The potential distribution theorems for the test particles provide a connection to the chemical potentials and the cavity distribution functions y(r) much used in molecular theory. These relations can be capitalized for establishing the closure relations for the Ornstein-Zernike equation. In this study, we formulate a class of closures with built-in flexibilities in order to satisfy the potential distribution theorems (or the related zero separation theorems) and thermodynamic consistency. The theory is self-contained within the Integral equation framework. We test it on the Lennard-Jones fluid over ranges of temperatures (down to T*=0.81) and densities (up to rho*=0.9). To achieve self-sufficiency, we exploit the connections offered by writing down n members of the mixture Ornstein-Zernike equations for the coincident oligomers up to n-mers. Then the potential distribution theorems generate new conditions for use in determining the bridge function parameters. Five consistency conditions have been identified (three thermodynamic and two based on zero-separation values). This self-consistency allows for bootstrapping and generation of highly accurate structural and thermodynamic information. The same procedure can potentially be extended to soft-sphere potentials other than the Lennard-Jones type. (C) 1997 American Institute of Physics.