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Claim. The answer to Problem 659 is yes. 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 (v3 on 5 February 2026), present in Section 4.3 a complete solution the authors attribute to Aletheia, a math research agent built upon Gemini Deep Think, and classify as an independent rediscovery. The lattice is the ring of integers of $K=\mathbb Q(\sqrt{-7})$ embedded in the plane, $\Lambda=\{m(1,0)+k(\tfrac12,\tfrac{\sqrt7}2):m,k\in\mathbb Z\}$, whose squared norm is the form $Q(m,k)=m^2+mk+2k^2$ of discriminant $-7$, so every squared distance in $\Lambda$ is an integer. The point set $P_n$ consists of the $n$ points of $\Lambda$ nearest the origin, which lie in a disk of radius $O(\sqrt n)$. Bernays' theorem, that the positive integers up to $X$ represented by a positive definite binary quadratic form number $O(X/\sqrt{\log X})$, bounds the distinct distances of $P_n$ by $O(n/\sqrt{\log n})$. For the local condition the solution lists the six similarity types of planar four-point sets with exactly two distances: a square, the isosceles trapezoid of four vertices of a regular pentagon, and four types containing an equilateral triangle. The trapezoid has the irrational squared-distance ratio $4\cos^2(2\pi/5)=(3-\sqrt5)/2$, which integer squared distances cannot realize; an equilateral triangle $0,u,v$ in $\Lambda$ would give $v/u=(1\pm\sqrt{-3})/2\in K$, impossible since $\sqrt{-3}\notin\mathbb Q(\sqrt{-7})$; and a square $0,u,w,u+w$ would give $w/u=\pm i\in K$, impossible since $i\notin\mathbb Q(\sqrt{-7})$. So every four points of $P_n$ determine at least three distances. The source card is feng_2026_semi_autonomous_mathematics_discovery_gemini_case.

Priority and standing. The agent ran from 2 to 9 December 2025 (the paper's Section 1.1). The paper's Remark 4.3 (Remark 4.2 of v1) says that Grayzel's write-up came after all of the agent's solutions were generated and evaluated, that a 2014 blog post of Sheffer gives essentially the same result in an argument due to Sheffer and Lund that does not treat the pentagon trapezoid, and that the agent's logs show it did not access that post. Grayzel published first, on the site's discussion thread on 13 January 2026 and on arXiv on 14 January 2026, with a different lattice, and the site's curator credits Grayzel; that credit is recorded on Grayzel's claim page and gives this page no reviewed evidence. The preprint is unrefereed, the only review recorded is the authors' own expert evaluation, and no formalization of this proof is recorded. The claim is therefore claimed.