Within the fullRSB solution developed over a decade ago by Charbonneau, Kurchan, Parisi, Urbani, and Zamponi (CKPUZ, 2014), three critical exponents — a, b, and c — describe the scaling behavior near the jamming point .

• One relation, b = (1 + c)/2, had already been proven analytically from the diffusion-drift balance in the scaling ansatz .
• The second relation, a + b = 1, was observed numerically to arbitrary precision for over a decade, but an analytic proof had remained elusive .

What the Proof Revealed

The identity a + b = 1 is derived directly from the scaling fullRSB equations. Crucially, this relation implies a = (1 - c)/2, which — together with b = (1 + c)/2 — means that all three exponents are determined by a single parameter c . This reduces the fullRSB solution's predictive degrees of freedom and solidifies the theory's internal consistency. It also independently yields the scaling relations α = 1/(2 + θ) and κ = 2 - 2/(3 + θ), previously predicted by the mechanical-marginal-stability arguments of Wyart and collaborators .

How Anthropic's Claude Contributed

The proof was not achieved through human insight alone. The paper explicitly states: "The proof was obtained through interaction with Claude (Sonnet 4.6 and Opus 4.7) and verified by us" .

Iteration and Collaboration

• Parisi and Zamponi engaged in an extended dialogue with Claude across approximately 40 conversation rounds .
• They used two Claude model versions: Sonnet 4.6 and Opus 4.7 .
• Claude did not produce a final polished proof in one shot. Instead, it iteratively derived nearly the complete proof through back-and-forth exchanges. The authors then reviewed, corrected, and verified each step .
• The full chat logs have been archived on Zenodo for public inspection .

Division of Labor

• Claude generated candidate algebraic manipulations, identified patterns in the fullRSB equations, and carried out the heavy symbolic reasoning .
• Parisi and Zamponi provided the conceptual framework, validated the mathematical correctness, and ensured the proof fit within the established physics . The preprint notes that Opus 4.7 essentially derived the proof by itself, with minimal supervision from the authors .

Publication and Significance

The preprint was posted to arXiv on June 2, 2026 (updated July 2, 2026) under the title "A proof of an identity for the critical exponents of jamming" . The authors list is Giorgio Parisi and Francesco Zamponi; Claude is acknowledged as the AI tool used in the methodology rather than a co-author.

News coverage in outlets including Il Fatto Quotidiano, Correio Braziliense, 36Kr, and Digg has framed this as a landmark example of a Nobel laureate using a large language model as a genuine research collaborator to break a long-standing theoretical impasse .

Why It Matters

The result closes a decade-old gap in the fullRSB theory of jamming — a fundamental problem in the physics of glasses, granular materials, and disordered systems. It also serves as a high-profile demonstration of how LLMs can contribute to advanced theoretical physics, not merely as assistants but as active participants in the derivation of non-trivial mathematical proofs.

Within the fullRSB solution developed over a decade ago by Charbonneau, Kurchan, Parisi, Urbani, and Zamponi (CKPUZ, 2014), three critical exponents — a, b, and c — describe the scaling behavior near the jamming point .

• One relation, b = (1 + c)/2, had already been proven analytically from the diffusion-drift balance in the scaling ansatz .
• The second relation, a + b = 1, was observed numerically to arbitrary precision for over a decade, but an analytic proof had remained elusive .

What the Proof Revealed

The identity a + b = 1 is derived directly from the scaling fullRSB equations. Crucially, this relation implies a = (1 - c)/2, which — together with b = (1 + c)/2 — means that all three exponents are determined by a single parameter c . This reduces the fullRSB solution's predictive degrees of freedom and solidifies the theory's internal consistency. It also independently yields the scaling relations α = 1/(2 + θ) and κ = 2 - 2/(3 + θ), previously predicted by the mechanical-marginal-stability arguments of Wyart and collaborators .

How Anthropic's Claude Contributed

The proof was not achieved through human insight alone. The paper explicitly states: "The proof was obtained through interaction with Claude (Sonnet 4.6 and Opus 4.7) and verified by us" .

Iteration and Collaboration

• Parisi and Zamponi engaged in an extended dialogue with Claude across approximately 40 conversation rounds .
• They used two Claude model versions: Sonnet 4.6 and Opus 4.7 .
• Claude did not produce a final polished proof in one shot. Instead, it iteratively derived nearly the complete proof through back-and-forth exchanges. The authors then reviewed, corrected, and verified each step .
• The full chat logs have been archived on Zenodo for public inspection .

Division of Labor

• Claude generated candidate algebraic manipulations, identified patterns in the fullRSB equations, and carried out the heavy symbolic reasoning .
• Parisi and Zamponi provided the conceptual framework, validated the mathematical correctness, and ensured the proof fit within the established physics . The preprint notes that Opus 4.7 essentially derived the proof by itself, with minimal supervision from the authors .

Publication and Significance

The preprint was posted to arXiv on June 2, 2026 (updated July 2, 2026) under the title "A proof of an identity for the critical exponents of jamming" . The authors list is Giorgio Parisi and Francesco Zamponi; Claude is acknowledged as the AI tool used in the methodology rather than a co-author.

News coverage in outlets including Il Fatto Quotidiano, Correio Braziliense, 36Kr, and Digg has framed this as a landmark example of a Nobel laureate using a large language model as a genuine research collaborator to break a long-standing theoretical impasse .

Why It Matters

The result closes a decade-old gap in the fullRSB theory of jamming — a fundamental problem in the physics of glasses, granular materials, and disordered systems. It also serves as a high-profile demonstration of how LLMs can contribute to advanced theoretical physics, not merely as assistants but as active participants in the derivation of non-trivial mathematical proofs.