OpenAI says AI proved three major conjectures
OpenAI says its internal models produced proofs in harmonic analysis, complexity theory and quantum field theory, including a refutation of the Exponential Time Hypothesis, and has handed them to outside experts to verify.
OpenAI announced three results it says its internal models produced. It is releasing each one differently, and independent review is still underway.
Mathematics. OpenAI says its models proved Sogge's 1991 local smoothing conjecture in every dimension. If the proof holds, it also settles the Kakeya conjecture and Stein's restriction conjecture in all dimensions. Previous work had proved local smoothing only in two spatial dimensions (Guth, Wang and Zhang, 2019) and Kakeya only in three (Wang and Zahl, 2025). OpenAI says the proof relies on a new auxiliary partial differential equation. The model output arrived on October 18 and was formalized in Lean the next day. Both the natural-language writeup and the Lean proof are on OpenAI's GitHub.
Computer science. OpenAI says its system proved the Exponential Time Hypothesis false, and that the proof was checked in Lean. Hundreds of papers have used ETH to derive conditional hardness results. Russell Impagliazzo and Ramamohan Paturi of UC San Diego and Ryan Williams of MIT lead the group that will verify and explain the result. The proof and its Lean formalization are public.
Physics. An external group is writing up a quantum field theory result before any AI-produced material is released. OpenAI's summary connects it to the existence of four-dimensional Yang–Mills theory, a Millennium Prize problem. No materials are available yet.
OpenAI says it does not want academic credit for the discoveries. It says participants are unpaid, will not sign nondisclosure agreements, and do not speak for the company. OpenAI acknowledges that choosing the first shepherds creates "asymmetry of information." None of the results has been independently confirmed so far.