# OpenAI Says Its Models Proved the Local Smoothing Conjecture in All Dimensions

By Simon Yoon

Canonical URL: https://www.tokenpost.com/news/technology/41755
Published: 2026-10-27T13:16:59.000Z
Updated: 2026-10-27T13:16:59.000Z
Section: Technology

> The company published a natural-language write-up and Lean formalization Oct. 26 and says the result also resolves the Kakeya and Stein restriction conjectures.

OpenAI says the result applies to the Euclidean wave equation’s local smoothing problem for every n ≥ 2; if valid, it would also settle the Kakeya and Stein restriction conjectures.

The company published a natural-language write-up and a Lean formalization on Oct. 26, presenting the material as raw model output for researchers to examine. OpenAI said the models generated the work Oct. 18 and that it was formalized in Lean the next day. The proof uses what the company described as a new auxiliary partial differential equation.

The formalized result covers dimensions n ≥ 2 and exponents 2 ≤ p < ∞. Its stated Sobolev order is s = max(0, (n−1)/2 − n/p) + ε, for ε > 0, with s = 0 permitted at p = 2.

The conjecture was proposed in 1991 and concerns how wave solutions focus over time. The two-dimensional case was proved in 2019. OpenAI's claim extends the result to every higher spatial dimension; its public materials include a readable paper and the Lean work.