  A correlation between maxima in virial coefficients (Bn), and "kissing"
numbers for hard hyper-spheres up to dimension D=5, indicates a virial equation
and close-packing relationship. Known virial coefficients up to B7, both for
hard parallel cubes and squares, indicate that the limiting differences
Bn-B(n-1) behave similar to spheres and disks, in the respective expansions
relative to maximum close packing. In all cases, the increment Bn-B(n-1) will
approach a negative constant with similar functional form in each dimension.
This observation enables closed-virial equations-of-state for cubes and squares
to be obtained. In both the 3D and 2D cases, the virial pressures begin to
deviate from MD thermodynamic pressures at densities well-below
crystallization. These results consolidate the general conclusion, from
previous papers on spheres and disks, that the Mayer cluster expansion cannot
represent the thermodynamic fluid phases up to freezing as commonly assumed in
statistical theories.
