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Counterexamples to Grad's conjecture

This paper constructs smooth, non-trivial one-parameter families of magnetohydrostatic equilibria on solid tori for every sufficiently large $N$ that possess only $C_N$ rotational symmetry, thereby providing counterexamples to Grad's conjecture regarding the necessity of plane-reflection, axial, or helical symmetries, with results further validated by a Lean 4 formalization.

Original authors: Javier Gómez-Serrano, Lukas Liehr, Mitchell A. Taylor

Published 2026-09-22

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CC BY 4.0

Original authors: Javier Gómez-Serrano, Lukas Liehr, Mitchell A. Taylor

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the hidden world inside a nuclear fusion reactor, a superheated gas called plasma is held in place not by walls, but by invisible magnetic fields. This gas is so hot that it would melt any container, so scientists must shape these magnetic fields into a perfect, invisible cage to keep the energy contained. The goal is to create a stable, self-sustaining state where the magnetic forces and the pressure of the gas balance each other perfectly. For decades, physicists believed that for such a stable state to exist, the magnetic cage had to possess a high degree of symmetry. They thought the cage had to look the same if you rotated it, flipped it like a pancake, or twisted it like a screw. This idea, known as Grad's conjecture, suggested that if you tried to build a magnetic cage that lacked these symmetries, the plasma would inevitably become unstable or the pressure would flatten out until the magnetic field vanished, ruining the confinement.

A team of researchers has now proven that this long-held belief is incorrect. They have mathematically constructed a new kind of stable magnetic cage that does not look the same when flipped or twisted in the ways previously thought necessary. Instead of a perfectly smooth, symmetrical shape, their solution is a magnetic bottle that repeats a specific pattern only a certain number of times as you go around the circle, like a gear with a specific number of teeth. This structure is stable, smooth, and holds the plasma in place without collapsing, even though it lacks the continuous symmetries that experts thought were required. The researchers did not just simulate this on a computer; they provided a rigorous mathematical proof that such a state must exist, and they even had a computer program verify every step of their logic to ensure there were no errors.

The story begins with the magnetohydrostatic equations, a set of rules that describe how a magnetic field and a gas pressure interact when nothing is pushing them from the outside. Imagine a balloon filled with gas; the gas pushes out, and the skin of the balloon pushes back. In a fusion reactor, the "skin" is the magnetic field, and the "gas" is the plasma. The researchers were looking for a solution where the magnetic field lines wrap around a central axis, creating a series of nested, doughnut-shaped layers of pressure. For a long time, it was assumed that for these layers to stay smooth and stable, the entire system had to look identical no matter how you rotated it or reflected it in a mirror. If the system lacked this perfect symmetry, the math suggested the pressure would have to become flat and the magnetic field would die out, making the reactor useless.

The team set out to find a counterexample to this rule. They focused on a specific type of magnetic cage shaped like a solid torus, or a doughnut. They asked a simple question: can you build a stable magnetic cage that repeats a pattern exactly $N$ times around the circle, where $N$ is a large number, but has no other symmetries? They wanted to see if the magnetic field could vanish only on a single, perfect circle in the center, while the pressure levels remained as smooth, nested doughnuts all the way out to the edge. Crucially, they wanted to ensure that the only symmetries left were these specific rotations, with no mirror flips or continuous twisting allowed.

To solve this, the researchers used a powerful mathematical technique called the Nash-Moser theorem. This method allows mathematicians to find solutions to complex, non-linear problems by starting with a simple, known solution and then carefully bending and twisting it into a new shape without breaking the rules. They began with a straight, simple magnetic field that was easy to understand. Then, they slowly curved this straight field into a circle, creating the doughnut shape. As they did this, they had to adjust the magnetic field and pressure at every step to keep the balance perfect. The challenge was that as they bent the field, the math became incredibly sensitive, and small errors could cause the whole structure to collapse.

The researchers discovered that by carefully choosing how the cross-section of the magnetic cage twisted as it went around the circle, they could maintain stability. They found that the cross-sections of the doughnut could be elliptical, like a flattened circle, and these ellipses could rotate as they moved along the path. By controlling the speed and direction of this rotation, they could create a magnetic field that was stable and smooth. They proved that for any sufficiently large number of repetitions, $N$ , they could construct such a solution. The resulting magnetic field vanishes exactly on a central circle, and the pressure levels form a perfect stack of nested tori.

