Claude gives us N=4 super Yang-Mills to nine loops

In this guest post, physicist and science writer Matt von Hippel shares what happened when he issued a challenge to AI companies regarding a problem in his former subfield of theoretical physics.It’s not often that you issue a challenge, only to see it beaten a month later. But we’re living in unusual times.Let me introduce myself: I’m Matt von Hippel. I used to be a theoretical physicist; these days I’m a science writer. Throughout, I’ve been a blogger, writing weekly at 4gravitons.com about physics and the people who do it.More and more, blogging about physics has meant blogging about AI. That’s a problem, because I’m definitely not an AI expert. I’ve dabbled in it, sure. I probably know more than your grandma. But I mostly have to step back and trust the experts. And frustratingly, the experts disagree! I’ve heard from smart, well-informed people who are confident that AI is a few years away from superintelligence, and that superintelligence will be capable of truly terrifying things. And I’ve heard from smart, well-informed people who are equally confident that LLM-based AI is close to a ceiling, that models like Claude won’t even be able to do impressive work in physics, let alone conquer the world.I’ve been reluctant to make my own predictions. Before forming an opinion, I wanted to see an LLM make progress on something familiar, something I knew was hard to do because I’d tried to do something similar myself.In addition to that, I wanted to see an LLM do something that I expected to be computationally hard. LLMs have made impressive strides in math, certainly, and this month alone has likely changed many peoples’ minds. But progress in math comes from new ideas, and ideas are mysterious things: one never quite knows how hard they are to find until they’re found. Computation felt more solid. I wanted to see an LLM tackle a challenge that seemed out of reach not because researchers didn’t know how to do it in principle, but because doing it seemed like the kind of thing that would take more computers and time than the researchers reasonably had access to. I wanted to see if those researchers were wrong: if a smarter, artificial researcher could use the same computers, and solve the problem anyway.So, I issued a challenge:“If AI companies want to impress people like me (or scare us, for that matter), then they need to tackle my old field. Show that an AI can take the kinds of computer resources an academic has access to, and solve one of the scattering amplitudes field’s big outstanding problems. Show that a computational limit everyone expected to be a problem doesn’t actually matter. Give us N=8 supergravity to seven loops, or N=4 super Yang-Mills to nine loops.”In short: can AI solve a frontier problem in my former subfield of theoretical particle physics? And can it do it on a budget?The challengeMy old field is a branch of theoretical particle physics called amplitudeology. When other particle physicists predict new particles, they make sure they can do the calculations to test those predictions. They compute formulas called scattering amplitudes, which let physicists use the momenta and energies of subatomic particles to calculate how likely they are to react in particular ways. If physicists can make more accurate predictions for these reactions, they can check whether results from experiments like the Large Hadron Collider match those predictions. A mismatch could be evidence for a new theory, one that could explain some of physics’ big lingering mysteries, like the nature of dark matter, or the balance between matter and antimatter in the universe.These scattering amplitude formulas are hard to compute, so hard that physicists almost always use approximations. They do partial calculations, cut off at a specific number of “loops,” a measure of how complicated interactions between particles are allowed to get. The more “loops” they include in their calculations, the closer they get to the real answer, and the harder, computationally, the calculation is to do.In practice, most scattering amplitude formulas have only been calculated to two loops. A few have three. The most precise prediction in particle physics you might have heard of used five.Amplitudeologists want to do better. They develop experimental new techniques, and test them on special “toy model” theories. By trying the technique with a toy model where the calculation is easier, rather than the more challenging particles of the real world, amplitudeologists can stress-test the new methods and see how far they can go.I posted challenges for two of those toy models. The one the folks at Anthropic chose to tackle was to go up to nine loops with a particular toy model theory, called N=4 super Yang-Mills.