De Giorgi’s conjecture for the Allen–Cahn equation in dimensions up to eight

Claimed by OpenAI. Claims every entire $C^2$ solution of $\Delta u=u^3-u$ on $\mathbb{R}^8$, valued in $(-1,1)$ with a strictly positive directional derivative, is a one-dimensional tanh profile. It also classifies stable $[-1,1]$-valued solutions on $\mathbb{R}^7$ as constant wells or planar transitions. No energy-growth or directional-limit assumption is imposed.

Scope relative to this problem: The source addresses the eight-dimensional branch with $\Delta u=u^3-u$, equivalent to the displayed Allen-Cahn equation, for entire $C^2$ solutions valued in $(-1,1)$ with a strictly positive directional derivative. It claims a one-dimensional tanh profile, hence every nonempty level set is a hyperplane; levels outside the profile range are empty. The separate stable-solution classification in $\mathbb{R}^7$ is an additional result. No energy-growth or directional-limit assumption is added, and no higher-than-eight-dimensional classification is claimed.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/De-Giorgis-conjecture-in-dimension-eight-September-26-2026/article.pdf