Perhaps most importantly, they showed that these solutions are not isolated accidents. They exist in a continuous family, meaning you can smoothly change the shape of the magnetic cage by adjusting a single parameter, and it will remain stable. This proves that these exotic, asymmetric states are not just theoretical curiosities but a genuine part of the physical landscape of plasma physics. The researchers also demonstrated that these solutions have no other symmetries. They are not symmetric under mirror reflection, nor do they have any continuous rotational symmetry other than the specific $N$ -fold rotation. This directly contradicts the previous belief that such symmetries were necessary for stability.

To ensure their result was beyond doubt, the team translated their entire mathematical proof into a computer-readable language called Lean. This allowed a computer to check every single logical step of their argument, from the basic definitions to the final conclusion. The computer verified that the proof was correct, leaving no room for human error or oversight. This level of verification is rare in mathematics and physics, and it gives the result a unique kind of certainty. The work shows that the universe of possible magnetic cages is richer and more diverse than previously thought. It opens the door to designing fusion reactors that do not need to be perfectly symmetrical, potentially allowing for more flexible and efficient designs.

The implications of this discovery extend beyond just fusion energy. It changes our fundamental understanding of how fluids and magnetic fields can interact in three-dimensional space. For years, physicists had to assume that symmetry was a requirement for stability, which limited the types of magnetic cages they could design. Now, they know that nature allows for stable, asymmetric configurations. This means that future fusion reactors might be able to use shapes that were previously dismissed as impossible. The researchers have shown that the rules of the game are more permissive than anyone realized, and that stability can be achieved even when the magnetic field breaks the traditional patterns of symmetry.

In the end, the paper provides a definitive answer to a question that had stood for decades. It shows that the magnetic fields holding a fusion reactor together do not need to be perfectly symmetrical to work. They can be twisted, rotated, and shaped in complex ways, as long as they follow a specific repeating pattern. The researchers have not only found these solutions but have proven they exist and are stable. This is a significant step forward in the quest to harness the power of the stars, offering new possibilities for how we might one day build a machine that creates clean, limitless energy. The work stands as a testament to the power of rigorous mathematics and computer verification in uncovering the hidden possibilities of the physical world.

Technical Summary: Counterexamples to Grad's Conjecture

Problem Statement
The paper addresses the existence of smooth, three-dimensional magnetohydrostatic (MHS) equilibria in solid tori that lack continuous Euclidean symmetries. The MHS equations describe the static equilibrium of an ideal, perfectly conducting plasma where the Lorentz force balances the pressure gradient ( $B \times \text{curl } B + \nabla P = 0$ ), the magnetic field is solenoidal ( $\text{div } B = 0$ ), and the boundary is a magnetic surface ( $B \cdot n = 0$ ).

The central question is Grad's conjecture, as precisely formulated by Constantin, Drivas, and Ginsberg. The conjecture posits that any smooth, unforced MHS equilibrium on a toroidal or cylindrical domain, possessing nested toroidal pressure surfaces and a non-vanishing magnetic field away from a central axis, must exhibit one of three specific symmetries: plane-reflection, axial, or helical symmetry. Furthermore, such equilibria are conjectured to be isolated in the moduli space of configurations, meaning they do not belong to a smooth one-parameter family of distinct solutions (excluding trivial rescalings and translations).

Methodology
The authors construct a counterexample by producing a smooth one-parameter family of solutions for every sufficiently large integer $N$ (representing the number of periods in the torus). The methodology relies on a rigorous application of the Nash-Moser inverse function theorem to a nonlinear system derived from the MHS equations.

