On May 20, 2026, OpenAI published a proof that an internal general-purpose reasoning model had disproved a long-standing conjecture by Paul Erdős about the planar unit-distance problem. The result, verified by a group of nine external mathematicians and made public the same day, marks a rare instance of an AI system producing an original, proof-level mathematical contribution. However, the disproof concerns only a specific conjectured growth rate; the broader problem of determining the exact asymptotic maximum remains unsolved.
The development arrives less than a year after OpenAI walked back a separate claim that GPT-5 had solved several Erdős problems—a misstep that undermined confidence. This time, the company released the proof manuscript and a companion note from independent experts, including some who had criticized the earlier episode, giving the result a credibility it otherwise would lack. For researchers tracking AI’s slow march into rigorous mathematics, the event is both a landmark and a reminder that single successes do not constitute a general reasoning leap.
The unit-distance problem, first posed by Erdős in 1946, asks: for n points placed in the plane, what is the maximum number of pairs exactly one unit apart? Erdős conjectured that the maximum ν(n) grows like n^(1+o(1))—slightly faster than n. The best known upper bound, established in 1984 by Spencer, Szemerédi and Trotter, is O(n^(4/3)). The OpenAI model proved the existence of a fixed δ > 0 such that for infinitely many n, ν(n) ≥ n^(1+δ), contradicting the n^(1+o(1)) conjecture.
The same day, nine mathematicians—including Noga Alon, Thomas Bloom, Fields Medalist Timothy Gowers, and Melanie Matchett Wood—uploaded a companion arXiv note verifying and refining the counterexample. They described the result as a milestone. Separately, Will Sawin of Columbia posted a preprint giving an explicit construction with δ = 0.014, yielding a lower bound of more than n^1.014 unit-distance pairs for arbitrarily large n, an improvement on the original inexplicit δ.
OpenAI said the proof came from a new general-purpose reasoning model that had not been specifically trained for mathematics. The proof manuscript states the initial solution was generated in a completely automated fashion before human review. The reasoning unexpectedly combined algebraic number theory—including infinite class field towers and Golod-Shafarevich theory—with discrete geometry, a connection that mathematicians described as surprising.
The result stands in contrast to October 2025, when OpenAI announced GPT-5 had solved several Erdős problems and then withdrew the claim after it became clear the solutions already existed in the literature. Thomas Bloom, who had publicly criticized that overclaim, co-authored the new companion note, lending the current work additional credibility. Still, the proof has not been accepted by a peer-reviewed journal, and the internal model is not publicly accessible, so the autonomy claim cannot be independently tested.
The disproof leaves the exact growth rate unresolved. The 1984 upper bound of O(n^(4/3)) remains untouched, and the gap between n^1.014 and n^(4/3) is substantial. OpenAI has not disclosed the model’s name or training details, and the external mathematicians note they improved the original AI output; how much human editing shaped the final manuscript is not specified.
For decision-makers allocating resources to AI research, the case offers a clear signal: large language models can assist in generating original proof-level mathematics when embedded in a rigorous human-verification pipeline. It does not yet imply that AI can independently crack open problems or replace trained mathematicians. The advance is real, but its domain remains narrow.