On September 25 (local time), Anthropic announced that Claude had completed a nine-loop calculation of six-particle scattering amplitudes in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory, surpassing the previous eight-loop record set in 2023 by physicists including Lance Dixon of the SLAC National Accelerator Laboratory.
Dixon himself participated in verifying Claude’s results; he noted that he had previously considered a direct nine-loop calculation too difficult and was surprised by Claude’s ability to accomplish the task directly.
According to Anthropic, Claude employed two distinct methods to complete the calculations—processes that required no continuous researcher intervention. Each method cost between $1,000 and $2,000 to run; one of the calculations utilized 96 CPUs running continuously for approximately one week.
Planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory is a highly idealized theoretical model used primarily to test computational methods in particle physics, rather than a complete physical model of the real world.
This achievement demonstrates that AI can independently perform complex calculations—tasks that previously required top-tier theoretical physicists and vast computational resources—provided that a methodological framework has already been established by experts. Dixon observed that while Claude primarily utilized tools and methods developed by human physicists over many years, the truly groundbreaking aspect lies in AI beginning to propose new physical principles and insights.
Leading AI model developers have recently been intensifying their efforts in scientific fields, including mathematics, the biological sciences, and physics.
Specific examples include Anthropic announcing that Claude raised the lower bound for the proportion of Riemann zeta function zeros satisfying the Riemann Hypothesis from 41.6% to 67.2%, and that Claude completed a full Lean formalization of Fermat’s Last Theorem after running largely autonomously for 11 days. Additionally, following OpenAI’s announcement of an internal system’s solution to the Navier-Stokes problem—accompanied by Lean formalization code—the company revealed that an internal model, training since August 28, had solved over 100 long-standing open mathematical problems. Separately, on September 16, an anonymous user known as “Captain Sude” posted on social media and GitHub, claiming to have completed a proof of the Liouville version of the Goldbach Conjecture using OpenAI’s GPT-6 Astra model.
However, it is important to note that the subject of this study was not the original Goldbach Conjecture, but rather the weaker Liouville-Goldbach theorem. The original Goldbach Conjecture (the “strong” conjecture) posits that every even integer greater than 2 can be expressed as the sum of two prime numbers—often referred to as “1+1.” The distribution of prime numbers within additive structures remains one of the most fundamental challenges in number theory and has yet to be proven.
In contrast, the weaker Liouville-Goldbach theorem merely requires that the total number of prime factors of the two addends be odd; there is no requirement for the addends themselves to be prime numbers. The mathematical community’s primary interest and recognition remain focused on the original conjecture. The Liouville version is a derivative, weaker proposition; while its verifiability makes it a useful testing ground for AI capabilities, it is not regarded as a substantive advancement toward solving the original conjecture. Indeed, Captain Sude acknowledged that while the result might seem “significant” to those deeply obsessed with number theory, it cannot be compared in any meaningful way to the Millennium Prize Problems.
Furthermore, the technical report relied solely on formal mathematical verification (using Lean4)—where a machine checks every step of the logical derivation to ensure the internal chain of reasoning is free of flaws—without receiving confirmation from mathematics experts or OpenAI itself.
As AI makes rapid strides in the scientific realm, friction has begun to emerge between traditional academic evaluation systems and artificial intelligence technologies.
On September 11, a group of 25 Fields Medalists, including Terence Tao, co-signed an open letter titled “AI’s Serious Misalignment in Mathematics.” They argued that by using the “solution of open problems” as a benchmark for model capability, technology companies risk overlooking values that are more critical to mathematical research—such as understanding, originality, the transmission of knowledge, and the attribution of academic credit. In response, OpenAI has established an independent mathematics advisory group comprising nine mathematicians to assist in evaluating the significance of findings, coordinating their release, and addressing issues related to academic standards.
Leading AI companies are pushing the boundaries of scientific research at a pace far exceeding that of traditional academia; however, speed of verification is not the same as science itself. The challenge now facing traditional scientific institutions is how to accommodate a researcher capable of working around the clock without rest, and how to update academic norms accordingly.