Coordinate Setup and Reduction: The problem is reduced to a single period cell of a torus with a curvature parameter $\varepsilon$ . The authors utilize flux coordinates where the pressure $P$ and magnetic field lines are built into the coordinate system. The domain is represented as a solid torus $Q = D^2 \times T$ .
Linearization and Inversion: A significant portion of the work is dedicated to inverting the linearized operator around a family of exact solutions at zero curvature (straight axis). These zero-curvature solutions, based on the work of Salat and Kaiser, possess elliptical cross-sections with varying orientation.
- The linearized equations are decomposed into "minus" (center and affine) and "plus" (elliptic) blocks.
- The authors construct a uniform inverse for the linearized operator across a range of cell frequencies. This involves splitting the domain into a "cap" near the magnetic axis (where equations are of Bessel type) and an "annulus" (where equations behave as radial evolutions).
- To handle the coupling between different frequencies and the loss of derivatives inherent in the nonlinear problem, the authors employ analytic weights that depend on the cell frequency and radial distance. These weights provide necessary damping on the annulus to control growth.
- The inversion process utilizes a Neumann series for high frequencies and a Schur complement to couple high and low frequencies, ensuring the inverse satisfies "tame" estimates required for the Nash-Moser scheme.
Symmetry Breaking: The construction explicitly breaks all continuous symmetries and discrete reflection symmetries. The solutions are designed to have an exact symmetry group of the cyclic group $C_N$ (rotations by $2\pi/N$ ). This is achieved by carefully selecting the orientation of the elliptical cross-sections along the axis using a specific function $\alpha_\lambda(\zeta)$ containing both even ( $\cos \zeta$ ) and odd ( $\lambda \sin 2\zeta$ ) harmonics. The odd harmonic prevents reflection symmetry, while the specific periodicity prevents continuous rotational symmetry.
Global Injectivity and Closure: After solving the equations on the cell for small curvature $\varepsilon$ , the authors set $\varepsilon = 1/N$ to close the cell into a torus. They prove global injectivity of the resulting map by analyzing the Jacobian and using tubular coordinates to rule out self-intersections, ensuring the solution is a valid embedding of a solid torus.

Key Contributions and Results
The main result, Theorem 1.1, establishes the existence of smooth equilibria that refute Grad's conjecture. Specifically, for every sufficiently large integer $N$ and a parameter $\lambda$ in a compact interval, there exists a smooth embedding $X_{N,\lambda}$ and smooth fields $B_{N,\lambda}, P_{N,\lambda}$ satisfying the MHS equations with the following properties:

Nested Pressure Surfaces: The regular pressure levels are nested embedded tori that foliate the domain, collapsing onto a round magnetic axis $\Gamma_N$ .
Magnetic Axis: The magnetic field vanishes exactly on the axis $\Gamma_N$ and nowhere else.
Exact Symmetry Group: The extended Euclidean stabilizer (allowing for isometries that reverse the sign of the field) is exactly the cyclic group $C_N$ of rotations. The solutions possess no plane-reflection, axial, or helical symmetries.
Non-Isolation: The solutions form a smooth one-parameter family depending on $\lambda$ . In the moduli space of embedded configurations (quotiented by reparametrizations, similarities, and field rescalings), this family is non-trivial and injective. Consequently, the equilibria are not isolated, directly contradicting the non-isolation hypothesis of Grad's conjecture.

Significance and Claims
The paper claims to provide the first construction of smooth, three-dimensional MHS equilibria with nested pressure surfaces that possess no continuous Euclidean symmetry and no reflection symmetry, even when allowing for field reversal.

Refutation of Grad's Conjecture: The results demonstrate that the conjecture's assertion that such equilibria must possess specific symmetries or be isolated is false. The constructed solutions exist in smooth families and exhibit only discrete cyclic symmetry.
Mathematical Rigor: The work provides a complete, rigorous proof using the Nash-Moser theorem, overcoming the technical difficulties of inverting the linearized operator in the presence of small divisors and loss of derivatives.
Formal Verification: A significant contribution is the Lean 4 formalization of the main theorem. The authors have verified the mathematical statements and proofs using the Mathlib library, ensuring the correctness of the complex analytical arguments.
Relation to Previous Work: The construction shares geometric features with the piecewise smooth equilibria of Enciso et al. and the formal expansions of Weitzner, but it achieves full smoothness ( $C^\infty$ ) and exact symmetry breaking without current sheets or forcing terms. The approach is noted to be closer in spirit to counterexamples for Schiffer's conjecture.

The authors emphasize that their examples are "exotic" in the sense that they break the expected symmetry patterns in plasma confinement theory, suggesting that the landscape of MHS equilibria is richer and less constrained by symmetry than previously conjectured.

Technical Summary: Counterexamples to Grad's Conjecture

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