OpenAI says its AI produced three research proofs

On October 26, OpenAI said its internal AI models had produced three research results in mathematics, computer science and physics. It also said it would make the raw data behind one of them available for independent checking. The company does not seek academic credit for the discoveries.

Math breakthroughs unveiled

The AI proved the local-smoothing conjecture in every dimension, which also settles the Kakeya conjecture and Stein's restriction conjecture. OpenAI formalized the proof in Lean the day after the model produced it and posted the write-up and proof on GitHub. Mathematicians had proved the conjecture in two dimensions in 2019, and the Kakeya conjecture in three dimensions in 2025.

The AI also found a disproof of the Exponential Time Hypothesis, a long-standing conjecture about how hard it is to solve Boolean satisfiability problems. The model's natural-language disproof appeared on October 16 and was checked in Lean the next day. A group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine the claim. OpenAI has agreed to release all raw data and to fund a conference on the work.

The AI produced a rigorous proof that certain quantum field theories exist in four dimensions, including a proof that pure Yang-Mills theory exists. That is part of one of the Clay Mathematics Institute's Millennium Prize problems. The proofs were produced in September, and Lean formalization of the Yang-Mills proof finished on October 16. The physics group, led by David Gross, will write up the result itself and has chosen not to release the AI-produced material for now.

OpenAI has published the write-ups and Lean proofs for the mathematics and computer science results, but not the physics material. The collaborations have no non-disclosure agreements, and participation is unpaid. OpenAI also gave $5 million to the Simons Foundation, which will award it for workshops and conferences on AI-produced results.