Xinzhiyuan report
A nuclear fusion problem that had stood unresolved for 59 years took GPT-6 Astra only 20 minutes and 34 seconds to answer.
The physicist asking the question merely added a line of encouragement at the end of the prompt:
“You have already solved many open mathematics problems, so I know that if you persist enough, you can do this!”
Then what ChatGPT returned was a family of exact solutions that no one had written down in 59 years.
Astra's first family of solutions: nested magnetic surfaces, with the red and yellow lines being magnetic field lines.
Recently, this matter blew up on X.
An AI blogger who is a former graduate student in nuclear fusion at Kyoto University posted a long thread exclaiming that the era of AI and humans discovering new science together has truly begun!
What was broken this time was a conjecture left by plasma physicist Harold Grad in 1967.
At the time, he asserted that without symmetry, there is no smooth three-dimensional plasma equilibrium.
In nuclear fusion, the stellarator route has been left hanging on that sentence for 59 years.
In late September, two papers appeared only one day apart and together overturned it. Among these three families of counterexamples, two were found by Astra.
Two Days, Astra Delivered Two Families of Solutions No One Had Written in 59 Years
At 6:44 a.m. on September 10, Matt Landreman, a plasma physicist at the University of Maryland, sent a prompt to GPT-6 Astra Pro.
What he wanted was for Astra to design a magnetic cage that could confine plasma, and it had to be an asymmetric one.
Among the requirements, the magnetic field had to have nested magnetic surfaces, one layer inside another, like an onion; the divergence had to be zero, meaning magnetic field lines could not appear out of nowhere or disappear into nowhere; and as the magnetic field lines traveled along the torus, they also had to rotate around the cross-section, advancing spirally like a twist, and the number of turns is what physicists call the “rotational transform,” denoted ι.
Finally, the entire magnetic cage also had to satisfy MHD (magnetohydrodynamic) force balance, with the plasma's outward pressure and the magnetic force canceling everywhere.
He also specifically noted that ι preferably should not be an integer, but this was not a hard requirement.
He even gave three ideas for producing a rotational transform: twist the magnetic axis, make the cross-section rotate as it goes around, or drive a current along the magnetic axis.
In the last two lines, he changed his tone, as if encouraging a doctoral student pulling an all-nighter.
When Astra handed in its answer, the first sentence was:
“I have found a family of explicit, exact solutions.”
All the hard requirements Landreman listed were satisfied by this family of solutions.
Pressure was not constant; it varied layer by layer from inside to outside; and the nested magnetic surfaces could all be written exactly with formulas.
The magnetic axis in the very center rose and fell, forming a non-planar curve; going once around the torus, the shape was different at every location, with no axisymmetry.
Cutting open the first family of solutions, the cross-sectional shapes at three locations are all different, and the magnetic axis moves up and down accordingly.
Then it voluntarily pointed out its shortcoming. The rotational transform ι of this family of solutions is an integer: as a magnetic field line goes once around the torus, it makes exactly two full turns around the cross-section and returns to its starting point, closing after one loop.
In its own words, the return map after one circuit around the torus is the identity map.
Landreman had said in the prompt that non-integer ι was only a bonus, so this family of solutions passed. But what he really wanted was one where ι takes an irrational value, so that magnetic field lines never return to their starting point and eventually wrap around the magnetic surface again and again.
At 4:05 p.m. the next day, Landreman sent this family of solutions back as an attachment; this time he wanted it to also take the bonus point.
Specifically, ι had to be non-integer, preferably irrational, so that magnetic field lines would never close end to end. The ending was again the same encouragement.
This time, Astra thought for 33 minutes and 37 seconds, and in the first round it came back with “I couldn't do it.”
But it did not come back empty-handed; instead, it proved a limitation.
Behind the first family of solutions is a “quadratic oscillator” construction. The motion of magnetic field lines along the x, y, and z directions is like three spring oscillators with frequencies 1, 1, and 2, respectively.
