De Giorgi's conjecture in dimension eight

What OpenAI claims

Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire C2 solution u:ℝ⁸ → (-1,1) of Δ u=u³-u that is strictly increasing in one direction depends on only one linear coordinate. A stronger theorem classifies all stable entire solutions v:ℝ⁷ → [-1,1] as constant wells or planar transitions, without an energy-growth assumption.

Paper

1. A positive resolution of De Giorgi's conjecture in dimension eight

   26 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We resolve De Giorgi's conjecture positively in dimension eight: every entire C2 solution u:ℝ⁸ → (-1,1) of Δ u=u³-u with an everywhere positive directional derivative is a planar heteroclinic. We prove that every stable solution v:ℝ⁷ → [-1,1] of this equation is a constant well or a planar heteroclinic, without an energy-growth assumption.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.