OpenAI's general-purpose reasoning model disproves Erdos unit distance conjecture

The disproof is a construction, not a heuristic argument. The model produced an infinite family of n-point sets in the plane whose unit-distance pair count grows like n^(1+delta) for some delta strictly greater than zero, beating the long-assumed square-grid scaling of n times a slowly-growing log factor ^[1]. The key move is to leave combinatorics entirely and import machinery from algebraic number theory: the construction rests on infinite class field towers in the style of Golod-Shafarevich, with the companion paper citing Ellenberg-Venkatesh and Hajir-Maire-Ramakrishna as the lineage of ideas ^[3]. Princeton's Will Sawin then made the gain explicit, pinning delta at roughly 0.014, so the new lower bound sits at about n^1.014 unit-distance pairs ^[7]. The Spencer-Szemeredi-Trotter upper bound of O(n^(4/3)) from 1983 is untouched, meaning the famous gap between lower and upper bounds is now narrower but still wide open ^[4]. One detail circulating on r/math underscores how exotic the object is: the smallest concrete instance of the construction reportedly needs on the order of 10^1,957,151 points, which is why no one will be drawing a picture of it.

The community-wide assumption since 1946 was that the extremal configurations should look essentially like square grids, which top out at roughly n times a small log factor. Harvard's Melanie Matchett Wood, quoted in Scientific American, frames this as a sociological failure as much as a technical one: mathematicians did not spend enough effort 'playing devil's advocate' against the conjecture, because everyone was busy trying to prove it ^[6]. Thomas Bloom, who maintains the canonical erdosproblems.com tracker, agrees that the lesson is about which toolbox to open: 'this shows that there is a lot more that number theoretic constructions have to say about these sorts of questions than we suspected' ^[5]. The model's advantage was not raw intelligence but breadth of search; r/math commenters noted that some of the algebraic ingredients were already known from adjacent work, but no human had bothered to wire them into the unit-distance problem because the conjecture felt settled.

Seven months ago, OpenAI claimed GPT-5 had solved ten Erdos problems; the claim was retracted within days after researchers showed the 'solutions' already existed in the literature, and both Demis Hassabis and Yann LeCun criticized the rollout publicly ^[1]. The May 2026 announcement is structured to avoid that failure mode. Nine external mathematicians, including Fields Medalist Tim Gowers and longtime AI-math skeptic Thomas Bloom, co-authored a 19-page companion paper on arXiv that digests the model's output into a human-readable proof ^[3]. Gowers, who appears in the OpenAI announcement video alongside Lijie Chen, Mark Sellke, Mehtaab Sawhney, and Sebastien Bubeck, told reporters he would have recommended the paper for acceptance at the Annals of Mathematics without hesitation if a human had submitted it ^[5]. University of Toronto's Daniel Litt went further, calling it 'the unique interesting result produced autonomously by AI so far' ^[6]. Gil Kalai compared the verification arc to the 1976 Appel-Haken computer-assisted proof of the Four Color Theorem, suggesting this is the closest historical analogue for how the mathematical community absorbs machine-produced results ^[4].

The r/MachineLearning thread, in contrast to the celebratory r/math reception, pushed back hard on what OpenAI did not disclose: the specific model name, how many candidate proofs were generated before this one was selected, the compute budget, and the prompt strategy. Public reporting confirms only that the proof came from a 'general-purpose reasoning model' whose success scaled with test-time compute, with the raw output running to hundreds of pages before human digestion and the released abridged chain-of-thought still spanning 150-plus pages ^[2]. Gowers himself flagged a methodological caveat that the press largely glossed: counterexamples are easier to imagine emerging from a 'try many things' search than positive proofs would be, so the result should not yet be read as evidence that AI can produce affirmative theorems on demand. There is also a brewing governance question raised on Reddit: mathematicians post preprints to arXiv as a gift to other researchers, not as training fodder for commercial labs that then capture financial value from the resulting results. Add the still-open Spencer-Szemeredi-Trotter ceiling and the absence of any Lean or Coq formalization of the construction, and the picture is less 'AI does math now' and more 'AI broadened the search and humans certified one specific output.'