C. MD Generation D. Angoricity Calculation In order to analyze numerical results, we perform MD simulations to obtain Z$_{dyn}$ , p$_{dyn}$ and φ$_{dyn}$ , which are herein considered real dynamics. The algorithm is described in detail in [6, 16, 24]. Here, a general description is given: A gas of non-interacting particles at an initial volume fraction is generated in a periodically repeated cubic box. Then, an extremely slow isotropic compression is applied to the system. The compression rate is Γ$_{0}$ = 5 . 9 t − 1 0 , where the time is in units of t$_{0}$ = R √ ρ/G . After obtaining a state for which the pressure p is a slightly higher than the prefixed pressure we choose, the compression is stopped and the system is allowed to relax to mechanical equilibrium following Newton’s equations. Then the system is compressed and relaxed repeatedly until the system can be mechanically stable at the predetermined pressure. To obtain the statistical average of Z$_{dyn}$ and φ$_{dyn}$ , we repeat the simulation to get enough packing samples having statistically independent random initial particle positions. Here, 250 independent packings are obtained for each fixed pressure (see Fig. 8). Since we obtain g (Γ , φ ) and 〈 p 〉 dyn for each volume fraction φ , we can calculate the inverse angoricity α by Eq. (1). The pressure 〈 p ( α, φ ) 〉 ens for a given φ is a function depending on α as: Figure 9 shows the result of the numerical integration of Eq. (16) for a particular φ = 0 . 614 as a function of α using the numerically obtained g (Γ , φ ) from Fig. 1. To obtain the value of α for this φ , we input the corresponding measure of the pressure obtained dynamically 〈 p ( φ ) 〉 dyn and obtain the value of α as schematically depicted in Fig. 9. The same procedure is followed for every φ (see Fig. 10) and the dependence α ( φ ) is obtained. The result is shown in Fig. 2B in the main text. We also check the inverse angoricity α ( φ ) using g (Γ , φ ) for different searching days (see Fig. 6) to ensure the accuracy and convergence to the proper value. From Fig. 11, we can see that, after 10 days searching, α ( φ ) is stable due to the fact that the density of state, 〈 p ( α, φ ) 〉 ens = ∫ ∞ 0 pg (Γ , φ ) e − α $^{Γ}$dΓ ∞ 0 g (Γ , φ ) e − α $^{Γ}$dΓ = ∑ pe − α Γ e − α Γ . (16) 11