Anthropic Says Claude Computed a Nine-Loop Particle Physics Amplitude

Anthropic on September 25, 2026 published a guest post on its research site reporting that two of its physicists used the company’s Claude model to compute the six-particle scattering amplitude in planar N=4 super Yang-Mills at nine loops, answering a public challenge issued to AI companies on August 7, 2026.

The Challenge

Matt von Hippel, a former theoretical physicist who now writes about physics at 4gravitons.com, set the terms in an August 7, 2026 blog post. He asked AI companies to show that an AI, using the computing resources an academic has access to, could solve one of the scattering amplitudes field’s big outstanding problems: determining whether N=8 supergravity diverges at seven loops, or finding the six-particle amplitude in N=4 super Yang-Mills at nine loops. “Give us N=8 supergravity to seven loops, or N=4 super Yang-Mills to nine loops,” he wrote.

Von Hippel argued that these problems are hard in a computational sense: each loop represents an increase in complexity, in calculations that typically scale exponentially or even factorially in the number of loops. In principle, he wrote, amplitudes researchers could solve any of them with no new ideas, using known methods, but they would need access to far more computing power.

At the end of August 2026, Liam Fitzpatrick and Siddharth Mishra-Sharma, two physicists at Anthropic, contacted von Hippel to say they had tackled one of the challenges, and after verifying the result with longtime amplitudes researcher Lance Dixon they walked him through the work. The post’s disclosure states that Anthropic invited von Hippel to write the guest post and compensated him for his time, that Anthropic staff gave feedback on drafts, and that the content and opinions are his own.

The Physics Problem

Scattering amplitudes are formulas that let physicists use the momenta and energies of subatomic particles to calculate how likely they are to react in particular ways. They are hard enough to compute that physicists almost always work with approximations cut off at a specific number of loops: the more loops included, the closer the result comes to the real answer and the harder the calculation becomes. Most scattering amplitude formulas have been calculated only to two loops, and a few to three; von Hippel notes that the most precise prediction in particle physics, for the electron’s anomalous magnetic dipole moment, used five.

N=4 super Yang-Mills is a toy-model theory used to stress-test new techniques. Yang-Mills theories describe three of the four fundamental forces of nature, and in the N=4 version of supersymmetry each particle has four supersymmetric partners. That surfeit of particles makes the theory unrealistic, von Hippel writes, but paradoxically easier to calculate with, because the balance between the particles means only certain combinations of variables are needed.

The calculations were done with a bootstrap technique. Rather than accounting for every possible particle interaction, the researcher starts with every possible answer tracked in a specialized alphabet, then applies constraints until only one possibility survives, with enough checks left over to catch mistakes. The previous result for this amplitude, at eight loops, was reached indirectly in 2023 by Dixon and Andy Liu through a related formula called a form factor and a symmetry called antipodal duality.

How Claude Ran the Calculation

According to the guest post, Fitzpatrick and Mishra-Sharma used Fable 5.1 working within Claude Science, a platform scientists can pay to use and which von Hippel describes as a harness that runs the Claude LLM with structured rules and prompts for more robust, scientifically useful behavior. After asking Claude which problem it was most likely able to tackle, they gave it a short prompt naming the task, then told it to continue working while they slept and to provide updates every four to six hours.

Claude performed the calculation two ways, the post reports: the original bootstrap and the indirect form-factor approach. Von Hippel writes that either approach would have cost an end-user around one or two thousand dollars, mostly from the expense of running Claude for so long. The bootstrap calculation, written in Python with the SymPy package, took around $100 of that budget, corresponding to 96 CPUs running for a week.

The result was released as computer-readable files on a result page dated September 16, 2026, in the format used for the earlier six-, seven-, and eight-loop amplitudes, with eight files over 100 MB hosted on Zenodo. The page states that the nine-loop symbol was computed in two representations: one built by bootstrapping the nine-loop form factor, mapping it onto the amplitude via antipodal duality, and fixing the remaining ambiguities with two-gluon flux-tube data; the other a separate direct bootstrap of the symbol in the space of hexagon functions.

The page states that the two representations agree on every coefficient compared, across all 107,053 nonzero coefficients that determine the second file. As a control, the same programs reproduced the published eight-loop amplitude’s symbol on 1,000 random words. The page also records the assumptions behind the amplitude as a function, notes that it has been computed once with no second independent computation, and states that the programs of the computation are not distributed.

Validation and a Concurrent Result

Dixon, a professor of particle physics and astrophysics at SLAC National Accelerator Laboratory and Stanford University, wrote in an addendum to the post that Fitzpatrick and Mishra-Sharma told him of the result on September 1, 2026 and asked him to validate it. He wrote that he spent the following two weeks validating it, mostly through the nine-loop form factor, a result his team had been working toward for a couple of years. The disclosure states that Dixon received Claude usage credits. In his account, Claude executed the complicated recipe he and his collaborators had laid out, developed all of the code from scratch, and presented the solution in the format they had already established; he had thought the amplitude would be too hard to compute directly.

The result was also within reach for humans. A few days after von Hippel heard from Anthropic, Song He, an amplitudeologist at the Chinese Academy of Sciences in Beijing, reported that his group had already obtained the majority of the result, using AI assistance based on GPT-6 for some of the constraints rather than a one-shot approach. On September 17, 2026, He, Jirong Jing, and Xiang Li of the Institute of Theoretical Physics and the University of Chinese Academy of Sciences published a dataset on Zenodo titled “The Symbols of Six-Gluon MHV Amplitudes through Nine Loops,” covering six-point BDS-like subtracted MHV amplitudes at two through nine loops in planar N=4 super Yang-Mills under a Creative Commons Attribution 4.0 license. Von Hippel wrote that Dixon, Song He, and their collaborators will publish the results with explanations and analysis for future researchers, and that Claude’s role is done for now.

Von Hippel’s Assessment

Von Hippel wrote that Claude Science accomplished the calculation in one shot, without any scientific oversight more sophisticated than repeated instructions to keep working, and that his biggest takeaway is that there is more low-hanging fruit in the field than experts expect. He contrasted the result with work from March 2026, when AI completed physics projects at a student level with substantial hand-holding, writing that he thinks the technology has genuinely improved, though he is unsure how far the result generalizes to the wider, more competitive field of real-world amplitude calculations.

He also wrote that Claude used known methods with somewhat more compute than researchers had tried before, and that the work did not deliver the glimpse of unexpected new methods he had hoped would inform debates about superintelligence. What he learned instead, he wrote, is that he had been too naïve about where the computational limit actually was.