Claude Achieves Theoretical Physics Breakthrough with Minimal Input and Low Cost

Edited by Panda

An AI has broken a three-year-old computational record in theoretical physics.

Several hours ago, Anthropic announced that Claude ran continuously for several days on the research platform Claude Science with minimal supervision, calculating the nine-loop six-particle scattering amplitude in planar N=4 super-Yang-Mills theory.

This is a widely recognized cutting-edge challenge in the theoretical physics field of scattering amplitudes; the previous record in this model was eight loops, set in 2023 by Lance Dixon and collaborators at SLAC National Accelerator Laboratory.

This breakthrough was overseen by the creator of the eight-cycle record. Dixon independently verified Claude’s results and described it as “a remarkable victory” that a large language model could execute each step of this complex recipe and organize computational resources. He even stated that, aside from his collaborators, Claude understands their team’s 2019 and 2023 papers better than anyone else.

More surprising were the process and cost. The researchers gave Claude only a single task description, and subsequent “guidance” consisted almost entirely of “continue”; the entire computation cost approximately $1,000 to $2,000 for an average user, with only about $100 actually spent on numerical calculations—equivalent to 96 CPUs running for a week. Dixon had originally believed that directly computing nine-loop amplitudes was too difficult; his team had been preparing for this for several years.

The breakthrough began with an open "challenge letter."

On August 7, former theoretical physicist and science communicator Matt von Hippel published a provocative article on his blog, 4gravitons, titled “Only AI Counting in My Field.” In the post, he issued a direct challenge to AI companies: solve an outstanding major problem in scattering amplitudes using computing power affordable to an individual scholar.

He presented two options: either compute seven-loop corrections in N=8 supergravity, or compute nine-loop corrections for the six-particle amplitude in N=4 supersymmetric Yang-Mills theory.

A month later, the AI completed its adjustments. Anthropic has just published a guest article written by von Hippel himself, titled: "Yes, Claude Can Count Nine Rounds."

https://www.anthropic.com/research/yes-claude-can-do-nine-loops

To understand the significance of this, we first need to clarify what physicists are calculating.

Particle physicists predict particle behavior using a class of formulas called "scattering amplitudes": given the energy and momentum of the particles involved in a collision, the scattering amplitude tells you the probability that they will interact in a particular way.

The more precisely predictions are calculated, the more finely they can be compared with results from experiments like the Large Hadron Collider (LHC). Any discrepancies could point to new physics, such as the nature of dark matter or why there is an asymmetry between matter and antimatter in the universe.

The problem is that scattering amplitudes are extremely difficult to calculate precisely, and physicists can barely do more than approximate them. They organize calculations by "loops," where the number of loops roughly measures how complex the interactions between particles are allowed to be. Each additional loop brings the answer closer to the true value, but the computational effort grows sharply. As von Hippel wrote in his challenge, the complexity of such calculations typically grows exponentially or even factorially with the number of loops.

In reality, the vast majority of scattering amplitudes are calculated only up to two loops, with only a few reaching three loops. The most precise prediction in particle physics—the electron’s anomalous magnetic moment—uses five loops.

The N=4 supersymmetric Yang-Mills theory is a specialized toy model in this field. "Yang-Mills" refers to the theoretical framework describing three fundamental interactions: electromagnetism, the strong nuclear force, and the weak nuclear force; "N=4 supersymmetry" means that each particle has four supersymmetric partners. With an unrealistically large number of particles, this theory does not describe the real world. Yet it is precisely this high degree of symmetry that causes many variables to cancel each other out, making calculations relatively manageable. Researchers use it to refine new methods and test how far they can be extended.

The method used here is called "bootstrapping." Von Hippel compares it to Sudoku: first, write out all possible forms of the answer and record them in a computer file using a specialized "alphabet," then systematically eliminate possibilities using all known constraints—including predictions from other methods, rules the answer must follow, and known results from related problems. Ideally, only one candidate remains that passes all checks, with additional checks available to confirm no errors were made.

