OpenAI has published a proof in which its latest reasoning model autonomously disproved a nearly 80-year-old problem in combinatorial geometry — the unit-distance problem: given n points on a plane, how many pairs can be exactly distance 1 apart? The prevailing belief that square-grid constructions were optimal has now been overturned by an infinite family of counterexamples generated by the model.

Key details:

  • The proof was independently verified by a group of external mathematicians, with Fields Medalist Timothy Gowers calling it a “milestone for AI in mathematics.”
  • This is the first time a significant open problem central to a branch of mathematics has been autonomously solved by AI — not by a math-specific system, but by a general-purpose reasoning model.
  • The solution bridges elementary geometry with unexpected ideas from algebraic number theory, demonstrating that modern AI can generate “original, ingenious ideas and carry them to completion,” according to the reviewing mathematicians.
  • The result is not a fluke: when re-run with more reasoning-time compute, the model reaches the proof in up to 48% of attempts.

OpenAI blog post · Proof paper (PDF) · Mathematicians’ remarks (PDF)