The thermodynamics of systems of hard spheres tethered to a perfect Face-Centered Cubic (FCC) lattice is investigated using event-driven molecular-dynamics simulations. The direct relation of the geometry of phase space to the particle-particle and, crucially, the particle-tether collision rates is used to examine its structure and compare between the FCC and fluid states. In tethered systems, the entropy can be determined by at least two routes, through integration of the tether collision rates with respect to the tether length or through integration of the particle-particle collision rates with the hard sphere diameter (or, equivalently, the density). If the entropy were an entirely analytic function of $r_T$ and $\sigma$, these two methods for calculating the entropy should lead to the same results; however, a non-analytic region exists as an extension solid-fluid phase transition of the untethered hard-sphere system, and integration paths that cross this region will lead to values for the entropy that depend on the particular path chosen. The difference between the calculated entropies appears to be related to the comunal entropy, and the location of the non-analytic region appears to be related to conditions where the regions of phase space associated with the FCC configuration becomes separated from the regions associated with the disordered fluid. The non-analytic region is finite in extent, vanishing below $r_T/a\approx0.55$, where $a$ is the lattice spacing, and there are many continuous paths that connect the fluid and solid phases that can be used to determine the crystal free energy with respect to the fluid.