Closing the Eleven Four-Species Impossible-Ecology Candidates: Classical M-matrices and grounded matrix-tree certificate

This record contains the manuscript and complete reproducibility package for “Closing the Eleven Four-Species Impossible-Ecology Candidates: Classical M-Matrices and Grounded Matrix-Tree Certificates,” by Kevin Fathi. Meng, Horvat, Modes, and Haas identified eleven four-species interaction topologies for which numerical searches found no positive, locally stable coexistence equilibrium in the generalized Lotka-Volterra model. Their numerical evidence established only an upper bound on the number of impossible ecologies. Subsequent work by Celik, Haas, Scholten, Wang, and Zucal proved three of the eleven cases analytically, leaving eight unresolved. This work gives exact impossibility certificates for all eleven candidates and thereby closes the remaining eight cases. At a positive equilibrium, the transformation C = B diag(x*) converts feasibility into prescribed signs for C and its row sums, while local stability requires C to be positive stable and therefore det(C) > 0. Five symmetric candidates are resolved using a classical nonsingular M-matrix obstruction for an obligate signature class. In each of the six asymmetric candidates, the feasible row cones become complete stars of K5 after adjoining an auxiliary root vertex. Determinant multilinearity and the grounded matrix-tree theorem produce exactly 125 negative spanning-tree terms and 131 zero terms. Consequently, det(C) < 0 for every feasible matrix, contradicting the necessary stability condition det(C) > 0. The same framework yields all-dimensional forbidden families. Complete obligate mutualism is recovered as a classical M-matrix corollary in the impossible-ecology formulation. An all-obligate directed cycle with every noncycle interaction deleterious is also proved impossible in every dimension. The reproducibility package includes the LaTeX manuscript, exact candidate sign data, official-source provenance and catalog hashes, symmetric-case certificates, two independently implemented integer determinant verifiers, and an exact stable witness for catalog entry 1390 as a positive control. All decisive calculations use exact arithmetic rather than numerical eigenvalue searches or optimization. The underlying M-matrix and matrix-tree results are classical. The proposed new contribution is their application to the eight previously unresolved ecological sign patterns, the resulting complete eleven-case classification, and the all-dimensional directed-cycle application. This is a research draft and has not yet undergone external peer review.