Then the Shannon entropy is defined as: where we have again assumed the “Boltzmann-like” constant in front to be one. The Shannon entropy density is obtained as: by linear fitting of the extensive part of the Shannon entropy. In general, the fluctuation entropy from Eq. (31) is greater than the Shannon entropy from Eq. (37) because Shannon entropy only counts configurational disorder and additional entropy could arise from freedom to move grains within the clusters of n particles without disrupting the Delaunay network. However, we discover that the fluctuation entropy ob- tained from Eq. (31) and Shannon entropy from Eq. (37) only differ by a scaling constant k = 0 . 1, ie., ks = s$_{shan}$ , see Fig. 3B. Beyond this multiplicative constant the agreement between both estimations of the entropy is very good. Due to finite size effects the Shannon entropy also gives a nonzero value of the entropy of FCC, s$_{fcc}$ = 0 . 6. This value is subtracted from the calculations. By fitting s$_{shan}$ with the thermodynamic entropy we obtain the final constant of integration in Eq. (29a), X$_{rlp}$ = 0 . 5 V$_{g}$ . While it is obvious that this constant is far from the infinite value which is expected and used in [23, 25] for the RLP limit, we notice that our final results are not very sensitive to the exact value of X$_{rlp}$ . For instance, by setting X$_{rlp}$ → ∞ the fitting of the fluctuation entropy is slightly off in comparison with s$_{shan}$ only in the vicinity of RLP but the values of X$_{c}$ and the entropy of fusion do not have appreciable change. Figure 4 in the main text shows the thermodynamic quantities in the first order phase transition of jammed matter. They are consistent with the general thermodynamic picture. Figure 10 plots the entropy versus w showing a linear dependence in the coexistence region. The entropy is an interpolation of the form: s$_{x}$ = xs$_{melt}$ + (1 − x ) s$_{freez}$ where x is the concentration of crystal clusters in the coexistence. Since X = ∂W/∂S , the linearity of s between 0.64 and 0.68 is a manifestation of the coexistence of two phases at a constant X$_{c}$ . 32 H ( n ) = − ∑ p$_{i}$ ln p$_{i}$, (36) s$_{shan}$ = lim n →∞ [ H ( n + 1) − H ( n )] , (37)