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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.

For ideal hard spheres, potential energy is zero for nonoverlapping configurations and infinite for overlaps. At fixed temperature and volume, the accessible fluid and crystal configurations therefore have no energetic preference of the ordinary bonding kind: the phase with larger entropy has lower Helmholtz free energy.

Ordering particle centers costs some configurational freedom, but at a high packing fraction it can give each particle more usable cage/free volume than a disordered crowded arrangement. The total number of allowed vibrational and positional configurations can then be greater in a crystal despite its visible long-range order. This is entropy-driven hard-sphere crystallization.

The crystal need not be at close packing. At the stated volume fraction near $0.55$, substantial free space remains for thermal motion. Close packing near $0.74$ is the geometric upper bound for equal spheres, not the fraction at which the crystal first becomes thermodynamically favorable. Treating crystallization as an inevitable decrease in total entropy would confuse visible positional order with the full phase-space count.