OpenAI · 375-01 · 2026-09-26 · Partial differential equations

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

preprints/De-Giorgis-conjecture-in-dimension-eight-September-26-2026

Family 375. De Giorgi's conjecture in dimension eight.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire $C^{2}$ solution $u : R^{8} \to (- 1, 1)$ of $Δ u = u^{3} - u$ that is strictly increasing in one direction depends on only one linear coordinate. A stronger theorem classifies all stable entire solutions $v : R^{7} \to [- 1, 1]$ as constant wells or planar transitions, without an energy-growth assumption.

We resolve De Giorgi's conjecture positively in dimension eight: every entire C2 solution $u : R^{8} \to (- 1, 1)$ of $Δ u = u^{3} - u$ with an everywhere positive directional derivative is a planar heteroclinic. We prove that every stable solution $v : R^{7} \to [- 1, 1]$ of this equation is a constant well or a planar heteroclinic, without an energy-growth assumption.

Preprint source and Lean scope

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