curator labels the problem proved and credits Mathialagan's theorem, not this argument, so the site's label is no review of it; the growth rate $k^{1/4}$ is also weaker than the $k^{1/2}$ that Mathialagan's theorem gives on its page. The claim is therefore claimed. The circle family satisfies the Pach–Sharir hypotheses with $k=3$ and $s=2$: three points lie on at most one circle, and two circles meet in at most two points.