On September 25, Anthropic reported that Claude, running Fable 5.1 in Claude Science, calculated a six-particle amplitude at nine loops in planar N=4 super Yang-Mills. The company described two established routes to the result, and physicist Lance Dixon said he checked it by comparing the amplitude with a related nine-loop form factor.
What Claude calculated
A scattering amplitude is a mathematical quantity used to describe how particles interact. This calculation concerned the six-particle, or hexagon, amplitude in planar N=4 super Yang-Mills (SYM), a theoretical framework used as a testbed for studying these calculations.
“Nine loops” identifies the calculation’s order in the theory. The result extends an earlier eight-loop calculation of the same amplitude by one loop. It does not, by itself, predict the outcome or cross section of real-world particle collisions.
Two established routes to the result
Anthropic described a hexagon-function bootstrap and an indirect form-factor approach. The bootstrap calculation used Python and SymPy; its reported computing workload was equivalent to 96 CPUs running for one week.
Anthropic physicists Liam Fitzpatrick and Siddharth Mishra-Sharma took up the challenge and set the task. The reported achievement was carrying out established methods at higher loop order, not introducing a new physical principle.
How Lance Dixon checked the calculation
Dixon said he checked the result mostly by converting the amplitude back to the related nine-loop form factor. Anthropic disclosed that Dixon received Claude usage credits.
The technical results list mathematical outputs and validation checks. The computation programs are not distributed.
How it compares with the earlier result
Lance Dixon and Andy Liu published the earlier eight-loop result in 2023. Their route used a form factor and antipodal duality; Anthropic described Claude’s calculation as using both a direct bootstrap route and an indirect form-factor approach.
| Calculation | Loop order | Described route |
| Dixon and Liu’s six-particle amplitude | 8 | Form factor and antipodal duality |
| Claude’s reported six-particle amplitude | 9 | Hexagon-function bootstrap and indirect form-factor approach |
Anthropic’s account also says Song He told Dixon that his group had calculated the symbol portion of a nine-loop amplitude. GPT-6 helped with some constraints, but did not provide the overall framework.
