Claude computes a nine-loop amplitude

Claude computed the nine-loop six-particle amplitude in planar N=4 super Yang-Mills. Lance Dixon of the SLAC National Accelerator Laboratory checked the result. The run did not propose a new physical principle. It used the bootstrap methods the field already had, at a cost the guest post describes as something an academic could afford.

In a guest post on Anthropic's site, physicist and science writer Matt von Hippel says he challenged AI companies to take the computer resources an academic can access and solve an open problem in scattering amplitudes. The list included N=4 super Yang-Mills at nine loops. The post is dated September 25, 2026. He writes that at the end of August, Anthropic physicists Liam Fitzpatrick and Siddharth Mishra-Sharma told him they had done one of the challenges. A Chinese tech site carried the story on the evening of September 26. The calculation in this piece follows the guest post and Dixon's addendum.

They used Fable 5.1 inside Claude Science. Von Hippel describes Claude Science as a harness: a program that drives Claude with structured rules and prompts so the behavior is more robust and more useful for science. He writes that they did not spend millions of dollars on compute. The prompt was to compute the six-particle hexagon amplitude in planar N=4 super Yang-Mills at nine loops. They then told it to keep going, to send updates every four to six hours, and that the person prompting it was going to sleep.

Scattering amplitudes estimate how particles react. More loops mean a finer approximation and a heavier calculation. Von Hippel writes that most formulas in practice stop at two loops, a few reach three, and the most precise prediction in particle physics he expects readers to have heard of used five. N=4 super Yang-Mills is not an explanation of dark matter or of anything in the real world. Amplitudeologists use it as a toy model for testing methods, because the extra supersymmetric particles make the calculation shorter. Yang-Mills theories themselves cover three of the four fundamental forces: electromagnetism, the strong nuclear force, and the weak nuclear force.

A bootstrap does not add up every particle interaction. It stores the rough shape of the answer in a specialized alphabet, then crosses out possibilities using predictions from other methods, rules the answer must obey, and related problems whose answers are easier. He compares it to Sudoku. In 2023 Dixon and Andy Liu used a related formula called a form factor, plus a symmetry they call antipodal duality, to reach the eight-loop amplitude. Von Hippel writes that Dixon had not computed nine loops, and neither had anyone else in the field.

Claude did the calculation both ways: the original bootstrap, and the indirect form-factor route. Either approach would have cost an end user about one or two thousand dollars, mostly because Claude ran for so long. The bootstrap in Python with SymPy took about $100, corresponding to 96 CPUs for a week.

In the addendum, Dixon says that on September 1 Fitzpatrick and Mishra-Sharma told him Claude had computed the nine-loop MHV six-particle amplitude in planar N=4 super Yang-Mills and asked him to validate it. He writes that the setup is fragile: one mistake in the recipe and it collapses like a failed soufflé. Many construction details are too tedious to publish in full, so Claude had to write the code itself. For the next two weeks he checked the result mostly through the nine-loop form factor. His group had been aiming at nine loops since 2023, and he had thought a direct amplitude calculation was too hard.

He writes that this does not crush him, for two reasons. His group was already training smaller custom transformers to predict higher loops, under the slogan that they can validate any candidate a machine provides. Claude is also a transformer, probably more than a million times larger than their custom model. And Claude used the methods he and his collaborators had built over years, and returned the answer in the format they had already set. While he checks Claude, Claude checks their earlier work. He also asserts that, aside from his co-authors, Claude may understand their 2019 and 2023 papers better than any human. That is his judgment, not a separate evaluation.

Dixon then writes that Song He of the Chinese Academy of Sciences told him Song's group had also computed the piece of the nine-loop amplitude called the symbol. The group used GPT-6 to help compute some of the constraints, not the overall framework. Within two weeks, Dixon says, he was scooped by a machine and then by humans plus a machine. The post's additional material lists the concurrent result by Song He, Jirong Jing, and Xiang Li. Von Hippel writes that the human authors will publish and explain the results, and that Claude's role is done for now. The line that Song's group had already gotten most of the result comes a few days after von Hippel heard from Anthropic. This piece does not turn that into a claim about who was absolutely first.

Von Hippel's conclusion is that he had hoped to see a model break a computational barrier in a surprising way, and instead saw something humans could also do. Claude used known methods, with a bit more compute than people had been willing to spend. His main takeaway is that more results are within reach than experts expect. The other is that Claude Science finished in one shot, with no scientific oversight finer than "keep going." He also says this does not answer the larger arguments about superintelligence. He wanted new methods for the calculation itself, and what he learned was that he had been naive about where the old limit sat.

Dixon calls it a triumph that a language model executed the existing recipe and organized the compute. The moment that would make him search harder, he writes, is when language models start to propose new physical principles and insights before humans do. The disclosure says Anthropic invited von Hippel to write the post and paid him, staff commented on drafts, and the views are his own. Dixon validated the result independently and received Claude usage credits.