The entropy of FCC, s$_{fcc}$ , is zero in the thermodynamic limit. Equations (29) and (31) require three constants of integration: X$_{rlp}$, X$_{melt}$ and the entropy of RCP: s$_{rcp}$ . We now introduce three extra constraints to close the system. First, there are two conditions for equilibrium between two phases in jammed matter [17, 21]: (a) “thermal” equilibrium where X$_{c}$ is the critical compactivity at the transition. (b) The equality of the chemical potentials of the two phases at the melting and the freezing RCP point: µ$_{melt}$ = µ$_{freez}$ due to the conservation of number of particles. This is analogous to the equality of the free energy density at the transition from Eq. (26), which allows to calculate the entropy at the freezing point via f$_{melt}$ = f$_{rcp}$ : and the entropy of fusion is then: The final constant of integration to be obtained is the compactivity at the RLP point. As a first order approximation, X$_{rlp}$ can be taken as infinite as has been shown in previous experimental and numerical studies [23, 25]. However, when we compare the obtained entropy Eq. (31) with an independent measure of the entropy using Shannon information theory (explained below) we find that there are slightly differences between both values of the entropy. Therefore, we consider X$_{rlp}$ as a fitting parameter to be obtained by fitting the result of Eq. (31) with the Shannon entropy which in principle does not require any integration constant to be calculated. We note that using the fitted value of X$_{rlp}$ instead of infinity does not change the final results, specifically the values of X$_{c}$ and the entropy of fusion, even though the obtained X$_{rlp}$ is “far” from infinite. We summarize the calculation as follows: (a) We assume a value of X$_{rlp}$ (which is later fitted with Shannon entropy) and integrate Eq. (29a) from RLP to φ$_{rcp}$ and obtain the compactivity X$_{rcp}$ = X ( φ$_{rcp}$ ). (b) Using Eq. (32) we obtain X$_{melt}$ (or X$_{c}$ ). (c) Using X$_{melt}$ , we integrate Eq. (29b) to obtain the X ( φ$_{j}$ ) in the ordered branch, thus completing the calculation of the compactivity equation of state. (d) We integrate Eq. (31b) up to φ$_{melt}$ to obtain the entropy of the melting point: s$_{melt}$ . (e) Using Eq. (33) we obtain the entropy at 30 X$_{melt}$ = X$_{rcp}$ ≡ X$_{c}$, (32) s$_{rcp}$ = s$_{melt}$ + V$_{g}$ ( ω$_{rcp}$ − ω$_{melt}$ X$_{c}$ ) , (33) Δ s$_{fus}$ ≡ s$_{rcp}$ − s$_{melt}$ = V$_{g}$ ( ω$_{rcp}$ − ω$_{melt}$ X$_{c}$ ) . (34)