〈 ( δW ) $^{2}$〉 = k$_{B}$X $_{2}$∂W ∂X . (27) σ 2 i = − k$_{B}$ V$_{g}$ ( X φ$_{j}$ ) 2 ∂φ$_{j}$ ∂X , i = 1 , 2 . (28) 1 X ( φ$_{j}$ ) = k$_{B}$ V$_{g}$ ∫ φ$_{j}$ φ$_{rlp}$ dφ φ $^{2}$σ 2 $_{1}$( φ ) + 1 X$_{rlp}$ , φ$_{rlp}$ ≤ φ$_{j}$ ≤ φ$_{rcp}$, (29a) 1 X ( φ$_{j}$ ) = k$_{B}$ V$_{g}$ ∫ φ$_{j}$ φ$_{melt}$ dφ φ $^{2}$σ 2 $_{2}$( φ ) + 1 X$_{melt}$ , φ$_{melt}$ ≤ φ$_{j}$ ≤ φ$_{fcc}$, (29b) 1 X = − φ 2 j V$_{g}$ ∂s ∂φ$_{j}$ . (30) s ( φ$_{j}$ ) = s$_{rcp}$ + V$_{g}$ ∫ φ$_{rcp}$ φ$_{j}$ dφ X ( φ ) φ $_{2}$, φ$_{rlp}$ ≤ φ$_{j}$ ≤ φ$_{rcp}$, (31a) s ( φ$_{j}$ ) = V$_{g}$ ∫ φ$_{fcc}$ φ$_{j}$ dφ X ( φ ) φ $_{2}$, φ$_{melt}$ ≤ φ$_{j}$ ≤ φ$_{fcc}$. (31b) 29 Here we assume that k$_{B}$ plays the role of the Boltzmann constant in thermodynamics. Its value could be set to unity without changing the obtained results since in this context it just defines the units of entropy (in the main text we set k$_{B}$ = 1 to simplify). This means that we measure the compactivity in units of volume V$_{g}$ and that the entropy, which has units of k$_{B}$ , is dimensionless. In terms of the Voronoi fluctuation σ and volume fraction φ$_{j}$ , Eq. (27) reads: Equation (28) applies to the disordered and ordered branches, σ$_{1}$ and σ$_{2}$ , separately. It does not apply to the coexistence region, since it requires the pure phases to calculate the fluctuations. Using Eq. (28) we can calculate the thermodynamic quantities by integration. We first obtain the compactivity from: where X$_{melt}$ = X ( φ$_{melt}$ ) is the compactivity of the packing at the melting point and X$_{rlp}$ = X ( φ$_{rlp}$ ) at RLP. The entropy density is then obtained by a second integration using the definition Eq. (23) which reads: We integrate Eq. (30) for each branch to obtain: