Elemental carbon exists in various aesthetically pleasing architectures. These forms include a group of synthesized allotropes with cubic modifications that have taken controversial or even unidentified crystal structures, which makes determining their physical properties difficult. In this study, four novel cubic carbon polymorphs (fcc-C10, fcc-C12, bcc-C 20, and fcc-C32) that exhibit lattice parameters within the same range as those of undetermined cubic carbon allotropes are proposed by employing a newly developed ab initio particle-swarm optimization methodology for crystal structure prediction. The four structures are all three-dimensional polymers consisting of unique, small C10, C12, C 20, and C32 cages with quite low density. Investigation of their electronic and mechanical properties illustrate that the cage-like cubic carbons are all semiconductors with excellent mechanical performance, specifically superhardness and high ductility. Moreover, we readily explain a long-standing controversial experimentally synthesized cubic carbon (viz., the so-called "superdense" carbon) using the previously proposed bcc C6 based on the coincident lattice constant and electron diffraction data between the theoretical and experimental results.