“Yang-Mills” is a technical name for a type of theory that explains most of the world around us. Three of the four fundamental forces of nature: electromagnetism, the strong nuclear force that holds the nuclei of atoms together, and the weak nuclear force that causes radioactive decay in things like bananas, are all Yang-Mills theories.The “N=4 super” comes from supersymmetry. Physicists have speculated that each particle has a “supersymmetric partner,” a particle with the same charge, but of a different type, matching matter particles like electrons to force particles like photons. At one time they were optimistic these particles could explain dark matter, via undiscovered partners of more familiar particles. Those speculations used “N=1” supersymmetry. In “N=4,” each particle has four supersymmetric partners, not just one.That surfeit of particles makes the theory very unrealistic. N=4 super Yang-Mills isn’t used as an explanation for dark matter, or for anything in the real world. Instead, amplitudeologists use it to hone their techniques, because N=4 is paradoxically easier to calculate with. The delicate balance between the different particles means only certain combinations of variables are needed, streamlining calculations.These calculations were done with an experimental technique called a bootstrap, which ended up bizarrely well-suited for use of AI. To bootstrap an amplitude, you don’t have to take into account every possible particle interaction. You just need to know roughly what the answer ought to look like, keeping track of every possibility in computer files in a specialized alphabet. Then you start checking everything you know: predictions from other calculation techniques, rules the answer has to obey, links to related problems where the answer was easier to find. It’s a bit like Sudoku, where you begin with a grid with all possible numbers, then cross them out as you go. In the end, you’re hoping to find that only one possibility satisfies all the checks, while having enough checks left over to make sure you didn’t make a mistake.That meant that Lance was already well set up to check if someone had handed him the next amplitude formula, with nine loops. It would be an interesting answer, not just as a validation of the bootstrap technique, but as a rare example of an amplitude with that many loops of complexity, an answer that could be worth studying in its own right.But he hadn’t computed it, and neither had anyone else in the field. The way he found the eight-loop answer was already a bit indirect, via a surprising link to a different but related formula called a form-factor, a kind of partial amplitude involving different particles that turns out to be a bit easier to calculate. He was expecting to find the next loop even more indirectly, potentially by a different kind of AI method. If people thought it was possible to just run the usual bootstrap method for one more loop, someone would have done it.Then people did itApparently, there are folks at Anthropic who read my blog.At the end of August, Liam Fitzpatrick and Siddharth Mishra-Sharma, two physicists at Anthropic, reached out to me to say they had tackled one of the challenges in my post. After verifying the result with Lance, they talked me through how they got it.True to the spirit of the challenge, they didn’t use millions of dollars in computer power. They used Fable 5.1, working within Claude Science, a platform scientists can pay to use. Claude Science is what folks in the biz call a “harness,” a program that uses the Claude LLM with structured rules and prompts in order to get more robust and scientifically useful behavior.Apparently, after asking Claude which problem it was most likely to be able to tackle, they gave it a simple prompt:“The problem is to compute the Six-particle (hexagon) amplitude in planar N=4 SYM at nine loops.”From there, they just kept telling it to keep going, with comments like:“I'm going to sleep and won't be available for another several hours. Keep working on this until I tell you to stop. Give me updates every 4-6 hours.”Claude ended up doing the calculation two different ways: the original bootstrap, and the indirect form-factor approach. Either approach would have cost an end-user around one or two thousand dollars, mostly due to the expense of running Claude for so long. The bootstrap calculation, done with the Python programming language with package SymPy, took around $100 of the budget, corresponding to running 96 CPUs for a week.Running 96 CPUs for a week might have felt like a lot when I was doing this kind of work ten years ago, but it’s pretty affordable now if you have a good reason.As it turned out, the result wasn’t all that far away for humans either. A few days after I heard from Anthropic, we heard from Song He, an amplitudeologist at the Chinese Academy of Sciences in Beijing. Song’s group had already gotten the majority of the result. They’d used some AI assistance, based on GPT-6, but not the kind of one-shot almost human-less approach Anthropic used.Everyone has been friendly here, which is a bit of a relief. The humans, Lance and Song and their collaborators, will get to publish the results, taking time to