AI Model Claude Helps Physicists Crack 10-Year-Old Jamming Conjecture in Complex Systems

Physicists and AI model Claude 'collaborate' to prove a 10-year-old jamming conjecture

A mathematical problem that had remained unsolved for more than 10 years in the physics of complex systems has finally been resolved through an unusual collaboration: one involving two theoretical physicists and an artificial intelligence system. In a study published in the Journal of Statistical Mechanics: Theory and Experiment, Giorgio Parisi, Nobel Prize winner in physics, and Francesco Zamponi, physicist at LaSapienza University of Rome, show how the AI model Claude contributed to finding the proof of a mathematical relation that had resisted researchers' efforts for years. Beyond its scientific significance, the result offers a concrete glimpse into how artificial intelligence is transforming the work of researchers. In physics, jamming describes the formation of a kind of "traffic jam" of particles: A system that is initially fluid suddenly becomes rigid while remaining disordered. Originally introduced to describe materials such as foams and granular matter, the concept has proved surprisingly general and is now also used in fields such as neuroscience and artificial intelligence. In 2014, Parisi, emeritus professor at LaSapienza University of Rome and recipient of the 2021 Nobel Prize in physics, Zamponi, professor of physics at LaSapienza University of Rome, and collaborators developed a theoretical description of jamming and noticed a surprising relationship: Two mathematical parameters of the model, denoted by a and b, always added up to 1, as numerical calculations showed with extraordinary accuracy. A surprising relationship This relationship, Zamponi explains, yields the same physical laws obtained through a different theoretical approach to jamming developed almost simultaneously by French physicist Matthieu Wyart (EPFL, Lausanne). In other words, it suggests that two very different ways of describing the phenomenon actually lead to the same conclusions. The result emerged clearly from numerical calculations from the beginning, but no one could explain why it was true. For years, researchers searched for a mathematical proof of the relation, convinced that some deeper structure of the theory lay behind its apparent simplicity. A persistent obsession After several unsuccessful years, the problem gradually faded into the background. Not for Parisi, however. "It really bothered him that we had never managed to prove it," Zamponi recalls. When the first generative AI models began to appear, Parisi identified this old problem as an ideal test case. Claude was chosen because it "seemed to have somewhat more advanced mathematical reasoning abilities," Zamponi says. The problem, after all, was well-defined: a clear conjecture, relatively simple mathematics, and an answer that was known numerically but had never been formally proved. The initial prompt was not to find the proof. Parisi asked the model to reproduce the numerical calculations developed by the group more than a decade earlier, in order to understand how far it could go in tackling a real mathematical problem. Once Claude was able to reproduce the result, the researchers' next question came almost naturally: If a+b equals 1, can you also prove why? "Quite quickly, Claude came up with an initial idea that was essentially correct," Zamponi says. The proof still contained errors and required several rounds of verification and revision by the authors, but the underlying intuition turned out to be the right one. Yet the surprise was not only the AI's result. For years, the researchers had been searching for a deep explanation of the relation, imagining that it concealed a new mathematical structure or an unknown symmetry. "We were hoping this would reveal some new understanding of the equations," Zamponi explains. Instead, the solution turned out to be much simpler: "The answer was right there, and we simply hadn't seen it." The proof therefore confirms that two very different theoretical approaches to jamming, developed independently by Parisi and collaborators on the one hand and Wyart and collaborators on the other, do in fact lead to the same physical laws.

Claude's Solution to Decades-Long Math Mystery Is 'Essentially Correct,' Physicists Say

As typically is the case with AI, whether it truly adds any value to human work depends on the user's intentions. For science, that means that, technically, AI can try offering an answer to a question, but it's up to the experts to check its work and decide if there's more to be explored. A couple years ago, physicists Giorgio Parisi and Francesco Zamponi reached a stalemate in their attempt to solve the jamming problem, a mathematical issue in physics concerning systems that suddenly turn rigid while remaining disordered. The pair from La Sapienza University of Rome in Italy tried asking the AI model Claude to come up with a solution. Claude's initial proof reportedly had a lot of errors, but the underlying approach appeared to be a step in the right direction. Parisi and Zamponi pursued this idea and landed on a surprisingly simple resolution, showing their reasoning in a paper published today in the Journal of Statistical Mechanics: Theory and Experiment. "Quite quickly, Claude came up with an initial idea that was essentially correct," Zamponi said in a statement. "The answer was right there, and we simply hadn't seen it." A mathematical traffic jam In physics, jamming refers to a process in which density increases in a granular material (think of a children's ball pit), resulting in a system becoming rigid, sort of like a "traffic jam" of particles. Back in 2014, Parisi, Zamponi, and other collaborators mathematically described jamming, finding in the process that two parameters of the model would always add up to one. "The result emerged clearly from numerical calculations from the very beginning, but no one could explain why it was true," according to the statement. "For years, researchers searched for a mathematical proof of the relation, convinced that some deeper structure of the theory lay behind its apparent simplicity." A fresh pair of eyes, kind of With quick advances in generative AI models, Parisi and Zamponi wondered if Claude's relatively more advanced mathematical reasoning abilities could offer anything useful for the jam in the jamming conjecture. Initially, the pair asked Claude to replicate the group's numerical calculations, then challenged the AI to try and come up with a proof for the two parameters always adding up to one. "We were hoping this would reveal some new understanding of the equations," Zamponi explained. In short, the proof "contained errors and required several rounds of verification and revision by the authors." However, the researchers were able to build upon the basic premises of Claude's suggestions to arrive at a more solid proof. AI and 'impossible' problems AI's increased use in mathematics seems to invoke both excitement and concern for experts. In an interview with Gizmodo, Princeton mathematician Will Sawin said that AI is definitely effective at searching the literature and finding patterns that humans might not have noticed before. In other words, it's not necessarily that an AI model generated an entirely novel idea that humans couldn't have found on their own, at least for now. Something similar appears to have been the case for Parisi and Zamponi. In the paper, they write that it's "difficult to say" why they hadn't realized what Claude seemingly did. However, the human pair admitted that they were "looking for something deeper" and overlooked a more "conceptually simple case," which Claude's suggestion pointed them towards.