Plasma conjecture disproof

"Counterexamples to Grad's conjecture" is a mathematical result by Javier Gómez-Serrano, Lukas Liehr and Mitchell A. Taylor, concerning magnetohydrostatic (MHS) equilibria in plasma physics. The GitHub repository lukasliehr/Grad-Conjecture contains a Lean 4 formalization of the paper's main theorem. The theorem constructs smooth families of embedded solid tori carrying smooth MHS equilibria with only a finite cyclic rotational symmetry, i.e. without continuous symmetry. The fields satisfy B×(∇×B)+∇P=0 and ∇·B=0, with B tangent to the boundary.

The formalized statement says that for any cell length L>0 there is a compact interval of parameters and an integer N0 such that for every N≥N0 the family has a pressure with a single round critical circle of radius NL, a magnetic field vanishing exactly on that circle, and a stabilizer equal to the cyclic group of N rotations. The family gives a continuous injective curve in the moduli space of embedded configurations whose interior members are non-isolated.

The Lean library covers Nash–Moser iteration, smooth parameter dependence, Fourier analysis, analytic weights, Banach calculus, topology and the MHS application. Two showcase files state the result: one imports only Mathlib and uses two sorry placeholders, and the other supplies full proofs. The proved result depends only on the standard axioms propext, Classical.choice and Quot.sound. A YouTube creator reported that the work used a GPT-6 model to find the counterexamples, but this is not confirmed in the provided repository text. This is a research artifact, not a product.