= Solution
The crystalline expression diverges at \b[$\phi=\phi_{\rm cp}\simeq0.907$], where the cage free area vanishes. This is physically different from the spurious $\phi=0.5$ 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.
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