AI model helps physics Nobel laureate out of a decade-old mathematical jam

Two theoretical physicists, among them a Nobel laureate, have solved a perplexing mathematical problem that had stumped them for ten years with the assistance of Claude, a large language model (LLM) developed by US-based company Anthropic. The significance of this breakthrough extends beyond fundamental physics, as it also demonstrates how AI is transforming the landscape of research in certain fields.

The enigma that kept Francesco Zamponi and Giorgio Parisi, from Italy's Sapienza Università di Roma, up at night, pertains to the field of complex systems and relates to a phenomenon known as jamming. To put it simply, jamming occurs when a fluid system abruptly transitions from a fluid state to a rigid, disordered state. To visualize this, imagine a box filled with floating spheres in zero gravity.

When there are few spheres, they have ample room to move, much like a classical gas. However, as more spheres are added, they eventually reach a critical density, locking up the system and rendering it unable to move.

Expanding the problem into more abstract terms, Zamponi explains that jamming is a highly general mathematical issue called constraint satisfaction. In this context, the spatial positions of the spheres serve as variables or degrees of freedom, while the strict rule that no two spheres can overlap acts as the constraint. As more variables are added, the difficulty of satisfying all constraints simultaneously increases.

This mathematical mapping has practical applications in neuroscience and artificial intelligence. In machine learning models or biological brains, synaptic weights represent degrees of freedom, while data classifications or memories that need to be learned serve as constraints. Just as physical spheres undergo a jamming transition, so do artificial neural networks.

In a "liquid" phase, the network efficiently configures its weights to satisfy all constraints, enabling perfect data classification. However, when faced with excessive data, it hits a wall, corresponding to the "solid" phase. It can no longer meet all conditions and begins making errors.

Interestingly, in 2014, Parisi and Zamponi uncovered a surprising connection in jamming theory. By collaborating with colleagues in the US and France, they discovered that two mathematical parameters, a and b, related to the scaling of contact forces and inter-sphere gaps as the system reaches the jamming point, mysteriously add up to 1. This parameter pair is crucial for characterizing the packing's physical structure, yet no formal mathematical proof of a + b = 1 had been achieved until now.

Parisi suggested using a generative AI to tackle the problem, as it was well-suited for machine assistance. Claude was selected due to its advanced coding and mathematical reasoning capabilities. However, the researchers initially prompted Claude to replicate their numerical calculations from a decade ago. Once it successfully reproduced those results, they asked Claude: "If a + b = 1, can you prove why?"

Claude initially provided an almost correct conceptual path but contained minor mathematical errors. After several iterations and verifications, the core intuition belonged to the AI, revealing that the solution was simpler than they had anticipated. The new proof connects the infinite-dimensional abstract theory with a more concrete physical framework developed by physicist Matthieu Wyart and his team at EPFL.