The finite-basis, pair formulation of the Spectral Theory of chemical bonding is briefly surveyed. Solutions of the Born-Oppenheimer polyatomic Hamiltonian that are totally antisymmetric in electron exchange are obtained from diagonalization of an aggregate matrix built up from conventional diatomic solutions to atom-localized problems. A succession of transformations of the bases of the underlying matrices and the unique character of symmetric orthogonalization in producing the archived matrices calculated "once-of-all" in the pairwise-antisymmetrized basis are described. Application is made to molecules containing hydrogens and a single carbon atom. Results in conventional orbital bases are given and compared to experimental and high-level theoretical results. Chemical valence is shown to be respected and subtle angular effects in polyatomic contexts are reproduced. Means of reducing the size of the atomic-state basis and improving the fidelity of the diatomic descriptions for fixed basis size, so as to enable application to larger polyatomic molecules, are outlined along with future initiatives and prospects.