{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-09-21-grad-conjecture-counterexamples/","md":"https://postcutoff.com/e/2026-09-21-grad-conjecture-counterexamples/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":"2026-09-21-grad-conjecture-counterexamples","date":"2026-09-21","date_precision":"day","short_title":"Grad's 1967 conjecture on 3D plasma equilibria falls","deck":"Two AI-assisted papers give three families of counterexamples","takeaway":null,"category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"arXiv:2610.07304 — Analytic non-axisymmetric 3D MHD equilibria without stellarator symmetry (Issan, Citrin, Landreman, Tracey; Gemini 4)","url":"https://arxiv.org/abs/2610.07304","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"arXiv:2609.24739 — Counterexamples to Grad's conjecture (Gómez-Serrano, Liehr, Taylor)","url":"https://arxiv.org/abs/2609.24739","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"arXiv:2609.26742 — Analytic toroidal 3D MHD equilibria and steady Euler flows with invariant surfaces (Landreman)","url":"https://arxiv.org/abs/2609.26742","type":"paper","group":"primary","domain":"arxiv.org"},{"n":4,"title":"Lean 4 formalization (GitHub: lukasliehr/Grad-Conjecture)","url":"https://github.com/lukasliehr/Grad-Conjecture","type":"code","group":"primary","domain":"github.com"},{"n":5,"title":"arXiv 2611.16688: An Analytic Toroidal MHD Equilibrium with Trivial Euclidean Symmetry Group (de Lacy, Sato)","url":"https://arxiv.org/abs/2611.16688","type":"paper","group":"primary","domain":"arxiv.org"},{"n":6,"title":"Japanese explainer on X (@yumamaeda1210, Oct 4)","url":"https://x.com/yumamaeda1210/status/2106672447617646846","type":"discussion","group":"reactions","domain":"x.com"},{"n":7,"title":"X: @0x_impai summary thread (~744k views)","url":"https://x.com/0x_impai/status/2103161612677107942","type":"discussion","group":"reactions","domain":"x.com"}],"official":5,"filed":"2026-09-30","updated":"2026-11-13","orgs":["University of Maryland","OpenAI","Anthropic"],"title":"Grad's 1967 conjecture on 3D plasma equilibria falls: two AI-assisted papers give three families of counterexamples","summary":"Two independent arXiv preprints posted a day apart (21 and 22 Sep 2026) construct smooth magnetohydrostatic plasma equilibria with nested toroidal pressure surfaces and non-constant pressure that have none of the symmetries Harold Grad conjectured were necessary, answering a question central to stellarator fusion theory. Matt Landreman's two explicit families were \"discovered using the artificial intelligence model GPT-6 Astra Pro\"; Gómez-Serrano, Liehr and Taylor's family was worked out with GPT-5.6 Sol, Claude Fable 5 and Claude Opus 5 and verified in Lean 4.","key_facts":["Third construction (arXiv 2611.16688, 12 Nov 2026, Kooper de Lacy and Naoki Sato, U. Tokyo): a real-analytic toroidal MHD equilibrium with nonvanishing B, nested pressure tori, rotational transform 2 and a trivial Euclidean symmetry group (no C_N either), joined to an axisymmetric one by an analytic non-isolated family; it 'constructively disproves one formulation of Grad's conjecture'. AI statement: 'The solution presented in this manuscript was discovered with the assistance of GPT-6 Astra Ultra.' Preprint; Mathematica code on GitHub","Grad's conjecture (1967): smooth 3D equilibria with nested flux surfaces and non-constant pressure require symmetry (plane-reflection, axial or helical)","arXiv:2609.24739 (21 Sep 2026), Gómez-Serrano, Liehr, Taylor: for each large N, a smooth one-parameter family on embedded solid tori whose only Euclidean symmetry group is the cyclic group C_N; magnetic field vanishes exactly on a round magnetic axis","Their LLM disclosure: authors set up the problem and a detailed roadmap in July 2026; GPT-5.6 Sol, Claude Fable 5 and Claude Opus 5 filled in technical details and produced the Lean code under guidance; GPT-6 Astra and Claude Fable 5.1 used in the late Lean stage; authors say the models did not contribute mathematical content of the article itself and they checked everything","Main theorem formally verified in Lean 4 (github.com/lukasliehr/Grad-Conjecture; depends only on standard axioms)","arXiv:2609.26742 (22 Sep 2026, rev. 28 Sep), Matt Landreman (U. Maryland): explicit analytic solutions in elementary functions with exact nested toroidal surfaces; one family with uniform rotational transform, one with sheared transform; also steady incompressible Euler flows","Landreman's acknowledgement: 'These solutions were discovered using the artificial intelligence model GPT-6 Astra Pro, which was also used to draft parts of the manuscript. All equations were confirmed manually by the author.'","An X post summarising both papers (@0x_impai, 24 Sep) reached ~744k views","Reach in Japan: a Japanese explainer post by @yumamaeda1210 (Oct 4, 2026) quoting the GPT-6 Astra Pro acknowledgement reached ~240k views in a day (fxtwitter)"],"key_numbers":[],"tags":["math","physics","fusion","stellarator","mhd","lean","gpt-6-astra"],"science":{"field":"physics","subfield":"plasma