A proof of an identity for the critical exponents of jamming

Within the full replica-symmetry-breaking (fullRSB) solution of dense hard spheres in infinite dimension, Charbonneau, Kurchan, Parisi, Urbani, and Zamponi (CKPUZ; J.Stat.Mech.P10009, 2014) introduced three critical exponents $a$, $b$, $c$ governing the matching region of the fullRSB profile near the jamming transition. These exponents satisfy two scaling relations. The first, $b=(1+c)/2$, was established analytically by the diffusion-drift balance in the scaling ansatz. The second, $a+b=1$, was observed numerically to arbitrary precision but could not be proven. The exponents $a,b,c$ of the scaling fullRSB ansatz are related to the physical exponents $\alpha, \theta, \kappa$ that control the gap, force, and overlap distributions by the relations $\alpha=a/b$, $\theta=(c-a)/(b-c)$, $\kappa=c+1$. Crucially, the relation $a+b=1$ yields the scaling relations $\alpha=1/(2+\theta)$ and $\kappa=2-2/(3+\theta)$ predicted on independent grounds by the mechanical-marginal-stability arguments of Wyart and collaborators. Here, we give an analytic proof of the identity $a+b=1$ from the scaling fullRSB equations. The proof was obtained through interaction with Claude (Sonnet 4.6 and Opus 4.7) and verified by us.