Different from traditional electronic structure methods, the contracted Schrodinger equation with reduced-density-matrix (RDM) reconstruction may be exact when only the 2-particle RDM is employed as the fundamental parameter. Although Rosina's theorem indicates that the 3 and the 4-RDMs are functionals of the 2-RDM, cumulant theory generates only those terms expressible as antisymmetrized products of lower RDMs. We present a formal solution for reconstruction where the approximate cumulant formulas are systematically corrected through contraction conditions. Using a part of the formal 3-RDM reconstruction, the CSE is compared with other methods through a quasi-spin model containing as many as eight-hundred fermions.