GPT-6 Astra Solves 59-Year-Old Fusion Conjecture in 20 Minutes

A 59-year-old nuclear fusion puzzle was solved by GPT-6 Astra in just 20 minutes and 34 seconds.

The physicist asking the question simply added an encouraging note at the end of the prompt:

You've solved many open math problems, so I know that with enough persistence, you can do it!

Then, what ChatGPT returned was an exact solution that no one had written in 59 years.

The first family solution found by Astra: concentric magnetic surfaces, with the red and yellow lines representing magnetic field lines.

Recently, this blew up on X.

An AI blogger with a background as a graduate student in nuclear fusion at Kyoto University wrote a long post lamenting that the era of AI and humans jointly discovering new science has truly begun!

The conjecture left behind by plasma physicist Harold Grad in 1967 was此次被攻破的.

At the time, he asserted that without symmetry, there could be no smooth three-dimensional plasma equilibrium.

The stellarator approach in nuclear fusion has been pursued for 59 years.

In late September, two papers, published just one day apart, together overturned it. Of these three families of counterexamples, two were discovered by Astra.

On September 10 at 6:44 a.m., plasma physicist Matt Landreman of the University of Maryland sent a prompt to GPT-6 Astra Pro.

What he wanted was for Astra to design an asymmetric magnetic cage capable of containing plasma.

Among these, the magnetic field must consist of nested magnetic surfaces, like an onion; the divergence must be zero, meaning magnetic field lines cannot appear or disappear out of nowhere; as the magnetic field lines follow the torus, they must also spiral around the cross-section, like a helix, with the number of turns referred to by physicists as the "rotational transform," denoted by ι.

Finally, the entire magnetic cage must satisfy MHD (magnetohydrodynamic) force balance, where the outward pressure of the plasma is everywhere balanced by the force of the magnetic field.

He also specifically noted that ι should preferably not be an integer, but this is not a strict requirement.

He even proposed three ways to create rotational transformations: twisting the magnetic axis, rotating the cross-section as it moves, or passing an electric current along the magnetic axis.

By the last two sentences, he changed his tone, as if encouraging a sleep-deprived PhD student.

When Astra submitted the exam, the first thing she said was:

I found an explicit, exact solution.

The hard conditions listed by Landreman are satisfied by all solutions in this family.

Pressure is not constant—it changes layer by layer from inside to outside; each nested magnetic surface can also be precisely described by formulas.

The central magnetic shaft moves up and down in a non-planar curve; as the doughnut rotates fully, its shape varies at every point and lacks axial symmetry.

The cross-sectional shapes at the three positions of the first family solution are different, and the magnetic axis moves up and down accordingly.

Next, it openly acknowledged its limitation: the rotational transformation ι in this family of solutions is an integer, such that a magnetic field line traveling once around the torus completes exactly two full rotations around the cross-section and returns to its starting point, forming a closed loop.

In its own words, the return map around a loop is the identity map.

Landreman mentioned in the prompt that ι not being an integer is merely a bonus, so this family of solutions passes. However, what he truly desired was for ι to take an irrational value, so that the magnetic field lines never return to their starting point, instead winding endlessly around the magnetic surface.

At 4:05 p.m. the next day, Landreman sent the family back as an attachment, this time asking it to also secure the bonus points.

Among these, ι must be a non-integer, preferably an irrational number, so that the magnetic field lines never close upon themselves. And as always, keep going!

This time, Astra thought for 33 minutes and 37 seconds, and returned the first round with the response: "Didn't get it done."

But it didn't return empty-handed; instead, it proved a limitation.

Behind the first-family solution is a "second-order oscillator" structure, where magnetic field lines move along the x, y, and z directions, resembling three spring oscillators with frequencies of 1, 1, and 2, respectively.

No matter how you adjust the frequency and parameters, these three frequencies will always lock into integer ratios, ι is always an integer, and the magnetic field lines will eventually close.

