OpenAI Model Disproves a Central Conjecture in Discrete Geometry

On the unit distance problem, OpenAI's general-purpose reasoning model produced a proof that breaks a long-standing conjecture. The problem, which asks for the maximum number of point pairs at distance 1 among n points in the plane, is a representative unsolved problem in discrete geometry posed by Erdős in 1946, and for a long time square-grid-like constructions were considered essentially optimal.

The new result presents a construction that, for infinitely many n, produces unit-distance pairs on the order of n^{1+δ}, disproving the existing n^{1+o(1)} conjecture. OpenAI did not provide an explicit δ in the original proof, but a follow-up improvement by Will Sawin showed that δ = 0.014 is achievable.

The key insight comes not from geometry but from algebraic number theory. The proof goes beyond a simple extension of Gaussian integers, drawing on tools such as infinite class field towers and Golod–Shafarevich theory to show the existence of the required number fields. Outside mathematicians have reviewed it, and a companion paper laying out the background and significance was also released.

This case shows that even a general-purpose reasoning model, not a math-specialized system, can produce original constructions and verifiable proofs on long-standing open problems. At the same time, it further fuels discussion of how AI might change problem selection, exploration, and verification in mathematical research.