Claude Independently Solves Frontier Theoretical Physics Problem, Unsupervised, for Under $2,000

Anthropic's two physicists, Liam Fitzpatrick and Siddharth Mishra-Sharma, invoked Fable 5.1 on the research workbench Claude Science, with an initial prompt of just one sentence: calculate the six-particle amplitude at nine loops in planar N=4 super Yang-Mills theory. After that, the two researchers' intervention was limited to telling it to continue, for example: "I'm going to sleep, unreachable for a few hours, keep going until I tell you to stop, report progress every 4 to 6 hours."

Claude ultimately worked through two routes. The first reproduced the Lance team's indirect approach; the second was the direct route Lance considered infeasible: directly bootstrapping the symbol of the nine-loop amplitude in the hexagon function space. The results of both routes matched one by one across all 107,053 nonzero coefficients. As a control, the same program also reproduced the published eight-loop results on 1,000 randomly drawn letter sequences.

On cost, the bootstrap computation was done in the Python language with the open-source symbolic computation library SymPy, at about 100 dollars, equivalent to 96 CPUs running for a week. Adding the cost of Claude's long-running operation, the entire project cost one to two thousand dollars in total. On September 16, Siddharth published the symbol and complete function of the nine-loop amplitude on his personal website as a computer-readable file, in the same format the Lance team had previously used when releasing the six-, seven-, and eight-loop results.

Scattering amplitudes describe the probability that particles produce a specific outcome after collision, and are the basis for comparing theoretical predictions with collider experiments. Physicists usually compute amplitudes using perturbation theory, adding quantum corrections level by level, where each level is one "loop." The higher the loop count, the more precise the result, and the computational complexity grows exponentially or even factorially. In actual research, most amplitudes are computed only to two loops, and a few can reach three loops.

Planar N=4 super Yang-Mills theory is a testing ground for amplitude researchers. The theory does not describe the real world, but the delicate balance among particle types greatly simplifies calculations, and researchers use it to hone new methods before extending them to theories closer to reality. The Lance team has advanced loop by loop using the bootstrap method since 2011: first delineating the function space where the answer lies based on physical principles, encoding each candidate function into a sequence using a specific "alphabet," that is, the "symbol" that preserves the main structure of the function, then imposing known constraints one by one to eliminate candidates until only a unique solution remains. The whole process is similar to solving a Sudoku.

By 2019, they had only reached six and seven loops. In 2021, Lance discovered that the amplitude for two gluons colliding to produce four gluons is numerically very similar to the "form factor" for two gluons colliding to produce one gluon and one Higgs boson, the latter being easier to compute. The team then confirmed that reversing the order of each letter sequence in the former's symbol, followed by a variable substitution, yields the latter, a phenomenon named "antipodal duality." In 2022, Lance and collaborators first computed the form factor to eight loops, and in 2023 used antipodal duality to work backward from it to the eight-loop amplitude. Since then, Lance planned to continue completing the ninth loop using this method.

The Anthropic team did not notify scholars in the relevant field in advance, contacting Lance only after the computation was done. Matt also felt somewhat caught off guard when he learned of it, and afterward suggested that Anthropic should contact relevant scientists in advance if it wanted to challenge other problems on the list.

The Lance team had originally been using a small model they trained themselves to try to predict results at higher loop counts, and they had said they were capable of verifying any candidate answer given by a machine. Lance worked backward from the amplitude to the form factor to check Claude's conclusion, and the two matched completely. Lance also noticed that everything Claude used were methods he and his collaborators had developed over many years, and the final result was presented in the format they had previously agreed upon. He believes Claude's understanding of their 2019 and 2023 papers may surpass that of everyone outside his co-authors.

Lance also stated that a great many details in the construction process had never been written into the papers, and Claude lacked references and had to write all the code from scratch. The biggest difficulty in this computation was actually reliability: the entire computational pipeline is very fragile, and any single error would cause total failure, and it would be very hard to locate where the problem occurred.

In China, in mid-September, researcher He Song and his team at the Institute of Theoretical Physics, Chinese Academy of Sciences, released a dataset on the open research data platform Zenodo, publicly disclosing the symbols and complete functions of amplitudes from two loops to nine loops. He Song has long worked on scattering amplitudes, and his team has for many years used the symbolic bootstrap method to study N=4 super Yang-Mills theory. In November 2025, He Song and collaborators published the result for the five-loop symbol of the seven-particle amplitude.

In the latest research, the He Song team used GPT-6 to compute some of the constraint conditions, while the overall computational framework was still personally built by human researchers. Based on this, Matt believes nine loops is not an unreachable goal. Subsequently, Lance, He Song, and their respective collaborators will continue to write papers to systematically explain and analyze the results, and AI's role temporarily comes to a pause.

When Matt issued the challenge, he expected to see AI find some unexpected new approach, but judging from the results, Claude still followed predecessors' mature methods, merely investing somewhat more computing power. Matt wrote that if anyone still thinks AI is riddled with errors and useless, this should change their view.

The speed of AI's development is also worth noting. In March of this year, in the physics research case released by Anthropic, AI was still like a student needing hand-holding and frequently making mistakes; half a year later, it was capable of independently completing a frontier computation usually undertaken by top experts. Matt reminds the public that when predicting AI's future, one should not assume that the present is the limit of capability and cost.

In Lance's view, the fact that a large model can fully execute a complex pipeline and schedule computing power is a remarkable achievement; when it can propose new physical principles and insights before humans, that will truly be worth physicists' reflection. Matt agrees with this, writing that if you cannot solve a problem yourself, but you believe an expert in another field or someone better at programming could solve it, then AI most likely can solve it too. But conversely, problems like quantum gravity depend on theoretical choices rather than technical capability, and fields that rely on experiments are a different matter. The problems listed on the challenge list still remain at the level of "toy models"; real-world amplitude calculations are closely related to collider physics and gravitational wave physics, with more participants and fiercer competition, and ready-made breakthroughs may be fewer. Matt suggests researchers in these fields act quickly to test whether existing AI research frameworks can complete their current computations in one go, while thinking in advance about how to verify the results. He estimates that AI might advance one more loop with a reasonable budget.