 Solution

The crystalline expression diverges at $\varphi ={\varphi }_{\mathrm{cp}}\simeq 0.907$, where the cage free area vanishes. This is physically different from the spurious $\varphi =0.5$ pole produced by extrapolating the low-density fluid approximation. A real compression path cannot be inferred by following that fluid pole and then abruptly switching curves. At low density it follows a fluid equation of state, while at sufficiently high density an ordered arrangement can leave more accessible rattling configurations than a disordered crowded arrangement. This is entropy-driven hard-sphere crystallization, here in two dimensions: allowed hard-core configurations have the same interaction energy, so configurational entropy drives the phase competition. An equilibrium first-order coexistence interval would have common pressure and chemical potential, giving a constant-pressure segment as mean density changes. The two asymptotic expressions alone neither locate that interval nor determine the detailed two-dimensional melting sequence. They require a consistent free-energy comparison and trustworthy approximations in the transition region.