Incidence-based generalizations of cycle covers, called contributors, extend the Harary-Sachs coefficient theorem for characteristic polynomials of the adjacency matrix of graphs. All minors of the Laplacian resulting from an integer matrix are characterized using their associated oriented hypergraph through a new minimal collection of contributors to produce the coefficients of the total-minor polynomial. We prove that the natural grouping of contributors via tail-equivalence is necessarily cancellative for any contributor family that reuses an edge. We then provide a new combinatorial proof on the non-0 isospectrality of the traditional characteristic polynomials of the Laplacian and its dual.