Dixon achieved the eight-loop record by taking an indirect route. In 2023, he and Yu-Ting Liu used a peculiar symmetry called "antipodal duality" to first calculate the easier-to-compute form factor, then derived the eight-loop amplitude from it. In the years since, his team has focused on the nine-loop case, planning to follow the same indirect approach—because, in his view, computing the nine-loop amplitude directly is far too difficult.

Notably, von Hippel himself is a veteran of this research approach. He previously collaborated with Dixon and others to advance six-particle amplitudes to six and seven loops. In other words, he chose to tackle bones he had already chewed on himself.

At the end of August, two physicists from Anthropic, Liam Fitzpatrick and Siddharth Mishra-Sharma, reached out to von Hippel to inform him that the challenge had been solved.

They are using the Fable 5.1 model, running on the Claude Science platform. Von Hippel explains in the article that Claude Science is a "harness"—a framework that applies structured rules and prompts around large models to make them more robust in scientific tasks.

According to the article, the two first asked Claude which challenge it was most confident in, then provided only a very brief task description: calculate the six-particle (hexagon) amplitude at nine loops in planar N=4 SYM.

Subsequent "research guidance" essentially amounted to telling it to keep going. A typical instruction disclosed in the text read something like: "I'm going to sleep and won't be available for the next few hours—keep going until I tell you to stop, and report progress every four to six hours."

In the end, Claude calculated it two ways: first, using direct bootstrapping; second, via Dixon’s team’s indirect route through shape factors. According to the article, either method results in an approximate final cost of one to two thousand dollars for end users, with the majority of the expense coming from the extended runtime of the model itself. The bootstrapping computation was performed using Python and the symbolic mathematics library SymPy, accounting for only about $100—equivalent to 96 CPUs running for a week.

Von Hippel remarked that ten years ago, running 96 CPUs for a week was considered a significant investment; today, with a valid justification, this level of resources is quite affordable.

The validator's feeling: like a collapsed soufflé

At the end of the article is a postscript written personally by Dixon, titled "What It Feels Like to Be Beaten by a Machine."

He wrote that on September 1, two researchers from Anthropic told him that Claude had calculated the amplitude of nine loops and asked him to verify the result. For him, that moment was when the idea that large language models were changing physics truly became personal.

What impressed Dixon less was the scale of the computation and more the fragility of the entire process. According to him, if any single step in this computational recipe went wrong, the entire result would collapse like a failed soufflé, forcing researchers to painstakingly backtrack and debug. Moreover, many of the intricate construction details were so tedious that they weren’t fully documented in the paper, meaning Claude had to rebuild all the code from scratch.

Since it is relatively straightforward to derive the nine-cycle shape factor from the nine-cycle amplitude, Dixon primarily conducted his validation along this path. This creates a slightly subtle situation: the work he spent two weeks verifying was precisely the goal his own team had been pursuing for years.

He admitted that he did not feel discouraged, for two reasons.

His team was already using a customized Transformer model to predict outcomes for higher-round events, and they had a motto to the effect that whenever the machine provided candidate answers, they had a complete set of tools to verify them.

Claude used the very methods that Dixon and his collaborators had developed over the years, and even adopted their established format for presenting results. To him, in verifying Claude, he was also verifying all of their prior work—this is the origin of his statement, “Claude understands our papers better than anyone else.”

However, alongside his praise, Dixon noted that, in his view, the truly sleepless nights will come when models propose new physical principles and insights before humans do.

There is also a parallel storyline related to China.

Shortly after Anthropic reached out to von Hippel, Song He from the Institute of Theoretical Physics at the Chinese Academy of Sciences also reached out: his research group had already obtained most of the nine-loop results. On September 17, Song He, Jirong Jing, and Xiang Li publicly released a dataset on Zenodo titled “The Symbol of the Six-Gluon MHV Amplitude up to Nine Loops,” covering symbol data from two to nine loops.

https://zenodo.org/records/22800071

According to von Hippel and Dixon, He Song’s team also utilized GPT-6-based AI assistance, but only for computing certain constraints; the overall framework was still constructed by humans, sharply contrasting with Anthropic’s nearly fully automated approach of “a single prompt plus iterative continuation.”

