We evaluate a number of current closure relations used in the integral equations for hard sphere fluids, such as the Percus-Yevick, Martynov-Sarkisov, Ballone-Pastore-Galli-Gazillo, and Verlet modified (VM) closures with respect to their abilities of satisfying the zero-separation theorems for hard spheres. Only the VM closure is acceptable at high densities (p similar to 0.7), while all fail at lower densities (lim 0

Keywords:MOLECULAR-DISTRIBUTION FUNCTIONS;MONTE-CARLO SIMULATION;BRIDGE FUNCTION;BODY-FLUIDS;MIXTURES;LIQUIDS