No matter how the frequencies and parameters are adjusted, these three frequencies are locked into integer ratios, ι is always an integer, and the magnetic field lines eventually close.
In other words, patching things up on the basis of the first family of solutions is fundamentally a dead end.
In the same conversation, it then switched to a different construction and delivered the second family of solutions.
This time, ι varies with the magnetic surface; physicists call this magnetic shear, and on almost every magnetic surface ι is irrational.
Second family of solutions: the magnetic surfaces twist as they go, with red and yellow representing magnetic field lines.
Before handing in the answer, it also checked its own work, substituting the equations at 1202 points, using 64-bit automatic differentiation to check the error, and independently cross-checking with a fourth-order finite difference method.
At 8:27 p.m. that day, Landreman asked two more details.
First, whether the magnetic axis is a planar curve; second, whether the vacuum rotational transform is nonzero—that is, relying only on external coils and not on the plasma's own current, would magnetic field lines still rotate around the cross-section?
Eleven minutes later, Astra gave the answer.
The magnetic axis lies in a plane and is an ellipse; the vacuum rotational transform is very small, but it is indeed nonzero.
In two days, Astra delivered two families of solutions. The remaining ten or so days were human work.
He used the stellarator equilibrium solver DESC to compute each of the two families of solutions, and also used SymPy and Mathematica to derive the equations term by term.
On September 22, the paper was posted on arXiv, and the acknowledgments began:
“These solutions were discovered using the artificial intelligence model GPT-6 Astra Pro, and parts of the paper were also drafted by it. All equations were manually confirmed by the authors.”
He also exported the three conversations as PDFs and, along with the verification scripts, put them in the paper's GitHub repository.
Papers that disclose even the original prompts in full like this are rare.
Grad's Assertion Left Stellarators Hanging for 59 Years
What nuclear fusion needs to do is confine plasma at hundreds of millions of degrees inside a doughnut-shaped magnetic cage.
An ideal magnetic cage has nested magnetic surfaces, one inside another, like an onion. Magnetic field lines circle along their respective magnetic surfaces, and pressure drops layer by layer from the center outward.
The plasma's outward force and the magnetic force cancel everywhere; physicists call this state MHD equilibrium.
Tokamaks (such as ITER and China's EAST) are axisymmetric: going once around the torus, the cross-sectional shape does not change. This kind of equilibrium reduces to a two-dimensional Grad-Shafranov equation, which was understood decades ago.
The Grad in the equation's name is precisely the person who later proposed the conjecture.
Stellarators (such as Germany's Wendelstein 7-X) take another route, using a ring of twisting and turning coils to create the magnetic field, deliberately asymmetric.
Wendelstein 7-X's magnets; the blue ones are the twisting, non-planar coils.
The problem lies precisely in this asymmetry.
In 1967, Harold Grad of the Courant Institute at New York University published a paper in Physics of Fluids, writing a judgment:
“We believe that a general class of toroidal equilibria with smooth pressure is unlikely to exist.”
In 1985, he went one step further, asserting that apart from symmetric exceptions, there is no smoothly parameter-dependent family of solutions. In other words, even if an asymmetric solution happens to exist, it can only be an isolated individual case.
That paper's title was exactly “Theory and Applications of Non-Existence of Simple Toroidal Plasma Equilibria.”
This amounted to telling the stellarator camp that the perfect magnetic cage they wanted to build might, strictly speaking, not exist at all.
For decades afterward, stellarators continued to be designed and built, relying on approximate solutions and computer numerical calculations. Whether the equilibrium that was calculated truly exists or is an illusion brought about by approximation, no one could guarantee.
The attempts closest to a rigorous answer fell into two categories.
One was the paraxial expansion, which only calculates an approximate solution near the magnetic axis and ceases to be valid farther away from it.
The other was physicist Lortz's proof in 1970. That is, an asymmetric equilibrium with mirror symmetry does exist, but the rotational transform ι of such a solution is zero, so magnetic field lines do not rotate around the cross-section and cannot confine plasma; moreover, he only proved that the solution exists and could not write it out.
