A mathematical problem that had remained unsolved for more than ten 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 (JSTAT), 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, Giorgio Parisi, Emeritus Professor at LaSapienza University of Rome and recipient of the 2021 Nobel Prize in Physics, Francesco 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 one, as numerical calculations showed with extraordinary accuracy.
A surprising relationship
This relationship, explains Zamponi, co-author of the new study together with Parisi, 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 very 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 Giorgio 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," says Zamponi.
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 proven.
The prompt given initially 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 one, can you also prove why?
"Quite quickly, Claude came up with an initial idea that was essentially correct," says Zamponi.
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.
