Appendix A first order phase transition at the random close packing of hard spheres Yuliang Jin and Hern´ an A. Makse Here, we describe the details of the MD simulations (Section I), geometrical interpreta- tions of the transition (Section II), and the calculations leading to the descriptive (Section III) and the thermodynamic (Section IV) view of the RCP transition. We use computer simulations to obtain jammed packings containing N = 10 , 000 mono- disperse spheres of radius R = 100 µ m with periodic boundary conditions. We first apply a modified Lubachevsky-Stillinger (LS) algorithm [11, 12] to generate packings of densities up to ∼ 0.72. In the algorithm, a set of random distributed points grow into nonoverlaping spheres at a fixed expansion rate γ . The spheres are considered perfectly elastic and evolve in time according to Newtonian dynamics. The configurations eventually arrive at out-of- equilibrium states with a diverging collision rate and a density φ$_{i}$ . Practically, we set the reduced kinetic pressure of the fluid defined as p = P V/Nk$_{B}$T to be 10 12 [12] as a criteria of the diverging collision rate. The final packing configurations depend on the expansion rate γ : large values of expansion rate result in random packings with very low packing densities, while small values of expansion rate result in packings with higher densities. Although the packings obtained from the modified LS algorithm are considered as “ge- ometrically jammed”, they are not jammed in the mechanical sense since the particles do not carry on any forces. In order to study mechanical stable packings characterized by a jamming transition, we model the microscopic interaction between deformable grains by the nonlinear Hertz-Mindlin normal and tangential forces [7, 14]. We use configurations from the modified LS as the starting point, φ$_{i}$ , and apply molecular dynamics to simulate Newton equations for the evolution of the particles following algorithms in [7, 14]. The aim of this part of the protocol is to generate mechanically stable jammed packings at the jamming transition φ$_{j}$ . For the packings obtained by the LS algorithm, we first reset the velocities of the particles to zero. At this point the system stress σ and mechanical coordination number Z$_{j}$ are zero since there is no deformation or overlapping between the 16 I. ALGORITHM