Grad’s 1967 conjecture on 3D plasma equilibria falls
Two AI-assisted papers give three families of counterexamples
Result confirmed
Importance: historic (5 of 5)Status
- Claim
Result confirmed
- Our reporting
- High confidence
- Verification
- Lean-verified
- Importance
- Historic (5 of 5)
- Last verified
- 13 November 2026
Your AI and this story
- GPT-6 Astra144 days after its cutoff
- Claude Opus 5.583 days after its cutoff
- Gemini 4 Hydrogen174 days after its cutoff
- Grok 4.7113 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 83 days before it.
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
Show 1 more
- 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)
What happened
In 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.
On 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.
Why it matters
It 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.
Follow-up: a Google team generalizes Landreman’s solutions with Gemini 4 (Oct 5, 2026)
Opal Issan, Jonathan Citrin, Matt Landreman and Brendan Tracey posted arXiv:2610.07304 on 5 October 2026. It adds a symmetry-breaking parameter to both of Landreman’s families, giving non-axisymmetric solutions that also break stellarator symmetry and reduce to his when the parameter is zero. The paper’s “Use of AI tools” section says: “This work was carried out using Gemini 4 running as a small team of agents in Google Antigravity, an agentic development environment. Given Landreman [1] and a list of the open questions it raised, the agents proposed the generalized solutions that remove the stellarator-symmetry assumption. Promising candidates appeared within hours. Gemini 4 Argon was also used to draft parts of the manuscript and write symbolic and numerical verification code. ... Every derivation was independently re-derived and checked symbolically by the authors.” Status: preprint, not peer-reviewed. Google cited the paper in its Oct 13 Gemini 4 Argon release post (see Google releases Gemini 4 Argon broadly).
Sources
7 sources from 3 sites. Numbers match the chips in the text.
7 sources: 5 primary, 2 reactions
Primary
- arXiv:2610.07304 — Analytic non-axisymmetric 3D MHD equilibria without stellarator symmetry (Issan, Citrin, Landreman, Tracey; Gemini 4)arxiv.org, paper
- arXiv:2609.24739 — Counterexamples to Grad’s conjecture (Gómez-Serrano, Liehr, Taylor)arxiv.org, paper
- arXiv:2609.26742 — Analytic toroidal 3D MHD equilibria and steady Euler flows with invariant surfaces (Landreman)arxiv.org, paper
- Lean 4 formalization (GitHub: lukasliehr/Grad-Conjecture)github.com, code
- arXiv 2611.16688: An Analytic Toroidal MHD Equilibrium with Trivial Euclidean Symmetry Group (de Lacy, Sato)arxiv.org, paper
Reactions
- Japanese explainer on X (@yumamaeda1210, Oct 4)x.com, discussion
- X: @0x_impai summary thread (~744k views)x.com, discussion
Changes
- Third counterexample family (de Lacy–Sato, analytic, trivial symmetry group, GPT-6 Astra Ultra) added (science agent)
- Added the Gemini 4 follow-up paper arXiv:2610.07304 (found through Google’s Argon release post) with its AI-use disclosure
- Science block: short label for the open-problems table
- Added the viral Japanese explainer post (~240k views)
- Filed (found via a viral X post)