OpenAI claims De Giorgi’s conjecture in dimension eight
Awaiting review
Importance: major (4 of 5)The takeaway
A manuscript in OpenAI’s 6 Oct 2026 math release (family 375, dated 26 Sept) claims to complete De Giorgi’s conjecture. Every bounded entire solution of the Allen–Cahn equation Δu = u³ − u in ℝ⁸ that is strictly increasing in one direction is a one-dimensional tanh profile. Dimensions 2 and 3 were known, as was Savin’s result up to dimension 8 under an extra limit assumption, and counterexamples exist from dimension 9. The paper also classifies stable solutions in ℝ⁷ without energy bounds. A Chinese group has already built on it. No expert assessment and no Lean proof are known as of 28 Oct.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 28 October 2026
Your AI and this story
- GPT-6 Astra159 days after its cutoff
- Claude Opus 5.598 days after its cutoff
- Gemini 4 Argon189 days after its cutoff
- Grok 4.7128 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 98 days before it.
Key facts
- Claim (abstract, verbatim): ‘We resolve De Giorgi’s conjecture positively in dimension eight: every entire C² solution u : ℝ⁸ → (−1, 1) of Δu = 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.’
- Why dimension eight is the endpoint: del Pino, Kowalczyk and Wei built non-planar monotone solutions in every dimension ≥ 9 (from non-affine entire minimal graphs, which first exist in ambient dimension 9). Since lower-dimensional solutions extend trivially, dimension 8 implies 4–7, so the claim would settle the conjecture in every dimension where it can hold
- Prior state: Ghoussoub–Gui (1998, n = 2), Ambrosio–Cabré (2000, n = 3), Savin (2009, n ≤ 8 assuming lim_{t→±∞} u(x′, t) = ±1); stable-solution rigidity in ℝ⁷ without growth assumptions was also open (a 6 Oct 2026 preprint by Liu–Luo–Wang–Wei–Wei–Wu needs O(R^{N−1}) energy growth)
- Follow-up: Zhouyu Long and Wenming Zou (arXiv 2610.28080, 20 Oct): ‘Using the framework developed by OpenAI, we prove that finite Morse index implies O(R^{n−1}) energy growth for bounded entire solutions … in dimensions 4 ≤ n ≤ 9’, giving planar rigidity for 4 ≤ n ≤ 7
- Formal status: no Lean formalization in the release; google-deepmind/formal-conjectures issue #6915 (opened 7 Oct, still open on 28 Oct) proposes marking DeGiorgi_le_eight and DeGiorgi_four…_eight solved on the basis of Theorem 1.1, with an AI-assisted statement comparison
- Reactions: none found from specialists (Savin, del Pino, Wei, Wang) on 28 Oct; community notes (oai-math-revised) list ‘Comments and discussion: None found’
What happened
In 1978 Ennio De Giorgi asked whether the transition layers of the Allen–Cahn equation, a basic model of phase separation, must be flat when they are monotone in one direction, in dimensions up to eight. The question mirrors the Bernstein problem for minimal surfaces. It was proved in dimensions 2 and 3. Savin proved it up to dimension 8 under an extra assumption, and it fails from dimension 9. The full case of dimensions 4–8 without extra assumptions stayed open.
The OpenAI release contains a long manuscript (about 280k characters of text) claiming the sharp case, dimension 8, which implies all lower ones. It also classifies stable solutions in dimension 7, a separate known open problem. Long and Zou have already used its framework in a follow-up paper.
Why it matters
If correct, this closes one of the best-known conjectures in elliptic PDE and phase-transition theory, almost fifty years old. Our 6 Oct coverage missed it and it surfaced as a lead on 26 Oct. Status: unrefereed AI output, no formal proof, no public expert assessment yet.
What is disputed or not yet verified
| Verification | Unrefereed preprint in the OpenAI release; no Lean formalization; used as a framework by one follow-up paper |
|---|
Sources
4 sources from 2 sites. Numbers match the chips in the text.
4 sources: 3 primary, 1 reaction
Primary
- OpenAI: A positive resolution of De Giorgi’s conjecture in dimension eight (26 Sept 2026)github.com, paper
- Long, Zou: On the Allen–Cahn equation (arXiv 2610.28080, builds on OpenAI’s framework)arxiv.org, paper
- formal-conjectures issue #6915: De Giorgi’s conjecture holds in dimension at most 8github.com, code
Reactions
- oai-math-revised: community notes on family 375github.com, discussion
Changes
- Filed (science agent; paper, openai/math CONTENTS, oai-math-revised notes, arXiv 2610.28080, GitHub issue)