An OpenAI model disproves a 1946 unit-distance conjecture

OpenAI says one of its general-purpose reasoning models has disproved a conjecture about the unit distance problem that Paul Erdős posed in 1946. The question is how many pairs of points in a plane can sit at exactly the same distance from each other, and for eighty years the working belief was that a tweaked square grid was essentially the best you could do. According to OpenAI, the model produced an infinite family of constructions that beats that bound by a polynomial factor, which is a structural improvement rather than a numerical nudge. The construction draws on algebraic number theory, which is not where geometers usually look for help.

The trust story matters as much as the math. OpenAI's previous "we solved Erdős problems" claim, from October 2025, fell apart when reviewers found the model had just rediscovered known results. This time the company published the result only after Noga Alon, Melanie Wood, and Thomas Bloom checked the proof, with Bloom having been the harshest critic of the earlier mistake. OpenAI also notes the original output was correct but messy, and that humans tightened it for the writeup.

The result came from a general-purpose model rather than a math-specific system or a scaffolded proof search. That framing is what the company is selling: not a specialised tool, but a reasoning model strong enough to surprise a field's working assumptions.

Why it matters

If you have been waiting for a defensible answer to "can current models do real math research," this is the cleanest example so far, and it is verified by reviewers who would have called out a bluff. Treat it as a single data point, not a trend, but it shifts the burden of proof in a debate that until now leaned mostly on hand-waving.