II. OTHER ASPECTS OF THE RCP TRANSITION A. Finite size analysis B. Isostatic and geometrical coordination numbers and symmetry breaking 19 It is important to determine the finite size effects of our results. Figure 5A shows the results for smaller systems of 500 particles compared with 10,000 spheres system used in Fig. 1. We find that both plots are consistent with each other. While Fig. 1 shows the average of Z$_{j}$ over the LS packings, Fig. 5A shows each point representing a single packing obtained with the indicated algorithms for different system sizes. We find that the transition is similar over the different protocols. While the isostatic coordination number has been well documented at RCP [4, 14, 18], the possibility of states with Z$_{j}$ = 6 along the coexistence region with φ$_{rcp}$ < φ$_{j}$ < φ$_{melt}$ requires more elaboration [6]. We recall, however that the limiting condition Z$_{iso}$ = 6 is necessary, but not sufficient, for a rigid isostatic aggregate: a kinematic condition for rigidity must hold [18] where the functions that describe the connections between the centers of contacting particles are independent. The presence of crystal-like regions suggests that this condition may not be satisfied. Thus, the packings with Z$_{j}$ = 6 in the coexistence region are not necessarily isostatic, except exactly at the freezing point. It is interesting to understand the geometrical rearrangements occurring during the RCP transition in light of the fact that the packings enter the coexistence region from RCP by keeping Z$_{j}$ = 6 constant. The particles modify the positions of the 6 contacts in average at RCP to create crystal-like regions without creating new contacts or destroying old ones. This implies that the arrangements of particles are such that particles in the second coordination shell come closer to the central particle and contribute to the first coordination shell, yet without producing a new contacting force since Z$_{j}$ is kept at 6 in the entire coexistence region. The new particles moving into the first coordination shell can be considered in geometrical contact but carrying no force. Thus following [7] we introduce the idea of geometrical contact, z$_{g}$ , as those particles in the first coordination shell that do not provide any force but still are close enough to the central particle to contribute to the geometrical contact network. The geometrical coordination number z$_{g}$ is different from the mechanical