OpenAI models produce proofs on the Kakeya conjecture, ETH and QCD

OpenAI has announced three mathematical, computer science and theoretical physics breakthroughs generated by its internal reasoning models, paired with three different models for academic review across the scientific community.

In harmonic analysis, an OpenAI model produced a proof on 18 October 2026 of Sogge’s 1991 local smoothing conjecture for the Euclidean wave equation in every spatial dimension, using a new auxiliary partial differential equation. Formalised in the Lean theorem prover on 19 October 2026 and published directly to OpenAI’s openai/math GitHub repository, the result implies both the Kakeya conjecture and Stein’s restriction conjecture in all dimensions, generalising the 2025 three-dimensional Kakeya proof by Wang and Zahl.

In computational complexity, OpenAI’s models disproved the Exponential Time Hypothesis (ETH), a foundational conjecture regarding Boolean satisfiability (SAT) hardness, with the refutation verified in Lean on 17 October 2026. OpenAI published the natural-language writeup and Lean proof (lean/docs/378.md), formed a working group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams, and committed to funding an open academic conference. In constructive quantum field theory, the models produced existence proofs in four dimensions for pure Yang-Mills theory (part of the Clay Millennium Prize problem) on 9 September 2026 and for interacting quantum chromodynamics (QCD) and a grand unified completion of the Standard Model on 18 September 2026, building on Balaban’s renormalisation approach. A physics working group led by Nobel laureate David Gross alongside Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov is writing up those results before public release.