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The claimed proof would also settle the Kakeya and Stein restriction conjectures, and it comes with a machine-checked Lean formalization
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.
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The Kakeya conjecture asks whether a set in Euclidean space containing a unit line segment in every direction must possess full Hausdorff dimension, even if its Lebesgue measure is zero.
Stein’s restriction conjecture deals with how functions behave when their frequency information is restricted to curved surfaces. It sits at the heart of modern Fourier analysis.
These problems are linked. OpenAI’s position is that proving local smoothing in every dimension implies both of the others in every dimension too.
Prior progress came piece by piece. The two-dimensional case of local smoothing was settled in 2019, and the three-dimensional Kakeya conjecture was resolved in 2025.
Part of a much larger release #
The local smoothing proof did not arrive alone. OpenAI says it is part of a broader release that includes over 700 AI-generated mathematical preprints.
What this means #
The most important detail may be the Lean formalization. A machine-checked proof changes the conversation: the question shifts from whether the AI made an error to whether the formal statement in Lean matches the conjecture mathematicians actually care about.
Formalization only guarantees that the stated theorem follows from its assumptions. Experts will want to confirm the formal statement captures Sogge’s conjecture faithfully, with no quiet simplifications.
Disclosure: This article was edited by Diego Almada Lopez. For more information on how we create and review content, see our