coordination number Z$_{j}$ which only counts those contacts with nonzero forces. By definition z$_{g}$ ≥ Z$_{j}$ . 

While Z$_{j}$ is easy to measure in computer simulations of soft particles as the number of contacts between overlapping particles, the geometrical coordination number, z$_{g}$ , can be measured by slightly inflating the spheres up to 4% of their diameters and counting the resulting contact particles, as discussed in [7]. In practice, the geometrical coordination number measures the particles surrounding a central one with a gap between them from zero or negative (giving Z$_{j}$ ) up to δ = 0 . 08 R as discussed in [7]. We notice that δ = 0 . 08 R is much smaller than the location of the second peak in the radial distribution function which occurs around δ ∼ 2 R . The value δ = 0 . 08 R is specific for a system of N =10,000. We expect this value to diminish in the thermodynamic limit. 

Figure 5B plots the geometrical and mechanical coordination z$_{g}$ and Z$_{j}$ as a function of φ$_{j}$ for the same packings as in Fig. 1. We find that along the disordered branch, z$_{g}$ ≈ Z$_{j}$ as expected [7]. However, in the coexistence region, Z$_{j}$ = 6 stays constant while z$_{g}$ keeps growing with φ$_{j}$ . The separation between z$_{g}$ and Z$_{j}$ is a signature of the onset of ordering at the freezing point. As explained above, the system starts to crystallize by allowing particles in the second coordination shell to come closer to the central particle and moving the Z$_{j}$ contacting particles towards a FCC arrangement. At the melting point, the condition Z$_{j}$ = 6 cannot hold any longer and the system transitions to the other branch with an increase of Z$_{j}$ up to 12. 

The distinction between z$_{g}$ and Z$_{j}$ is not only important for a characterization of the transition. It is also important to interpret the experimental results. Due to the uncertainty in detecting the exact position of the particles in any experiment, the exact mechanical coordination might be very difficult to obtain. Thus, a small uncertainty in the determination of the contacting particles δ = 0 . 08 R will produce z$_{g}$ as shown in Fig. 5B. One way to obtain the actual mechanical coordination from experimental data is to use the experimentally obtained coordinates of the particles as initial positions of a MD simulation using Hertz- Mindlin forces to relax the configurations and find the exact force balance network of the packing. Codes to develop this procedure are available at www.jamlab.org. We also provide most of the packings used in this study as well as the code to calculate the entropy. 

