(Fig. 2B). 

Interestingly, the ordered phase has two significant peaks in the probability distribution P ( q$_{6}$ ) of the local order parameter of each particle, q$_{6}$ (Fig. 2C). The peaks correspond to FCC and HCP [19] signaling that both crystalline configurations are present in the ordered structure. From the available data we cannot rule out the possibility of another transition from HCP to FCC before φ$_{j}$ ∼ 0 . 74. The Gaussian distributions P ( q$_{6}$ ) obtained for φ$_{j}$ within 0.55 ∼ 0.64 show no preferred lattice structure in the random branch. While the relative peak positions in P ( q$_{6}$ ) do not change, the percentage of crystal and random phase found in the packing progresses from one to the other in the coexistence region. Microscopically, the existence of the two pure phases is starkly represented by the two separated distributions of local Voronoi volume fractions P ( φ$_{vor}$ ) for which the same phenomenology of P ( q$_{6}$ ) applies (Fig. 2D, φ$_{vor}$ = V$_{g}$/V$_{vor}$ , where V$_{vor}$ is the Voronoi volume [7, 15] of each particle of volume V$_{g}$ ). This descriptive analysis is further supported in Appendix-Section III. 

Having identified the structure of the phases we now develop a thermodynamic viewpoint of the RCP transition to rationalize the obtained results. Transitions in equilibrium physical systems are driven by a competition of energy and entropy. Instead, a transition in athermal jammed matter is driven by the minimization of the system’s volume W by compactification and entropy maximization of jammed configurations [21, 22]. In accordance with the second law of thermodynamics, the granular system tends to minimize its Gibbs-Helmholtz “free energy” F = W − XS rather than W alone. The compactivity of the system X = dW/dS is a measure of how much further compaction a packing can undergo; the lower the volume the lower the compactivity [21]. Thus, we map the packing problem to a thermodynamic problem where the volume W replaces the energy and X takes the role of temperature. The principle of free energy minimization can thus be applied. 

The equations of state, φ$_{j}$ ( X ) and S ( φ$_{j}$ ), can be calculated from the fluctuations of the Voronoi volumes in the disordered and ordered phases [23], σ$_{1}$ ( φ$_{j}$ ) and σ$_{2}$ ( φ$_{j}$ ) respectively, in analogy to the standard Boltzmann statistical mechanics ( σ 2 ≡ 〈 w 2 vor 〉 − 〈 w$_{vor}$ 〉 2 and ω$_{vor}$ = 1 /φ$_{vor}$ = V$_{vor}$/V$_{g}$ is the reduced Voronoi volume). Figure 3A shows clearly the existence of the two pure phases and a discontinuity between both branches. We obtain the compactivity by integration of σ$_{1}$ and σ$_{2}$ using Einstein fluctuation theory [23–25] (see Appendix-Section IV A for more details, we set k$_{B}$ = 1 for simplicity, X is given in units of 

