Nobel Laureate and Claude Join Forces to Crack the “Jamming” Conjecture
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 between two theoretical physicists and an artificial intelligence system. In a study published in the Journal of Statistical Mechanics: Theory and Experiment (JSTAT), Nobel laureate Giorgio Parisi and physicist Francesco Zamponi show how Anthropic’s AI model Claude provided the key idea needed to prove a mathematical relation that had resisted researchers’ efforts for over a decade.
What Is Jamming?
The problem centers on “jamming”, a phenomenon where particles that move freely in a fluid-like way suddenly become rigid while still disordered. Picture a container filled with sand, bubbles, or grains: as the density rises, the material hits a critical point where movement stops, turning from a fluid into a solid that can withstand mechanical stress.
Originally introduced to describe foams and granular materials, the jamming concept has turned out to be surprisingly general. Today, it is also used to understand phenomena in neuroscience and artificial intelligence.
The Puzzle That Stumped Physicists for a Decade
In 2014, Parisi, who received the 2021 Nobel Prize in Physics for his work on complex systems, Zamponi, and their collaborators developed a theoretical model describing jamming. During this work, they found an unexpected relationship: two mathematical parameters in the model, denoted as a and b, always add up to 1.
Numerical calculations confirmed this relationship with remarkable accuracy, again and again. But there was a problem: no one could explain why it was true.
“We observed it was true, but we couldn’t prove it,” Zamponi explained.
The situation became even more intriguing when French physicist Matthieu Wyart and his group at EPFL reached the same relationship through a completely different theoretical approach around the same time. This suggested that two very different ways of describing jamming were actually leading to the same conclusions, but the mathematical bridge connecting them remained out of reach.
“It really bothered [Parisi] that we had never managed to prove it,” Zamponi recalled.
Enter Claude: From Code Monkey to Collaborator
When generative AI models began to appear, Parisi saw an opportunity. He chose Anthropic’s Claude because it “seemed to have somewhat more advanced mathematical reasoning abilities”.
The researchers did not ask Claude for a proof right away. Instead, Parisi’s first prompt was to have Claude write C++ code using the “shooting method” to solve a nonlinear differential equation with high precision. This was basically a programming task: verify the numerical result to sufficient accuracy.
Claude handled the coding, adjusted precision from double to quadruple precision, and pushed numerical results to many decimal places. At one point, Parisi even wrote the equation incorrectly, and Claude correctly identified that the flawed equation had no solution.
The Breakthrough
The real turning point came when Parisi gave Claude a new directive: “I can handle the rest myself. You should notice that a + b ≈ 1 with extremely high precision. Some have conjectured this relationship is exact. I want you to do an analytic calculation and prove it.”
From that moment, Claude’s role shifted from programmer to mathematical collaborator.
“Quite quickly, Claude came up with an initial idea that was essentially correct,” Zamponi said.
The proof’s core involved constructing a special auxiliary function that, after algebraic manipulation, led to a key identity. Combined with known physical conditions, this yielded the conclusion: a + b = 1.
Human Scientists Were Not Out of the Picture
While Claude provided the crucial insight, the proof was not flawless. The initial draft contained errors that required several rounds of verification and revision by the human researchers.
In one instance, Claude confidently used an extremum principle to argue that a certain function was never negative. Zamponi directly pointed out that the argument was wrong, as there was no contradiction at the minimum point. Claude acknowledged the mistake: “Your colleague is right…I made a sign error.”
The error correction went both ways. In another case, the researchers made a small mistake in an asymptotic calculation, and Claude caught it.
However, the truly decisive human contribution came when Parisi reframed the entire problem. He pointed out that Claude was trying to prove something that wasn’t generally true, since the equation had multiple solutions and most oscillated. The real question wasn’t whether the function was always non-negative, but whether there existed one non-negative solution.
Parisi then provided the conceptual path forward: instead of focusing on the limiting equation, return to the original equation and define a function that evolves with scale. If the evolution preserves non-negativity and starts from a non-negative initial condition, the proof succeeds.
Claude followed this guidance, translating it into a standard reaction-diffusion equation and applying a well-known extremum principle to complete the proof.
What This Reveals About AI-Assisted Research
The collaboration offers a concrete glimpse into how artificial intelligence is changing scientific research. The researchers documented the entire process, including Claude’s contributions, the errors it made, and the human corrections, and made it publicly available.
“The answer was right there, and we simply hadn’t seen it,” Zamponi reflected.
The surprising part was the simplicity of the solution. For years, researchers had been searching for a deep explanation, imagining the relationship hid a new mathematical structure or unknown symmetry. Instead, Claude’s proof showed that the solution was straightforward—humans had simply been looking in the wrong direction.
The Emerging Research Paradigm
This case shows a new model of scientific discovery: AI systems can now take part in the search for mathematical structures and proof construction, rather than just handling routine calculations. However, human scientists remain essential for:
- Setting the research question
- Identifying when the AI’s approach is flawed
- Reframing problems when the initial approach fails
- Providing strategic direction
- Validating final results
The partnership documented by Parisi and Zamponi suggests that the most productive path forward may not be AI replacing human scientists, but AI and humans working together as collaborators, each contributing their own strengths to the research effort.
by PAUL ZIMMER