Anthropic's Claude Computes a Checked Nine-Loop Amplitude

In a guest post published by Anthropic on September 25, physicist and science writer Matt von Hippel reports that Claude Science computed a six-particle scattering amplitude at nine loops. Lance Dixon, a physicist at SLAC National Accelerator Laboratory who has worked on earlier versions of the calculation, checked the result.

The result concerns a deliberately simplified theory, not a prediction for an experiment. Claude applied techniques physicists already knew, while another research group independently reached most of the result with AI assistance. The demanding part was carrying the calculation through to an output a specialist could inspect.

Nine Loops, but Not a Prediction About Real Particles

A scattering amplitude is a mathematical ingredient physicists use to calculate the likelihood of a particle interaction. Researchers often approximate an amplitude by adding successive orders of perturbation theory. Each order accounts for more complicated contributions; physicists refer to these orders by their number of “loops.” Nine loops here describes the depth of the physics calculation, not the number of times Claude revised an answer.

Higher orders can provide a more detailed approximation within a theory’s useful range, but the calculations become much harder. Von Hippel chose nine loops as a challenge because researchers had already pushed the relevant six-particle amplitude to eight loops. The next order looked computationally daunting, even though the field had methods that might, in principle, reach it.

The theory is planar N=4 super-Yang-Mills. Its supersymmetry and planar limit give researchers mathematical structure they can use to constrain a calculation. They also make it a poor description of the particles and forces measured in the real world. Physicists study such theories partly to develop and test techniques that would be much harder to investigate in more realistic settings.

The precise target was the six-gluon maximally helicity-violating amplitude, often shortened to the six-particle or hexagon amplitude. Its posted nine-loop results are specialized mathematical data. They are not a newly measured collision rate, evidence for supersymmetric particles, or a direct prediction for the Large Hadron Collider.

Claude Used Two Established Routes to the Answer

The central technique was a bootstrap. It starts with a structured space of potential answers and applies known mathematical constraints until the possibilities are sufficiently restricted. Representing that space, implementing the constraints and checking the output are difficult parts of the work.

Von Hippel says Anthropic physicists Liam Fitzpatrick and Siddharth Mishra-Sharma put the problem to Claude Science after asking which of his proposed challenges the system was most likely to handle. He describes Claude Science as a harness that runs a Claude model with structured prompts and rules for scientific work; the post identifies the model used as Fable 5.1.

According to the account, the researchers gave Claude the nine-loop six-particle problem and repeatedly instructed it to continue, including during periods when they were unavailable. Humans selected the challenge, operated the system and asked it to persist. Claude’s reported accomplishment was the substantial technical work it carried out within that setup, not independently deciding what question physics needed to answer.

Von Hippel reports two approaches. One extended the direct bootstrap for the amplitude. The other first bootstrapped a related object called a form factor, then used an established relationship to transfer information to the amplitude and resolve what remained undetermined. The result files and methodological description distinguish the form-factor route from the separate direct bootstrap.

Neither route required Claude to invent a new method of theoretical physics. An existing recipe can still demand extensive coding, symbolic manipulation and decisions about how to make a large calculation tractable. Von Hippel’s original challenge was partly about the gap between knowing a route in principle and completing it with accessible resources.

What the Check Establishes

Dixon had worked on the earlier eight-loop result and could assess the new calculation against the field’s established framework. Von Hippel reports that Dixon verified Anthropic’s result. An expert check is stronger evidence than an answer that merely looks plausible, though it does not mean every part of the work received a separate, independent recomputation.

The posted materials contain two representations of the nine-loop symbol, a structured way of recording much of the amplitude’s mathematical content. The result page says the form-factor-based representation and the direct-bootstrap representation agree on every coefficient compared, including the 107,053 nonzero coefficients used to determine one of the distributed files. Its checks also include reproducing coefficients from the published eight-loop result on a sample of 1,000 words.

These cross-checks have boundaries. The result page records that some coefficients are represented through calculations modulo primes rather than fully reconstructed as rational numbers. It lists assumptions and outstanding tests, and says the programs used for the computation are not distributed. Other researchers can inspect the released outputs and validation records, but those materials do not by themselves let them rerun Claude’s entire workflow.

A symbol is also distinct from the full mathematical function. The page presents a function-level result separately; the two-route coefficient comparison described above concerns the symbol. That distinction shows which parts have mutually agreeing constructions and where the published record describes further assumptions.

The Reported Cost Was $1,000 to $2,000 per Approach

Von Hippel estimates that either approach would have cost an end user roughly $1,000 to $2,000, mostly for running Claude over an extended period. That is an estimate for each route, not a quoted price for buying a completed nine-loop result or a guarantee that another team could repeat the project for the same amount.

The conventional computing was comparatively cheap in his account. The direct bootstrap used Python and SymPy, and von Hippel puts its computation cost at about $100, corresponding to 96 CPUs running for a week. As reported, the limiting expense was largely the sustained model-assisted work around the calculation, not an extraordinary conventional-computing bill.

A stated end-user estimate cannot capture every cost of defining the problem, building on years of published physics or having an expert available to check the answer. It does show that von Hippel’s challenge did not require a reported multimillion-dollar compute campaign.

Human Researchers Were Reaching the Same Frontier

Claude was not alone at the nine-loop frontier. Von Hippel reports that Song He’s group had independently reached most of the result, also with AI assistance. “Most” matters: it would overstate the evidence to say the group had already completed an identical result, just as it would overstate Claude’s achievement to portray nine loops as uniquely beyond human researchers.

Anthropic’s case shows a long-running model workflow executing known specialist techniques and producing a result that Dixon checked. Song He’s group was also making substantial progress through another AI-assisted effort. Together, they show how established research programs may advance when researchers can delegate more of their computational implementation.

Final Thoughts

Claude Science helped complete a difficult calculation in a tractable theoretical setting. Dixon’s check, the two approaches and the released validation data give the result more substance than an uninspectable answer.

The case does not establish that Claude discovered a new physics method or can make comparably useful predictions for real-world particle experiments. Whether similarly sustained workflows can produce checkable results when the mathematical structure is less accommodating and the route to an answer is not already known remains an open question.