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Claim. Theophilus Agama, New bounds for the Heilbronn triangle problem, arXiv:2006.05269, first posted 5 June 2020; version 13 of 6 May 2026 is the one the release preprint of OpenAI's claim discusses. For $s$ points in the unit disk, the quantity $\alpha(s)$ of Problem 507, the paper asserts the upper bound $\alpha(s)\ll s^{-3/2+\varepsilon}$ for small $\varepsilon>0$ (its Theorem 1.2) and the lower bound $\alpha(s)\gg(\log s)/s^{3/2}$ (its Theorems 1.3 and 4.1), by what it calls the geometry of compression. Together the two bounds assert $\alpha(n)=n^{-3/2+o(1)}$, an estimate of the quantity the problem asks to estimate, so the claim is full and its value answered. The asserted upper bound is stronger than the best published one, the exponent $7/6+o(1)$ of Cohen, Pohoata and Zakharov [CPZ24] recorded on the problem page, and the asserted lower bound is stronger than the release's $n^{-2+\eta}$.

Rejection. The release preprint records that Theorem 4.1 of version 13 counts the center of the circle together with the boundary points, so that its point set contains a collinear triple, the two endpoints of a diameter and their midpoint, whose triangle has area zero; and that omitting the center leaves the every-triple estimate unproved, since the argument estimates only the triangles formed by the center and adjacent boundary points. The lower bound on which the estimate rests is therefore unproved as printed, and the claim is recorded as rejected on that record. The release says that its objection concerns the argument and not the possibility of the asserted bound. No record of the upper bound's argument is recorded here beyond the preprint itself.

Depends on. No page of this wiki.

Acceptance. None recorded. The preprint has no journal record, the site's page labels the problem OPEN with the bounds $(\log n)/n^2\ll\alpha(n)\ll n^{-7/6+o(1)}$ as the best known and does not mention it, and no outside review of it is recorded.