Grad’s conjecture
For every smooth magnetohydrostatic equilibrium in Euclidean -space, where is an embedded solid torus, the regular level sets of the pressure are nested tori, and the magnetic field vanishes exactly on a round magnetic axis, the equilibrium admits a nontrivial continuous Euclidean symmetry. Equivalently, the domain and fields should possess one of the continuous symmetries described by Grad, such as axial or helical symmetry, possibly together with the relevant reflection symmetry.
References
Primary source
Additional references
- Counterexamples to Grad's conjecture — arXiv — Javier Gómez-Serrano, Lukas Liehr, Mitchell A. Taylor
Progress summary
A September 2026 preprint claims to overturn Grad’s conjecture with certified counterexamples, but the claim has not yet been independently verified.
Grad’s conjecture predicts continuous symmetry for a class of magnetohydrostatic equilibria in an embedded solid-torus setting. Earlier literature described the Euclidean case as unresolved, while some restricted analytic cases were proved.
Known results
- Analytic localizable equilibria are axisymmetric, with a rotationally symmetric domain (2026).
- For generic adapted Riemannian metrics, equilibria with nonconstant pressure have no continuous Killing symmetries; this is not a Euclidean counterexample (Cardona, Duignan, and Perrella, 2024).
September 21, 2026 counterexample claim
On September 21, 2026, Javier Gómez-Serrano, Lukas Liehr, and Mitchell A. Taylor’s Counterexamples to Grad's conjecture reported one-parameter families with symmetry group for every sufficiently large , together with a Lean 4 certification. This would refute the conjecture in the stated setting, but the claim remains unverified.
Current status (as of September 2026): A preprint claims a certified counterexample for sufficiently large in the specified solid-torus setting; independent verification is not recorded, so the conjecture is not yet settled.
Solutions 0
No solutions have been posted yet.