FIG. 1. The RCP transition. We plot the mechanical coordination number Z$_{j}$ versus the volume fraction φ$_{j}$ for each packing. We identify: (i) a disordered branch which can be fitted with the statistical model of [7] as shown, (ii) a coexistence region, and (iii) an ordered branch. Error bars are calculated over 523 packings obtained from initial LS configurations. The 3d plots visualize how the transition occurs in terms of arrangements of contacting particles. White particles are random clusters, light blue are HCP and green are FCC clusters. Further microscopic information regarding the transition is provided in Appendix-Section II B. 

FIG. 2. Descriptive viewpoint of the RCP transition. (A) Global orientational order parameters Q$_{l}$ versus φ$_{j}$ for different packings signaling the freezing point at φ$_{rcp}$ . A linear fit is possible in the coexistence region. (B) Global third-order invariants W$_{l}$ versus φ$_{j}$ signaling the melting point at φ$_{melt}$ . (C) Probability distribution of local orientational order parameter P ( q$_{6}$ ) versus q$_{6}$ (vertical axis) for packings with φ$_{j}$ (horizontal axis). For packings with φ$_{j}$ within 0.68 ∼ 0.72, the distributions have two significant peaks centered at q fcc 6 = 0 . 57 and q hcp 6 = 0 . 48, which correspond to FCC and HCP arrangements, respectively [19]. Color bar indicates the values of P ( q$_{6}$ ). Since the peaks are very pronounced, we plot P ( q$_{6}$ ) up to the indicated value. (D) Probability distribution of local volume fractions of the Voronoi volumes of each particle, P ( φ$_{vor}$ ) versus φ$_{vor}$ (vertical axis) for different φ$_{j}$ (horizontal axis). The plot indicates a clear discontinuity between both branches. Color bar indicates the values of P ( φ$_{vor}$ ) which are plotted up to the indicated value. 

FIG. 3. Equations of state of the RCP transition. (A) Volume fluctuations of the Voronoi cell of a particle as a function of φ$_{j}$ . The data indicates a discontinuity between the ordered and disordered branches which are fitted by functions as indicated. These fittings are used in the integrations of Eq. (1). The larger fluctuations in volume observed in the order state compared to the disorder state at similar φ$_{j}$ are due to the fact that the system packs better in the former and thus displays larger fluctuations when the system volume is the same. (B) Entropy obtained from fluctuation theory in Eq. (2), s , and Shannon entropy from information theory, s$_{shan}$ , versus volume fraction φ$_{j}$ . Both entropies agree (up to a multiplicative constant, k = 0 . 1, as indicated) confirming our calculations. The extended branch denotes a metastable state ending at point b at an hypothetical Kauzmann density, φ$_{K}$ , in analogy with the physics of glasses [6] (see Appendix-Section II C). 

FIG. 4. Thermodynamic viewpoint of the RCP transition. All the observables are con- 