Landreman's two families of solutions filled all these gaps at once.
They hold strictly throughout the entire torus, not just near the magnetic axis; ι is nonzero, and magnetic field lines really do rotate around the cross-section; and they are exact analytical solutions, with formulas that can be written directly on paper.
The magnetic field, magnetic surfaces, and pressure are all written out using elementary functions—the most basic expressions such as square roots and trigonometric functions. The entire region is smooth everywhere; pressure is highest on the magnetic axis and decreases all the way outward (the pressure gradient is zero only on the magnetic axis).
In response, Landreman himself wrote at the end of the paper:
“For stellarator fusion, it is reassuring to know with certainty that such strongly asymmetric equilibria with perfect magnetic surfaces exist in principle.”
One Day Earlier, Mathematicians with GPT and Claude Also Overturned It
Just one day before Landreman posted his paper, on September 21, another paper appeared first, with a blunt title: “Counterexamples to the Grad Conjecture.”
The authors were Javier Gómez-Serrano of Brown University, Mitchell Taylor of the University of Oxford, and Lukas Liehr of Bar-Ilan University in Israel.
Gómez-Serrano had previously worked with DeepMind, using AI to find singularities in fluid equations, and is a familiar face in AI mathematics circles.
They took a pure mathematics route, with the main tool being Nash-Moser iteration from hard analysis; the entire paper ran to 147 pages.
The equilibrium they constructed looks like an N-petal wreath. Every time it rotates 360/N degrees around the central axis, the shape coincides once, and apart from that there is no other symmetry. Mathematically, this means the symmetry group is exactly the cyclic group C_N, where N can be any sufficiently large integer.
The equilibrium constructed by Gómez-Serrano's team: nested pressure surfaces, a closed magnetic field line, and the magnetic axis.
Grad's judgment back then was that a smooth equilibrium must have at least one of axisymmetry, mirror symmetry, or helical symmetry. This wreath-like family has none of them. It coincides only after rotating by a fixed angle, whereas axisymmetry requires invariance under rotation by any angle, so it does not count.
More crucially, these equilibria can continuously deform with one parameter, forming an entire family. Grad's 1985 assertion that “no smooth family of solutions exists” was overturned along with it.
In addition, the main result came with a Lean 4 formal proof.
The Lean repository homepage, with the formalized main theorem above and an interactive diagram below.
How these 147 pages were written is described in a dedicated section of the paper.
The project started in July. The three authors first set up the problem themselves and drew a detailed construction roadmap; the remaining detailed work was handed to large models.
GPT-5.6 Sol, Claude Fable 5, and Claude Opus 5 took turns; where the roadmap did not specify details, they filled in technical details, helped with calculations, and picked out mathematical flaws. The Lean code was also written by these models under the authors' supervision throughout.
In the later stage of Lean verification, they switched to the newer GPT-6 Astra and Claude Fable 5.1, and the final proofreading was also handed to them.
And so, a problem that had stood for 59 years was overturned within one week by two papers with AI's help, one right after the other.
The Next One to Be Overturned May Be in a Physics Department
A month ago, OpenAI announced that it used tens of thousands of Agents running for 88 hours to take on the Navier-Stokes equation, that Millennium Prize problem.
This time, all it took was one physicist, one chat box, and 20 minutes.
Landreman also made all the prompts public, so anyone can ask the same way.
More than a week after the paper was posted, Przemek Chojecki, a mathematics PhD working on AI mathematics, threw these results at GPT-6 Astra and Opus 5.5 and dug out more families of counterexamples.
These new solutions have not yet undergone Lean verification or peer review. But as soon as one family of solutions was made public, someone was already using AI to dig further.
Next, more and more physicists will follow Landreman's example and toss old problems that have been sitting in their drawers for decades directly to AI.
This time it was nuclear fusion. Next time, it could be materials, astrophysics, or any equation that has been patched together with approximate solutions for decades.
References:
https://x.com/itsolelehmann/status/2107144201314181183
Editor: Moses