A mathematical relation that resisted two theoretical physicists for 10 years — the identity a + b = 1 in the theory of jamming — has now been formally proved with help from Claude, the large language model built by Anthropic. Giorgio Parisi, who shared the 2021 Nobel Prize in Physics, and Francesco Zamponi of the Sapienza Università di Roma published the proof in the Journal of Statistical Mechanics: Theory and Experiment alongside the full transcript of their conversation with the model.

The unsolved problem dated to 2014, when Parisi and Zamponi, working with colleagues in the US and France, spotted that two parameters governing the scaling of contact-force and gap distributions at the jamming point mysteriously summed to unity in numerical calculations. The parameters characterize the physical structure of disordered packings. A decade of attempts at a formal proof failed.

What is jamming, and why did the proof matter?

"Jamming describes a sudden transition at which a fluid system becomes completely rigid, yet remains disordered," Zamponi explains. Picture spheres floating in zero gravity: at low density they behave like a gas, but past a critical density the system locks up. If compressed or filled quickly, it freezes into a disordered packing rather than a crystal.

The problem reaches beyond granular physics. Jamming maps onto constraint satisfaction, a general mathematical framework in which sphere positions act as variables and the no-overlap rule as the constraint. The same structure appears in neuroscience and machine learning, where synaptic weights are the degrees of freedom and the data to be learned are the constraints.

"Just like physical spheres, an artificial neural network undergoes a jamming transition," Zamponi observes. "In the 'liquid' phase, the network easily configures its weights to satisfy all constraints... But when tasked with too much data, it hits a wall, corresponding to the 'solid' phase."

How did the researchers use the model?

Asking generative AI for help was Parisi's idea. The problem fit machine assistance well: a clear conjecture, a known numerical answer, no analytical proof. The pair chose Claude for what they judged to be stronger coding and mathematical reasoning than comparable models.

They did not request the proof outright. "We first prompted Claude to replicate the numerical calculations our group had developed a decade ago," Zamponi says. "Once it had successfully reproduced those exact results, we took the natural next step and asked it: 'If a + b = 1, can you prove why?'"

The model's first conceptual path was essentially correct but contained minor mathematical errors and required several rounds of iterative refinement and verification by the physicists. Zamponi credits the AI with the decisive insight: "We had spent years looking for a complicated solution – like deeply hidden structural symmetry – but Claude showed us that the solution was far simpler; it was right there in front of us, but we had just missed it."

The proof also bridges two theoretical frameworks. It connects the pair's infinite-dimensional theory — more abstract but rigorous — with a concurrent framework built by Matthieu Wyart's team at the École Polytechnique Fédérale de Lausanne, which rests on concrete physical notions but carries assumptions. Zamponi says the result confirms both starting points yield the same physical laws.

What does this mean for research practice?

Zamponi frames the stakes broadly: generative AI could prove as consequential as the industrial revolution or the birth of the Internet. As a "telescope for the mind," it could let scientists test ideas at unprecedented speed and lower barriers between disciplines by unlocking literatures that take too long to master.

He also flags risks for research managers and funders. LLMs, he argues, encourage poor-quality, pseudo-scientific output that strains peer review. "We desperately need to figure out how to filter and review this influx," he says, adding that graduate and undergraduate curricula need new pedagogical frameworks for integrating such tools responsibly.

Transparency is the pair's proposed standard for AI-assisted results. "If an AI sparks a genuinely new idea, we believe that its 'thought processes' should be public record," Zamponi says — the reason the full Claude conversation appears alongside the paper.

What comes next?

Zamponi is already applying the same workflow to random sequential absorption (RSA) of hard hyperspheres, a classic protocol that adds spheres one by one at random positions and never lets them move. Unlike jamming, RSA is irreversible and reaches a saturation limit rather than a collective transition. He describes it as a vital tool for understanding void-space geometry and packing efficiency, with direct links to optimal error-correcting codes and the curse of dimensionality.