OpenAI says one of its unreleased internal reasoning models has disproved a central conjecture in discrete geometry that had stood since 1946 — the first time, the company claims, that a prominent open problem central to a subfield of mathematics has been resolved autonomously by a general-purpose AI rather than a specialized prover.
The target was Paul Erdős's planar unit distance problem: among n points placed in the plane, how many pairs can sit exactly one unit apart? For nearly 80 years the best-known construction was grid-like, yielding roughly n^(1+C/log log n) unit-distance pairs, and the mathematical community largely believed no construction could do fundamentally better. OpenAI's model found an entirely new infinite family of point configurations that beats that ceiling, proving that for infinitely many n you can achieve at least n^(1+δ) pairs for a fixed positive exponent δ. Princeton professor Will Sawin subsequently pinned the constant down to δ ≈ 0.014 — a polynomial improvement, not a marginal one.
Not a theorem prover
The detail that matters for builders is provenance. OpenAI says the proof came from a general-purpose reasoning model, not a math-only system, and not one scaffolded to search proof strategies or targeted at the unit distance problem specifically. The model was given a collection of Erdős problems as a test and produced a proof that resolves the conjecture. Notably, it imported heavy machinery from algebraic number theory — infinite class field towers and Golod–Shafarevich theory, tools built to study factorization in integer extensions — and turned it onto an elementary-sounding question about distances in the plane. That cross-domain transfer is closer to research behavior than retrieval-and-recombination.
Verified by humans, including a Fields Medalist
The result was independently checked by leading external mathematicians, with companion notes from Sawin and from Thomas Bloom, who curates the Erdős problems database. Fields Medalist Tim Gowers weighed in directly: "There is no doubt that the solution to the unit-distance problem is a milestone in AI mathematics." An accompanying paper, "An explicit lower bound for the unit distance problem," lays out the construction.
What it signals for production
This is one result, and human verification was still required to confirm and refine it — the cost and exact compute footprint behind the run remain unconfirmed, despite secondhand speculation about a sub-$1,000 figure. But for teams deploying frontier reasoning models against open-ended R&D, the signal is concrete: a general-purpose model generated a novel, correct, cross-disciplinary proof that human experts had not found in eight decades. That moves the credible use case for these models beyond coding and summarization toward autonomous discovery loops in research-grade settings, where the bottleneck shifts from generation to verification — and where the value of fast, trustworthy human or automated checking goes up accordingly.
