OpenAI claims proofs of the Kakeya conjecture in all dimensions and Yang–Mills existence
In the same release, OpenAI said its models proved Sogge’s local smoothing conjecture in every dimension, implying the Kakeya and Stein restriction conjectures, and an existence proof for four-dimensional Yang–Mills theory and QCD.
Source published · In the Daily Brief of 27 October 2026
Filed under Mathematics
Sogge’s 1991 local smoothing conjecture says that extreme focusing of waves can only be brief. It implies the Kakeya conjecture and Stein’s restriction conjecture. Guth, Wang and Zhang proved it in two spatial dimensions in 2019, and Wang and Zahl settled the three-dimensional Kakeya conjecture in 2025. OpenAI says its models proved local smoothing in every dimension using a new auxiliary partial differential equation, obtained on 18 October and formalised in Lean on 19 October. The write-up and the Lean proof are public.
The physics result is a claimed rigorous construction of interacting four-dimensional gauge theories with fermions, including QCD and a grand unified completion of the Standard Model, building on Balaban’s approach. It includes the existence of pure Yang–Mills theory, part of the Clay Millennium problem. A group led by David Gross, with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, will write up the result before any material is released.