An internal OpenAI reasoning model has produced a proof disproving Paul Erdős's unit distance conjecture, an open problem in combinatorial geometry that has resisted attack since 1946. OpenAI announced the result this week on its company blog, alongside a companion paper from nine external mathematicians who verified the argument.
The conjecture sounds almost trivial. Place n points on a plane. How many pairs can sit exactly one unit apart? Erdős suggested in 1946 that the answer grows barely faster than linearly in n, and the rescaled square grid was widely believed to be essentially optimal. An OpenAI model found an infinite family of point arrangements that do measurably better, giving at least n^(1+δ) unit-distance pairs for some fixed positive δ. Princeton's Will Sawin later pinned down a value of roughly 0.014 in a refinement now circulating alongside the result.
Not a calculator, not a chess engine
What makes this awkward to dismiss is the method. The model didn't grind through cases. It reached for algebraic number theory, specifically infinite class field towers and Golod-Shafarevich theory, to build the construction. The technical paper traces the argument from a problem most undergraduates can state to machinery most working mathematicians have never touched.
This wasn't a system tuned for combinatorial geometry, or scaffolded to search through proof strategies, OpenAI says. It's a general-purpose reasoning model. It got the problem statement and produced a proof. That's the claim.
What the mathematicians actually said
The companion paper is more interesting than the announcement. Nine mathematicians including Fields medalist Tim Gowers, Noga Alon, Thomas Bloom, Daniel Litt, and Melanie Matchett Wood wrote up a digested version of the proof along with reflections.
"If a human had written the paper and submitted it to the Annals of Mathematics and I had been asked for a quick opinion, I would have recommended acceptance without any hesitation."
That's Gowers, in the same paper. Coming from a Fields medalist about an AI-generated proof, it's the kind of endorsement hard to walk back. Arul Shankar went further, writing that current models "go beyond just helpers to human mathematicians" and can produce original ingenious ideas. Bloom's involvement matters too: he was one of the mathematicians who publicly demolished an earlier overhyped OpenAI math claim. Now he's vouching.
The reliability footnote
OpenAI also ran the model multiple times with different settings to show the result wasn't a fluke. The proof rate climbed with more reasoning compute, reaching roughly 48% of runs at the high end. That's a strong signal it isn't lottery-ticket math, but it also means more than half the attempts failed.
What OpenAI didn't release: the model name, the parameter count, or any way to reproduce the work outside the company. The proof is checkable and the construction explicit. The system that produced them stays a black box.
Bloom has indicated the number theory connections may unlock other long-stuck problems in discrete geometry. Sawin's refinement is the first follow-up; further sharpenings are likely in coming weeks.
