g (Γ , φ ), does not change significantly. For volume fraction much larger than φ$_{c}$ , the system’s input pressure 〈 p ( φ ) 〉 dyn reaches the plateau at low α of the function 〈 p ( α, φ ) 〉 ens (see Fig. 10) and the corresponding α ( φ ) becomes much smaller (the angoricity A ( φ ) becomes much larger), leading to large errors in the value of A as φ becomes large. This might explain the plateau found in A when ( φ − φ$_{c}$ ) > 2 × 10 − 2 as shown in Fig. 2B. Using α ( φ ) for each volume fraction, we calculate 〈 Z 〉 ens by: 〈 Z ( φ ) 〉 ens = ∫ ∞ 0 Zg (Γ , φ ) e − α $^{Γ}$dΓ ∞ 0 g (Γ , φ ) e − α $^{Γ}$dΓ = ∑ Ze − α Γ e − α Γ , (17) the average force 〈 F 〉 ens by: 〈 F ( φ ) 〉 ens = ∫ ∞ 0 F g (Γ , φ ) e − α $^{Γ}$dΓ ∞ 0 g (Γ , φ ) e − α $^{Γ}$dΓ = ∑ F e − α Γ e − α Γ , (18) where F is the average force for each ensemble packing and the force distribution P$_{ens}$ ( F/F ) by: P$_{ens}$ ( F/F ) = ∫ ∞ 0 P ( F/F ) g (Γ , φ ) e − α $^{Γ}$dΓ ∫ ∞ 0 g (Γ , φ ) e − α $^{Γ}$dΓ = ∑ P ( F/F ) e − α Γ ∑ e − α Γ . (19) Equations (17)–(19) are then compared with the dynamical measures for a test of ergodicity in Fig. 3 in the main text. E. Entropy Calculation Here we present the calculation of the “jamming temperature” T$_{J}$ and the corresponding jamming “heat” capacity C$_{J}$ . From the power-law relation p = Γ /V ∝ ( φ − φ$_{c}$ ) $^{a}$, we have: ln p = ln p$_{0}$ + a ln( φ − φ$_{c}$ ) , (20) where p$_{0}$ is the constant depending on the system and the slope tan θ = a . Figure 4 indicates that the jammed system always remain at the positions of maximal entropy δS = 0 in the direction ( − sin θ ,cos θ ), perpendicular to the jamming power-law curve. In order to further analyze this result, we plot the entropy distribution along the direction ( − sin θ ,cos θ ) in Fig. 12. We see that the entropy of the corresponding jammed states remains at the peak of the distributions along ( − sin θ ,cos θ ), verifying the maximum entropy principle in this particular direction. We notice that some deviations are found in the vicinity of φ$_{c}$ . The maximization of entropy is not on Γ or V alone, but on a combination of both. This means 12