An OpenAI model has disproved a central conjecture in discrete geometry

An internal OpenAI model has successfully solved the planar unit distance problem, marking the first time a prominent open problem central to a subfield of mathematics has been solved autonomously by artificial intelligence. First posed by Paul Erdős in 1946, the unit distance problem asks how many pairs of points can be at an exact distance apart when placed in the plane. For decades, mathematicians believed that square grid constructions were essentially optimal for maximizing these pairs, leading to a widely accepted conjecture regarding an upper bound. The OpenAI model disproved this longstanding belief by providing an infinite family of examples yielding a polynomial improvement. The proof was produced by a new general-purpose reasoning model rather than a system trained specifically for mathematics or targeted at this particular problem. A group of external mathematicians checked the proof and authored a companion paper detailing the background. According to the text, the proof makes use of unexpected ideas from algebraic number theory, such as infinite class field towers and Golod-Shafarevich theory, applying concepts previously studied in number fields to an elementary geometric question. Fields medalist Tim Gowers called the result a milestone in AI mathematics, while number theorist Arul Shankar noted that current AI models are capable of generating original, ingenious ideas. This breakthrough illustrates how advanced reasoning systems can connect ideas across distant areas of knowledge and assist human researchers with complex problems.