physics / magnetohydrodynamics (fusion, stellarators); PDE analysis","problem":"Grad's conjecture on the non-existence of smooth non-symmetric 3D MHD (magnetohydrostatic) equilibria with nested flux surfaces","result":"Three families of smooth, non-symmetric toroidal equilibria with nested pressure surfaces and non-constant pressure: one C_N-symmetric family proved with Nash–Moser methods and certified in Lean (Gómez-Serrano–Liehr–Taylor), and two explicit analytic families (Landreman).","open_since":"1967","ai_system":["GPT-6 Astra Pro","GPT-5.6 Sol","Claude Fable 5","Claude Opus 5","GPT-6 Astra","Claude Fable 5.1"],"human_role":"Landreman: solutions discovered by GPT-6 Astra Pro, equations checked by hand. Gómez-Serrano–Liehr–Taylor: human roadmap, LLMs filled in technical details and wrote the Lean proof under guidance.","verification":{"key":"lean","label":"Lean-verified","text":"Preprints (not yet peer-reviewed); the Gómez-Serrano–Liehr–Taylor main theorem is Lean 4-verified; Landreman's equations are explicit and hand-checked"},"status":"confirmed","shock":"A 59-year-old question underlying stellarator design was settled twice in two days, once by solutions an AI model found outright.","short":"Grad's conjecture"},"body_md":"## What happened\n\nIn 1967 Harold Grad argued that, apart from special cases, smooth three-dimensional plasma equilibria with nested toroidal magnetic surfaces and a pressure gradient cannot exist without a symmetry. The question matters for stellarators, fusion devices that are deliberately non-symmetric and whose design codes assume nested surfaces.\n\nOn 21 September 2026 Javier Gómez-Serrano, Lukas Liehr and Mitchell A. Taylor posted a construction of such equilibria for every sufficiently large N, with only the discrete rotation group C_N as symmetry, and a Lean 4 certification of the main theorem. A day later Matt Landreman posted explicit analytic families written with elementary functions, which he says were discovered with GPT-6 Astra Pro.\n\n## Why it matters\n\nIt removes a long-standing theoretical doubt about smooth non-symmetric equilibria, which is relevant to stellarator design and gives exact test cases for equilibrium codes. It is also a clear case of a frontier model producing a physics result outright, with the author disclosing it.","disputed":[],"related":[{"id":"2026-08-05-sendov-conjecture-proved","url":"https://postcutoff.com/e/2026-08-05-sendov-conjecture-proved/","date":"2026-08-05","date_precision":"day","short_title":"Sendov's 1958 conjecture on polynomial roots proved with GPT-5.6 Pro","deck":"Tao simplifies and formalises it","takeaway":"It is a classic, well-known conjecture closed by AI, with the leading expert on the problem verifying and formalising the result.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":2,"official":1,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]},{"id":"2025-09-17-deepmind-unstable-singularities-fluids","url":"https://postcutoff.com/e/2025-09-17-deepmind-unstable-singularities-fluids/","date":"2025-09-17","date_precision":"day","short_title":"DeepMind and mathematicians use neural networks to find new unstable singularities in fluid equations","deck":null,"takeaway":"It was the groundwork of the AI-plus-computer-assisted-proof approach to fluid blow-up that culminated in the disputed 2026 Navier–Stokes claims.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":1,"filed":"2026-09-29","updated":"2026-10-19","orgs":["Google DeepMind","New York University","Stanford University","Brown University"]}],"people":[],"posts":[{"id":"x-yumamaeda1210-2106672447617646846","title":"“AIが、核融合研究で「人間が60年近く考えてきた問題」の答えを発見した。 使われたのは GPT-6 Astra Pro。 ざっくり説明すると、…”","url":"https://postcutoff.com/p/x-yumamaeda1210-2106672447617646846/"},{"id":"2026-09-24-0x-impai-grad-conjecture","title":"'A 59-year-old conjecture in fusion physics just fell'","url":"https://postcutoff.com/p/2026-09-24-0x-impai-grad-conjecture/"}],"videos":[],"models":[],"changes":[{"date":"2026-11-13","type":"edit","text":"Third counterexample family (de Lacy–Sato, analytic, trivial symmetry group, GPT-6 Astra Ultra) added (science agent)"},{"date":"2026-10-13","type":"edit","text":"Added the Gemini 4 follow-up paper arXiv:2610.07304 (found through Google's Argon release post) with its AI-use disclosure"},{"date":"2026-10-07","type":"edit","text":"Science block: short label for the open-problems table"},{"date":"2026-10-04","type":"edit","text":"Added the viral Japanese explainer post (~240k views)"},{"date":"2026-09-30","type":"filed","text":"Created (found via a viral X post)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-09-30","run":null,"sources_read":null,"updated":"2026-11-13","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":144,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":83,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":174,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":113,"in_training_data":false}],"short_url":"https://postcutoff.com/s/grad-1967-conjecture-3d"}