Jamming’s Missing Identity: How Parisi and an AI Found the Proof
09 Jun 2026, Yanjiang
A proof linking the gap and force exponents in jammed glass was found by Parisi and Zamponi, aided by an AI language model.
Imagine trying to prove that two seemingly unrelated properties of a jammed pile of oranges — the way inter‑particle gaps vanish and the way contact forces spread — are governed by the same underlying mathematical number. That number, a combination of exponents, is not a free parameter but something fixed by the internal logic of the glassy state. For years, physicists had every reason to believe the identity held. Numerical experiments screamed its truth. Yet the analytic proof remained stubbornly out of reach. Now, in a preprint (arXiv:2606.03300), Giorgio Parisi and Francesco Zamponi have closed that gap — with an unexpected collaborator at their side.
Parisi, a Nobel laureate in physics for his work on complex systems, and Zamponi, his long‑time colleague at Sapienza University of Rome and the International Research Center of Complexity Sciences at Beihang University in Hangzhou, have been exploring the deep mathematics of jamming for more than a decade. The jamming transition — the point at which a disordered collection of particles suddenly locks into a rigid structure — is a universal phenomenon, appearing in granular materials, emulsions, and even glasses. Understanding it requires not simulating real‑world grains but ascending to a dimension where the problem becomes analytically tractable: infinite spatial dimension.
In that theoretical playground, hard spheres remain strictly impenetrable — but their collective physics simplifies enough to become analytically tractable. Their physics is encoded in the structure of a replicas’ order parameter, a profile called full replica symmetry breaking (fullRSB). The idea is delicate: to compute the free energy of a glass, one imagines infinitely many copies of the system, then lets the copies interact in a controlled way. Near jamming, this replica profile develops a characteristic shape, governed by three numbers — exponents labelled a, b, and c — that control how the overlap between replicas scales away from the transition.
For the fullRSB description of jamming to be internally consistent, these exponents must satisfy precise relations. One relation, b = (1 + c)/2, emerged from a neat diffusion‑drift balance in the equations and was proven quickly. The other, a + b = 1, was something else entirely. “It was observed numerically to arbitrary precision,” Parisi and Zamponi write, “but could not be proven.” And this particular identity matters enormously, because it links the replica exponents to the physical exponents that experimentalists can actually measure: the gap distribution exponent alpha, the force distribution exponent theta, and the structural exponent kappa. With a + b = 1, one recovers the predictions alpha = 1/(2 + theta) and kappa = 2 ‑ 2/(3 + theta) — predictions that had already been argued on independent grounds by Wyart and collaborators using mechanical‑marginal‑stability ideas, but that now sat on a unified microscopic foundation. Without the proof, the entire replica‑based theory of jamming felt like a cathedral missing its keystone.
So how did the proof finally emerge? Not through a flash of human insight alone. The authors worked in dialogue with Claude, a large language model, specifically its Sonnet 4.6 and Opus 4.7 versions. The preprint states plainly that “the proof was obtained through interaction with Claude and verified by us.” One imagines a conversation in which the machine, having internalized the intricate equations of the fullRSB ansatz, suggested manipulations that a human might dismiss as too tedious or too far from intuition — and the humans, with their deep physical understanding, selected the ones that actually worked.
The proof itself is analytic, not numerical; it involves taking the scaling equations that define a, b, and c and rearranging them until the relation a + b = 1 drops out as an inevitable consequence. The authors have deposited their full conversation with the model in a public Zenodo repository, allowing anyone curious to trace exactly which steps came from Claude and which they supplied. But the outcome is a testament to a new mode of mathematical physics: not replacing the scientist, but augmenting her reach.
Think of it less like outsourcing the thinking to a black box, and more like having a tireless colleague who can explore every forking path of a formal manipulation, handing back a bouquet of promising algebraic identities for the human to inspect. The historical arc is striking: Parisi himself, decades ago, invented the machinery of replica symmetry breaking now being deployed, often through strokes of insight that seemed almost oracular. Now the oracle has a computational sidekick.
The proof’s significance extends beyond the specific identity. The jamming transition belongs to a broad class of critical phenomena in disordered systems that sit outside the traditional framework of equilibrium phase transitions. The exponents it yields are not mere academic curiosities; they control the mechanical response of real materials — why some powders flow and others clog, why certain colloidal gels suddenly stiffen. By securing the identity, the authors have tightened the theoretical net around these phenomena.
And the method? A proof found with machine assistance is still a proof. It must stand on its own logical legs, and in this case the authors have checked it exhaustively. The episode nudges us to reconsider what mathematical physics will look like when the tools of automated reasoning become as natural as the blackboard once was — not replacing the taste and intuition of a Giorgio Parisi, but giving them a faster hand.
— Yanjiang
Yanjiang is an online editor of LoomSci.com.
References
- Giorgio Parisi and Francesco Zamponi, A proof of an identity for the critical exponents of jamming, arXiv:2606.03300