OpenAI Disproves the Unit Distance Conjecture
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An OpenAI internal reasoning model produced a counterexample to Erdős's 1946 unit distance conjecture, the first time an AI system has autonomously resolved a prominent open problem in mathematics. For nearly 80 years, mathematicians believed square grids were essentially optimal for placing n points to maximize unit-distance pairs. The new construction beats grids using an infinite unramified tower of totally real number fields with 3-power Galois groups, producing n-point sets with more than n^(1.014) unit distances. A human-verified companion paper was prepared by nine external mathematicians including Noga Alon, Tim Gowers, and Melanie Matchett Wood.
First autonomous resolution of a central open problem: The proof was discovered without human guidance on the mathematical substance. External mathematicians then verified, digested, and rewrote it for publication.
Deep number theory, not search: The construction uses Golod-Shafarevich theory and Galois cohomology rather than enumerative or brute-force search. The model selected and combined tools from algebraic number theory that are far from the typical training distribution for math problem solving.
Verified by the field's experts: Nine external mathematicians, including some of the conjecture's harshest prior critics, verified the proof. The companion paper presents a digested, human-verified version of the AI-generated counterexample.
Why it matters: A model autonomously closing an 80-year-old open problem changes the prior on what frontier reasoning systems can contribute to research mathematics. It also raises practical questions about credit, verification, and how the math community will integrate AI-discovered results.
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