In other words, trying to patch up the first family of solutions won't work from the root level.

In the same conversation, it then switched to a different construction and presented the second family of solutions.

This time, ι varies with the magnetic surface; physicists call this magnetic shear, and ι on almost every magnetic surface is an irrational number.

Second solution: The magnetic surfaces twist as they move; the red and yellow colors represent magnetic field lines.

Before submitting, it independently verified the calculations by substituting the equation at 1,202 points, checking the error using 64-bit automatic differentiation, and then cross-verified once more using fourth-order finite differences.

At 8:27 PM, Landreman asked two more details.

First, is the magnetic axis a planar curve? Second, is the vacuum rotational transformation non-zero—that is, without relying on the plasma’s own current, will the magnetic field lines still rotate around the cross-section, driven solely by the external coils?

Eleven minutes later, Astra provided the answer.

The magnetic axis lies in a plane and is elliptical; the vacuum rotation transformation is very small but not zero.

In two days, Astra delivered two solutions. The remaining dozen or so days are up to humans.

He used the stellarator equilibrium solver DESC to compute each family of solutions and then verified the equations term by term using SymPy and Mathematica.

On September 22, the paper was posted on arXiv, with the acknowledgments beginning with:

These solutions were discovered using the AI model GPT-6 Astra Pro, and parts of the paper were also drafted by it. All equations have been manually verified by the authors.

He also exported the three dialogues as a PDF and uploaded them, along with the verification script, to the paper's GitHub repository.

It is uncommon to find papers that fully disclose the original prompts.

What fusion needs to do is confine plasma at hundreds of millions of degrees inside a doughnut-shaped magnetic cage.

An ideal magnetic cage consists of nested magnetic surfaces, like an onion. Magnetic field lines circle along their respective surfaces, with pressure decreasing layer by layer from the center outward.

The outward force of the plasma is everywhere balanced by the force of the magnetic field; physicists call this state MHD equilibrium.

Tokamaks (such as ITER and China's EAST) are axisymmetric, meaning their cross-sectional shape remains unchanged along the toroidal loop. This equilibrium is described by a two-dimensional Grad-Shafranov equation, which has been understood for decades.

The "Grad" in the equation's name is the very person who later proposed this conjecture.

Stellarators, such as Germany’s Wendelstein 7-X, take a different approach, using a series of twisted, coiled magnets to generate a magnetic field that is intentionally asymmetric.

The magnets of Wendelstein 7-X, the blue ones, are the twisted, non-planar coils.

The issue lies in this asymmetry.

In 1967, Harold Grad of the Courant Institute at New York University published a paper in Physics of Fluids, writing the following statement:

We believe that it is unlikely that a general class of toroidal equilibria with smooth pressure exists.

In 1985, he took another step forward, asserting that, except for symmetric exceptions, there are no families of smooth solutions depending on parameters. In other words, asymmetric solutions, even if they happen to exist, can only be isolated cases.

The title of the paper is "Theory and Applications of the Nonexistence of Simple Toroidal Plasma Equilibrium."

This is equivalent to telling the stellarator camp that the perfect magnetic cage they’re trying to build may not exist in strict terms.

For decades afterward, stellarators continued to be designed and built using approximate solutions and computer-based numerical calculations. No one could be certain whether the equilibrium achieved was real or merely an artifact of the approximations.

The two closest attempts to the exact answer are:

One type is the paraxial approximation, which yields approximate solutions only near the magnetic axis and becomes invalid farther away from it.

Another class is the proof by physicist Lortz in 1970, which showed that an asymmetric equilibrium with mirror symmetry does exist; however, for this solution, the rotational transformation ι equals zero, meaning the magnetic field lines do not twist around the cross-section and thus cannot confine the plasma. Moreover, Lortz only proved the existence of such a solution without explicitly deriving it.

Landreman and these two clans have filled all these gaps at once.

