Grad's 1967 conjecture on 3D plasma equilibria falls: two AI-assisted papers give three families of counterexamples
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
- 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
Science result
- Field
- physics / 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 (open since 1967)
- 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).
- 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
- 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
- Why surprising
- A 59-year-old question underlying stellarator design was settled twice in two days, once by solutions an AI model found outright.
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.
Changelog
- 2026-09-30: created (found via a viral X post; missed by earlier runs because the sweep had no arXiv or science-social source)
Related posts (1)
- 'A 59-year-old conjecture in fusion physics just fell' 0ximai @0x_impai · x · 2026-09-24
Viral (~744k views) summary of the two Grad's-conjecture papers, including that two of the three families were found with GPT-6 Astra.
Related events
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- DeepMind and mathematicians use neural networks to find new unstable singularities in fluid equations ★★★
Sources (4)
- paperarXiv:2609.24739 — Counterexamples to Grad's conjecture (Gómez-Serrano, Liehr, Taylor)
- paperarXiv:2609.26742 — Analytic toroidal 3D MHD equilibria and steady Euler flows with invariant surfaces (Landreman)
- codeLean 4 formalization (GitHub: lukasliehr/Grad-Conjecture)
- discussionX: @0x_impai summary thread (~744k views)
id: 2026-09-21-grad-conjecture-counterexamples · updated 2026-09-30 · open in the interactive timeline