Extensive numerical solutions of the hypernetted-chain (HNC) and Rogers-Young (RY) integral equations are presented for the pair structure of a system of two coupled replicae (1 and 2) of a "soft-sphere" fluid of atoms interacting via an inverse-12 pair potential. In the limit of vanishing inter-replica coupling epsilon(12), both integral equations predict the existence of three branches of solutions: (1) A high temperature liquid branch (L), which extends to a supercooled regime upon cooling when the two replicae are kept at epsilon(12) = 0 throughout; upon separating the configurational and vibrational contributions to the free energy and entropy of the L branch, the Kauzmann temperature is located where the configurational entropy vanishes. (2) Starting with an initial finite coupling epsilon(12), two "glass" branches G(1) and G(2) are found below some critical temperature, which are characterized by a strong remnant spatial inter-replica correlation upon taking the limit epsilon(12) -> 0. Branch G(2) is characterized by an increasing overlap order parameter upon cooling, and may hence be identified with the hypothetical "ideal glass" phase. Branch G(1) exhibits the opposite trend of increasing order parameter upon heating; its free energy lies consistently below that of the L branch and above that of the G(2) branch. The free energies of the L and G(2) branches are found to intersect at an alleged "random first-order transition" (RFOT) characterized by weak discontinuities of the volume and entropy. The Kauzmann and RFOT temperatures predicted by RY differ significantly from their HNC counterparts.