They hold strictly throughout the entire doughnut, not just near the magnetic axis; ι is non-zero, and the magnetic field lines truly wrap around the cross-section; moreover, it is an exact analytical solution, with formulas that can be written directly on paper.

The magnetic field, magnetic surfaces, and pressure were all expressed using elementary functions—basic expressions such as square roots and trigonometric functions. The entire region is smooth everywhere, with pressure highest at the magnetic axis and decreasing outward (the pressure gradient is zero only at the magnetic axis).

In conclusion of the paper, Landreman himself wrote:

For stellarator fusion, it is reassuring to know that a stable, perfectly magnetic-surface equilibrium with strong asymmetry can, in principle, be determined.

On September 21, the day before Landreman published his paper, another paper was posted online with the straightforward title: "A Counterexample to the Grad Conjecture."

The authors are Javier Gómez-Serrano from Brown University, Mitchell Taylor from the University of Oxford, and Lukas Liehr from Bar-Ilan University in Israel.

Gómez-Serrano previously collaborated with DeepMind to use AI to find singularities in fluid equations and is a well-known figure in the AI mathematics community.

They took a purely mathematical approach, using the hardcore analysis technique of Nash-Moser iteration as their primary tool, and the entire paper spans 147 pages.

The balance they created resembles a flower with N petals. It overlaps with itself every 360/N degrees around the central axis, and has no other symmetries. Mathematically, this is called a symmetry group that is exactly the cyclic group C_N, where N can be any sufficiently large integer.

The balance constructed by the Gómez-Serrano team: nested pressure surfaces, a closed magnetic field line, and the magnetic axis.

Grad's judgment was that a smooth equilibrium must possess at least one of axial symmetry, mirror symmetry, or helical symmetry. These wreaths possess none of these. They only coincide after rotation by a fixed angle, whereas axial symmetry requires invariance under rotation by any angle, so they do not qualify.

More importantly, these balances can be continuously deformed by a single parameter, forming an entire family. Grad's 1985 assertion that "no smooth family of solutions exists" has also been disproven.

In addition, the main result is accompanied by a formal proof in Lean 4.

Lean warehouse homepage, with the formalized main theorem above and an interactive diagram below.

How these 147 pages were written is explained in a dedicated section of the paper.

The project kicked off in July. The three authors first built the framework themselves and created a detailed construction roadmap; the remaining detailed work was handed over to the large model.

GPT-5.6 Sol, Claude Fable 5, and Claude Opus 5 took turns handling tasks not detailed in the roadmap—filling in technical specifics, performing calculations, and identifying mathematical issues. The Lean code was also written by these models under the author’s close supervision.

In the later stages of Lean validation, they switched to the updated GPT-6 Astra and Claude Fable 5.1, and final proofreading was entrusted to them.

Thus, a 59-year-old problem was overturned within a week by two papers, aided by AI.

A month ago, OpenAI announced that it used tens of thousands of agents running for 88 hours to solve the Navier-Stokes equation, one of the Millennium Prize Problems.

This time, all it took was a physicist, a dialogue box, and 20 minutes.

Landreman also made all the prompts publicly available, so anyone can use them.

More than a week after the paper was posted, math PhD Przemek Chojecki, who specializes in AI mathematics, fed these results to GPT-6 Astra and Opus 5.5 and uncovered additional families of counterexamples.

These new solutions have not yet been validated by Lean or peer review. But as soon as one set of solutions was published, someone already used AI to further explore them.

Next, more and more physicists will follow Landreman’s example, handing over long-unsolved problems gathering dust in their drawers directly to AI.

This time, it’s fusion. Next time, it could be materials, astrophysics, or any equation that has been patched together with approximate solutions for decades.

Reference materials:

https://x.com/itsolelehmann/status/2107144201314181183

This article is from the WeChat public account "New Intelligence Yuan," authored by ASI Revelation, edited by Moses.