confining stress, or equivalently in the hard-sphere limit, σ/Y → 0 $^{+}$) covering a range of φ$_{j}$ from φ$_{rlp}$ = 0 . 55 to crystallization at φ$_{fcc}$ = 0 . 74. 

The mechanical coordination number averaged over all the particles in a packing, Z$_{j}$ , characterizes different states of granular matter [2, 7, 15]. Therefore, our study begins by plotting Z$_{j}$ versus φ$_{j}$ for all generated packings. Figure 1 suggests the existence of a transition occurring at RCP evidenced by the abrupt plateau in Z$_{j}$ . This transition could be thought of as an analogue to the classical hard sphere liquid-solid phase transition in thermal equilibrium [16, 17]. Such an analogy becomes apparent if one identifies Z$_{j}$ of the jammed packing with the kinematic pressure of the equilibrium hard sphere system [6] and it is in agreement with a recent conjecture regarding the definition of RCP [8]. 

Figure 1 identifies two branches and a coexistence region: (i) An ordered branch of crystallized states with φ$_{j}$ ranging from 0.68 to a FCC lattice at 0.74. (ii) A disordered branch within 0 . 55 ∼ 0 . 64 which can be fitted with the statistical theory of [7]: φ$_{j}$ = Z$_{j}$/ ( Z$_{j}$ + $_{2}$√$_{3) as shown in the figure.}$ (iii) A coexistence region between 0.64 to 0.68 displaying a plateau at the isostatic coordination number, Z$_{iso}$ = 6 [4, 14, 18]. The intersection between the disordered branch and the coexistence line identifies the “freezing point” of the transition providing a definition of RCP. Using the theoretical results of [7], freezing occurs at Z$_{iso}$ = 6 and φ$_{rcp}$ = 6 / $_{(6 + 2}$√$_{3)}$ ≈ 0 . 634. The corresponding “melting point” appears at the other end of the coexistence at φ$_{melt}$ = 0 . 68, signaling the beginning of the ordered branch. Finite size analysis is shown in the Appendix-Fig. 5A: the results for 500 and 10,000 spheres are consistent with each other. Other geometric aspects of the transition are discussed in Appendix-Section II. 

To reveal the nature of the newly found phases we start with a descriptive viewpoint and then turn to a thermodynamic analysis to model the transition. In order to investigate if the concept of phase transition applies to the trend observed in Z$_{j}$ , one commonly looks at the global ( Q$_{l}$, W$_{l}$ ), and local ( q$_{6}$ ) orientational order parameters for a signature of varying amounts of crystallization present in the packings as defined elsewhere [19] (see Appendix- Section III A and Fig. 2 for definitions). The salient feature of Q$_{l}$ is that its zero value means disorder and non-zero value means crystallization. Therefore, the presence in Fig. 2A of an increase in Q$_{l}$ from zero at φ$_{rcp}$ defines the beginning of the coexistence region. Typically, a first-order transition is marked by a nonzero third-order invariant W$_{l}$ [19, 20] which we find appears at the melting point φ$_{melt}$ signaling the onset of the ordered branch 

