OpenAI says its models proved the local smoothing conjecture in every dimension

OpenAI says its AI models have proved the local smoothing conjecture in every dimension. The problem has been open since 1991. According to OpenAI, the result also resolves the Kakeya conjecture and Stein’s restriction conjecture across all dimensions. What OpenAI is claiming The announcement came on October 26, 2026. OpenAI says its models produced the proof on October 18, 2026. One day later, on October 19, the proof was formalized in Lean. Lean is a theorem prover, software that checks a proof step by step and refuses to accept any step it cannot verify. OpenAI published the formal proof on GitHub. It also released a natural language explanation, so human mathematicians do not have to read raw Lean code to follow the argument. The key ingredient, per OpenAI, is a newly introduced auxiliary partial differential equation. A partial differential equation describes how a quantity changes across several variables at once, such as space and time. Why these three conjectures matter The local smoothing conjecture was proposed by Christopher Sogge in 1991. It belongs to harmonic analysis and concerns how solutions to wave equations behave when you average them over time. The Kakeya conje...