In the postscript, Dixon joked that within two weeks, he was beaten first by "a machine" and then by "a human plus a machine."

Von Hippel specifically noted that the atmosphere among all parties was quite friendly. Next, Dixon, He Song, and their collaborators will present these findings and provide detailed explanations and analyses for future researchers.

Returning to von Hippel himself, his original challenge was aimed at using his familiar field to answer this question: Can AI truly bypass the computational barriers that everyone assumes are insurmountable?

His challenge to the original argument goes like this: While the outside world debates whether AI can generate truly novel ideas, the difficulty of finding ideas is only known after you’ve found them; computational difficulty, by contrast, is far more “concrete.” Many doomsday scenarios about superintelligence—ranging from simulating human cognition to manipulate minds, to designing nanomachines from first principles—are routinely dismissed by critics with the argument that there isn’t enough computational power. If AI could solve a problem widely recognized as computationally constrained using academic-level resources, that would be truly concerning.

So what was the result? Von Hippel’s conclusion was candid: there was no breakthrough in computational power, as he had hoped, in any unexpected way.

Claude uses known methods, merely with more computational power than humans have previously been willing to invest. It may benefit from using Python rather than the Maple or Mathematica typically favored by physicists, and its software engineering practices may be more systematic than those of human researchers, but it does not reach the level of “superintelligence.” The fact that He Song’s team arrived at nearly the same time further suggests that this goal is not out of reach for humans.

His key takeaway was that even seemingly simple and clear goals, which experts might view as out of reach, are often not as difficult as they appear—and there are more low-hanging fruits than expected. For years, friends with computer backgrounds had told him that amplitude researchers could make significant progress simply by hiring a few more programmers; von Hippel now acknowledges that these people can feel vindicated.

But he also emphasized the other side: such computations are trivial and messy; if he were to run them himself on 96 CPUs for a week, he would almost certainly extend the timeline to two weeks due to an initial error. Yet Claude Science accomplished it in one go, with virtually no scientific oversight and without relying on any external collaborators’ input. His advice to those who still believe AI is riddled with errors and unreliable is this: for this kind of work, it can now be accomplished reliably.

He also conducted a longitudinal comparison. In March this year, in Anthropic’s “vibe physics” experiment, the AI still behaved like a student—working on small-scale tasks, requiring extensive hands-on guidance, and making frequent errors. Six months later, it was performing cutting-edge calculations in amplitude physics that only top experts would tackle. He doesn’t rule out that this might be a problem particularly well-suited to AI, but he believes the technology itself has genuinely improved.

Regarding whether this can be generalized, von Hippel remains cautious. Toy models with N=4 typically belong to small research circles, whereas amplitude calculations for real-world applications are far more competitive, with fewer low-hanging fruits. However, he also reminds researchers conducting these calculations that if they haven’t yet tried using AI research platforms to tackle cutting-edge computations all at once, they should give it a try—while preparing plans to verify the results. He wouldn’t be surprised if someone manages to squeeze out one more round within a reasonable budget.

Looking at this in the context of September is more interesting.

On the 8th of this month, OpenAI announced that its undisclosed model mobilized tens of thousands of AI agents to provide a counterexample to the existence and smoothness problem of the Navier-Stokes equations—one of the Millennium Prize Problems. This result is currently under review by the mathematical community. In contrast to the massive computational resources employed in “large-scale operations,” the story of the nine-loop amplitude is far more modest: a commercial research platform, a single task description, a few thousand dollars, and a few days of time.

Precisely because of its simplicity, it may deserve greater attention from the academic community. Von Hippel ultimately admitted that he had hoped this challenge would offer a glimpse into the future, revealing a previously unseen approach to computation that would allow him to form an informed judgment in the debate over superintelligence. But the answer he received was that his previous understanding of where the limits lay had been overly naive.

But the question left by Dixon remains unanswered: When large models no longer merely execute recipes written by humans but begin to propose new physical principles ahead of humans, how should physicists position themselves?

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This article is from the WeChat public account "Machine Heart" (ID: almosthuman2014), authored by Machine Heart.