1 X ( φ$_{j}$ ) = 1 V$_{g}$ ∫ φ$_{j}$ φ$_{rlp}$ dφ φ $^{2}$σ 2 $_{1}$( φ ) + 1 X$_{rlp}$ , φ$_{rlp}$ ≤ φ$_{j}$ ≤ φ$_{rcp}$, (1a) 1 X ( φ$_{j}$ ) = 1 V$_{g}$ ∫ φ$_{j}$ φ$_{melt}$ dφ φ $^{2}$σ 2 $_{2}$( φ ) + 1 X$_{melt}$ , φ$_{melt}$ ≤ φ$_{j}$ ≤ φ$_{fcc}$, (1b) s ( φ$_{j}$ ) = s$_{rcp}$ + V$_{g}$ ∫ φ$_{rcp}$ φ$_{j}$ dφ X ( φ ) φ $_{2}$, φ$_{rlp}$ ≤ φ$_{j}$ ≤ φ$_{rcp}$, (2a) s ( φ$_{j}$ ) = V$_{g}$ ∫ φ$_{fcc}$ φ$_{j}$ dφ X ( φ ) φ $_{2}$, φ$_{melt}$ ≤ φ$_{j}$ ≤ φ$_{fcc}$, (2b) V$_{g}$ and entropy is dimensionless): where X$_{melt}$ = X ( φ$_{melt}$ ) and X$_{rlp}$ = X ( φ$_{rlp}$ ) are the compactivities at the melting point and at RLP, respectively. Once X ( φ$_{j}$ ) is obtained from Eq. (1), the entropy density, s = S/N , is calculated by integration [23] (see Appendix-Section IV A): where we have used that the entropy of FCC is zero in the thermodynamic limit. There are three unknown integration constants in Eqs. (1) and (2): X$_{rlp}$ , X$_{melt}$ and the entropy of RCP, s$_{rcp}$ . To close the system, we consider the conditions for equilibrium between the phases [21]: (a) “thermal” equilibrium X$_{melt}$ = X$_{rcp}$ ≡ X$_{c}$ , where X$_{c}$ is the critical compactivity at the transition, and (b) the equality of the free energy density (or chemical potential), f = F/N , at the melting and the freezing RCP point: f$_{melt}$ = f$_{rcp}$ . This implies, ω$_{rcp}$ − ( X$_{c}$/V$_{g}$ ) s$_{rcp}$ = ω$_{melt}$ − ( X$_{c}$/V$_{g}$ ) s$_{melt}$ , where w = 1 /φ$_{j}$ = W/ ( NV$_{g}$ ) is the reduced volume of the system. The third integration constant X$_{rlp}$ can be considered infinite [7, 23, 25] since RLP is the highest volume of the system. While the precise value of X$_{rlp}$ does not affect our conclusions, a more accurate finite value can be obtained by fitting the entropy Eq. (2) with an independent measure obtained by cluster analysis from information theory (Shannon entropy, s$_{shan}$ ) as developed in [23] (Appendix-Section IV B). Figure 3B shows that the entropy from the thermodynamic integration Eq. (2) and s$_{shan}$ agree well (up to a multiplicative constant) supporting the framework of Eqs. (1)-(2). The entropy is composed of two branches plus the coexistence region (green line in Fig. 3B). Figure 4A displays a discontinuity in s ( X ) at X$_{c}$ = 0 . 031 V$_{g}$ revealing the first-order nature of the transition which is accompanied by an “entropy of fusion” Δ s$_{fus}$ ≡ s$_{rcp}$ − s$_{melt}$ = 3 . 0. The volume fraction is discontinuous at X$_{c}$ (Fig. 4B) where the system jumps from RCP to the melting point releasing an amount of volume given by the “enthalpy of fusion” Δ h$_{fus}$ = 5