Why a proved theorem still needs reproducible claim custody
On May 20, 2026, OpenAI announced that an internal reasoning model had produced a counterexample to the Erdős planar unit-distance conjecture.
The problem is easy to state: given $n$ points in the plane, how many pairs of points can be exactly distance $1$ apart?
For nearly eighty years, the prevailing expectation was that square-grid-type constructions were essentially optimal up to a slowly growing exponent. OpenAI’s announcement changed that. Its internal reasoning model produced an infinite family of examples giving a polynomial improvement, and the proof was checked and written up in mathematical form by external mathematicians.
In this article, “the remarks paper” refers to the companion PDF by Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang, and Matchett Wood, linked from OpenAI’s announcement.
