AI claims to disprove unit-distance conjecture

OpenAI said on May 20 that one of its internal reasoning models had produced a counterexample to the Erdős unit distance conjecture, a central question in discrete geometry first posed in 1946. The company published a research post, a proof document and a companion set of remarks by outside mathematicians. The claim then spread on X this week, including in a May 23 post that pointed readers to code and an AI-run proof attempt rather than to a peer-reviewed journal article. The underlying problem asks how many pairs of points among \(n\) points in the plane can be exactly distance 1 apart. For decades, mathematicians believed the best constructions were essentially “square grid” examples, giving about \(n^{1+o(1)}\) unit distances. OpenAI said its model found an infinite family of examples with a polynomial improvement, which would disprove that conjectural bound if the proof holds up. (openai.com) ### What, exactly, is being claimed here? Paul Erdős’s 1946 question concerns the maximum number of unit-distance pairs determined by \(n\) points in the Euclidean plane. The companion remarks posted to arXiv state a theorem asserting that there is an \(\varepsilon>0\) and a sequence of planar point sets whose number of unit distances is at least \(|P_i|^{1+\varepsilon}\). That is stronger than the long-conjectured \(n^{1+o(1)}\) behavior and therefore amounts to a counterexample to the conjecture. (openai.com) Will Sawin of Princeton separately posted an arXiv paper on May 20 giving an explicit lower bound: more than \(n^{1.014}\) unit-distance pairs for arbitrarily large \(n\). His abstract says that sharpens the recent OpenAI result by replacing an unspecified exponent greater than 1 with a concrete one. ### Did anyone outside OpenAI check the proof? Nine outside mathematicians are listed on the companion remarks: Noga Alon, Thomas Bloom, Tim Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang and Melanie Matchett Wood. (arxiv.org) The remarks describe themselves as a “human-verified” and “human-digested” version of the OpenAI-generated counterexample. OpenAI’s own post says the proof was checked by a group of external mathematicians. (arxiv.org) Tim Gowers is quoted by OpenAI calling the result “a milestone in AI mathematics.” OpenAI also quotes Arul Shankar as saying the paper shows current AI models are capable of “original ingenious ideas” and carrying them through. Those endorsements are part of OpenAI’s release package, not a journal decision. ### Why are mathematicians paying attention to the method? OpenAI said the proof uses ideas from algebraic number theory on an elementary geometric problem. (openai.com) The companion remarks say the argument relies on ideas connected, in retrospect, to Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna, and sketch a construction using algebraic number fields with many small-norm primes. The remarks also quote part of the model’s chain of thought saying, “Number fields deserve a closer look,” while noting that using number fields for counterexamples was not wholly new, but had not been made to work in this setting. ### What is still unresolved? No peer-reviewed journal publication was available as of May 23. The public record instead consists of OpenAI’s announcement, the proof materials it released, the companion remarks on arXiv, and Sawin’s explicit refinement on arXiv. (openai.com) That means the claim has undergone outside reading and public scrutiny, but not the full journal process. The May 23 X post that helped circulate the claim appears to have pointed readers to code and an AI-run proof attempt, but the post itself was not fully retrievable through the web tool. (arxiv.org) A separate GitHub repository indexed this week describes itself as tracking the public OpenAI package for the May 20 announcement, and other GitHub projects have since appeared to explain or audit the result. (openai.com) ### Where does this go next? May 20 is the date on both the OpenAI announcement and Sawin’s arXiv paper, and the companion remarks were posted within days. The next concrete step is broader mathematical vetting of those public documents, including any formal journal submission and further independent exposition by named researchers such as Alon, Gowers and Sawin. (openai.com) (github.com)