OpenAI Says Its Models Produced Proofs Challenging Math and Computing Theories

OpenAI says its models have produced mathematical proofs that challenge established theories in harmonic analysis and computer science, and the work also includes a claimed existence proof for quantum field theories, including quantum chromodynamics. Researchers will now examine the results, verify their implications and explain their significance to the wider scientific community. The company has adopted different release strategies: it has published the mathematical and computing proofs, while an external physics group will prepare its own account of the research.

One proof addresses the local smoothing conjecture, a problem in harmonic analysis proposed by Christopher Sogge in 1991. OpenAI says its models proved the conjecture across every spatial dimension, and that the result also resolves the Kakeya conjecture and Stein’s restriction conjecture. OpenAI generated its new proof on Oct. 18, 2026, and formalized it in Lean the following day. The company has published both the natural-language proof and its formal counterpart on GitHub.

OpenAI also says its models disproved the Exponential Time Hypothesis, a longstanding conjecture in computational complexity. The models produced the disproof on Oct. 16, 2026, before formalizing it in Lean the following day. However, refuting ETH would not automatically deliver faster algorithms for existing computing problems.

Russell Impagliazzo and Ramamohan Paturi of UC San Diego will lead the review alongside Ryan Williams of MIT. OpenAI has agreed to fund a conference examining the claim and released its proof materials for independent evaluation.

The physics work concerns the mathematical foundations of quantum field theory. OpenAI says its models produced existence proofs for interacting theories involving gauge fields and fermions in four dimensions. Physicist David Gross will lead an external group that includes Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov. OpenAI says all three collaborations will operate independently, with participants free to publish their findings.