A paper submitted to arXiv on October 5, 2026, presents two exact, smooth, three-dimensional ideal magnetohydrodynamic (MHD) equilibrium families that break stellarator symmetry. The authors say Gemini 4 agents helped propose the extensions. The solutions are intended as benchmarks for testing MHD equilibrium codes, not as a demonstration of fusion power. The paper is by Opal Issan, Jonathan Citrin, Matt Landreman and Brendan Tracey.
The work concerns Grad’s 1967 conjecture about symmetry in smooth toroidal equilibria with nonconstant pressure. It extends two earlier solution families associated with Matt Landreman; both earlier families had stellarator symmetry.
Two exact families broaden the solution set
An ideal MHD equilibrium describes a static balance of plasma pressure and magnetic forces, with a divergence-free magnetic field. The new solutions have nested toroidal flux surfaces and nonconstant pressure. Both are non-axisymmetric and break stellarator symmetry, a specific discrete reflection symmetry.
The two families have different rotational-transform properties. One has a constant rotational transform, ι = 2. The other has a sheared profile, with ι′(ψ) ≠ 0. In each case, setting the newly introduced symmetry-breaking parameter to zero recovers the corresponding earlier stellarator-symmetric solution.
Gemini 4 proposed extensions; the authors report checking them
The authors say Gemini 4 operated as a small team of agents in Google Antigravity. The agents received Landreman’s earlier work and its open questions, then proposed generalized solutions. The authors report that they independently re-derived and checked every derivation, used symbolic and numerical verification code, and conducted adversarial review. They take responsibility for the paper.
Why the solutions matter for fusion research
The exact solutions give researchers reference cases for checking three-dimensional MHD equilibrium solvers, including in solution spaces that do not impose stellarator symmetry. A solver’s results can be compared with these closed-form solutions.
That is a mathematical tool for research and code testing. The paper does not report a reactor demonstration or electricity generation.
