Ensemble calculations.— The density of jammed states g (Γ , φ ) is calculated in the framework of the potential energy landscape (PEL) formulation introduced by Goldstein [18] and Stillinger-Weber [19, 20] to describe supercooled liquids. In the case of frictionless jammed systems, the mechanically stable configurations are defined as the local minima of the potential energy surface (PES) of the system [4, 11] (see Fig. 1 inset for a schematic representation). In the simulations, two spherical soft particles in contact interact via a normal Hertz force [6, 21], F$_{n}$ ∝ ( δr ) $^{δ}$, where δr is the normal overlap between the spheres under deformation and δ = 1 . 5, in a periodically repeated cube [the interparticle potential energy is E ∝ ( δr ) δ $^{+1}$, see Materials and Methods Section III A]. The Hertz potential is chosen for its general applicability to granular materials. The results are expected to be independent of the form of the potential. Details of the algorithms [22, 23] to find the local minima of the PES (zero-order saddles) are in the Materials and Methods Section III B. Figure 1 shows g (Γ , φ ) versus Γ for different volume fractions. MD calculations.— The pressure 〈 p 〉 dyn as a function of φ is calculated by performing MD simulations. Packings are prepared by compressing a gas of particles from an initial (unjammed) low volume fraction to a final jammed state. This procedure simulates a dy- namical packing preparation [24]; details appear in Materials and Methods Section III C. We obtain (Fig. 2A) where φ$_{c}$ = 0 . 6077 is the volume fraction corresponding to the isostatic point J [4] following Eq. (3) and p$_{0}$ = 10 . 8MPa. This critical value and the exponent, a = 1 . 65, are slightly different than the values obtained for larger systems ( a = δ )[4]. However, our purpose is to use the same system in the dynamical calculation and the exact enumeration for a proper comparison. Calculation of angoricity.— For each φ we use g (Γ , φ ) to calculate 〈 p ( α ) 〉 ens by Eq. (1). Then, we obtain α ( φ ) by setting 〈 p ( α, φ ) 〉 ens = 〈 p 〉 dyn for every φ (see Figs. 9 and 10 and Materials and Methods Section III D). The resulting equation of state α ( φ ) is plotted in Fig. 2B and shows that the angoricity follows a power-law, near φ$_{c}$ , of the form: where the angoricity exponent is γ = 2 . 5. The result is consistent with γ = δ + 1 . 0, suggesting that A ∝ Γ ∝ F$_{n}$r . Angoricity is a measure of the number of ways the stress A ∝ ( φ − φ$_{c}$ ) $^{γ}$, (5) 〈 p 〉 dyn = p$_{0}$ ( φ − φ$_{c}$ ) 1 . 65 , (4) 4