{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-de-giorgi-conjecture-dimension-eight/","md":"https://postcutoff.com/e/2026-10-06-openai-de-giorgi-conjecture-dimension-eight/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-06-openai-de-giorgi-conjecture-dimension-eight","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims De Giorgi's conjecture in dimension eight","deck":null,"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI: A positive resolution of De Giorgi's conjecture in dimension eight (26 Sept 2026)","url":"https://github.com/openai/math/blob/main/preprints/De-Giorgis-conjecture-in-dimension-eight-September-26-2026/article.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"Long, Zou: On the Allen–Cahn equation (arXiv 2610.28080, builds on OpenAI's framework)","url":"https://arxiv.org/abs/2610.28080","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"formal-conjectures issue #6915: De Giorgi's conjecture holds in dimension at most 8","url":"https://github.com/google-deepmind/formal-conjectures/issues/6915","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"oai-math-revised: community notes on family 375","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/De-Giorgis-conjecture-in-dimension-eight-September-26-2026/AGENT_NOTES.md","type":"discussion","group":"reactions","domain":"github.com"}],"official":3,"filed":"2026-10-28","updated":"2026-10-28","orgs":["OpenAI"],"title":"OpenAI's release claims De Giorgi's 1978 conjecture in its sharp dimension eight, and classifies all stable Allen–Cahn solutions in ℝ⁷ with no energy-growth assumption","summary":"\"A positive resolution of De Giorgi's conjecture in dimension eight\" (OpenAI, 26 Sept 2026, released 6 Oct): Theorem 1.1: if u ∈ C²(ℝ⁸, (−1, 1)) solves Δu = u³ − u with ∂₈u > 0, then u(x) = tanh((e·x − c)/√2); \"Neither energy growth nor minimization is assumed\". Theorem 1.2: every stable solution v: ℝ⁷ → [−1, 1] is a constant well or a planar heteroclinic, \"without an energy-growth assumption\". Prior work: Ghoussoub–Gui (n = 2), Ambrosio–Cabré (n = 3), Savin (n ≤ 8 under pointwise limits ±1), del Pino–Kowalczyk–Wei counterexamples (n ≥ 9). Follow-up: Long and Zou, arXiv 2610.28080, \"Using the framework developed by OpenAI\".","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'"],"key_numbers":[],"tags":["math","pde","allen-cahn","de-giorgi-conjecture","minimal-surfaces","openai-math-release"],"science":{"field":"mathematics","subfield":"nonlinear PDE (Allen–Cahn equation, phase transitions, minimal surfaces)","problem":"De Giorgi's conjecture (1978): monotone bounded entire solutions of Δu = u³ − u in ℝⁿ, n ≤ 8, are one-dimensional; also stable rigidity in ℝ⁷","result":"Claimed proof in the sharp dimension 8 without Savin's limit assumption; classification of stable solutions in ℝ⁷ without energy growth","open_since":"1978","ai_system":["OpenAI internal model (math release)"],"human_role":"Autonomous AI output released by OpenAI","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed preprint in the OpenAI release; no Lean formalization; used as a framework by one follow-up paper"},"status":"pending","shock":""},"body_md":"## What happened\n\nIn 1978 Ennio De Giorgi asked whether the transition layers of the Allen–Cahn equation, a basic model of phase separation, must\nbe flat when they are monotone in one direction, in dimensions up to eight. The question mirrors the Bernstein problem for minimal\nsurfaces. It was proved in dimensions 2 and 3. Savin proved it up to dimension 8 under an extra assumption, and it fails from\ndimension 9. The full case of dimensions 4–8 without extra assumptions stayed open.\n\nThe OpenAI release contains a long manuscript (about 280k characters of text) claiming the sharp case, dimension 8, which implies\nall lower ones. It also classifies stable solutions in dimension 7, a separate known open problem. Long and Zou have already used\nits framework in a follow-up paper.\n\n## Why it matters\n\nIf correct, this closes one of the best-known conjectures in elliptic PDE and phase-transition theory, almost fifty years old.\nOur 6 Oct coverage missed it and it surfaced as a lead on 26 Oct. Status: unrefereed AI output, no formal proof, no public expert\nassessment yet.","disputed":[{"label":"Verification","text":"Unrefereed preprint in the OpenAI release; no Lean formalization; used as a framework by one follow-up paper","sources":[]}],"related":[{"id":"2026-10-26-ai-assisted-results-october-26-listing","url":"https://postcutoff.com/e/2026-10-26-ai-assisted-results-october-26-listing/","date":"2026-10-26","date_precision":"day","short_title":"AI-assisted results in the 26 Oct 2026 sweep","deck":"Claude-written Lean proof of a frame-indifferent rank-one convex function that is not quasiconvex, a sculptor and Claude Code prove Takemura's conjecture, Claude's contribution to deterministic root-finding, and Zenodo records","takeaway":"Monday's arXiv announcement (papers submitted 22–23 Oct) brought four results with their own entries: OpenAI's propeller conjecture reproved by Heilman, Tokareva's conjecture disproved by MiniMax agents, the triangle lower tail with GPT-6 Astra and Claude, and graph-connectivity communication complexity with ChatGPT. This page lists the next tier, from Lean proofs written entirely by Claude to papers where the AI only checked the text.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":22,"official":22,"filed":"2026-10-26","updated":"2026-10-26","orgs":["Anthropic","OpenAI","various researchers"]},{"id":"2026-10-24-oai-math-revised-community-hub","url":"https://postcutoff.com/e/2026-10-24-oai-math-revised-community-hub/","date":"2026-10-24","date_precision":"day","short_title":"oai-math-revised: mathematicians launch a community hub to revise OpenAI's AI-written math manuscripts (guest post on Tao's blog)","deck":null,"takeaway":"This is the first organized attempt by research mathematicians to digest and credit the OpenAI catalogue.","category":"research","category_label":"Research","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":6,"official":5,"filed":"2026-10-25","updated":"2026-10-27","orgs":["OpenAI","mathcommunity"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":79,"official":13,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-28","type":"filed","text":"Created (science agent; paper, openai/math CONTENTS, oai-math-revised notes, arXiv 2610.28080, GitHub issue)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-28","run":null,"sources_read":null,"updated":"2026-10-28","human_review":null,"version":{"date":"2026-10-28"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":"https://postcutoff.com/s/openai-de-giorgi-conjecture"}