Closing Five-Species Impossible Ecologies: Exact census of 248,436 sign topologies by row-cone certificates, signed-permutation pencils, and integer witnesses

Closing Five-Species Impossible Ecologies provides an exact resolution of the remaining five-species coexistence candidates identified by Meng, Horvat, Modes, and Haas in their numerical study of generalized Lotka-Volterra interaction networks. The original study isolated 30 five-species sign topologies for which numerical searches found no positive, locally stable coexistence equilibrium. Because failure to find an equilibrium does not prove that none exists, the list represented an upper bound rather than a completed classification. This work decides all 30 cases exactly. Twenty-four topologies are proved impossible, while six are shown to be possible through explicit integer interaction matrices satisfying the required sign, feasibility, and stability conditions. Stability of each possible case is verified using exact Routh-Hurwitz arithmetic rather than floating-point eigenvalue calculations. The impossibility proofs use four complementary mechanisms: signature-switched M-matrix obstructions, exhaustive determinant analysis of feasible row cones, signed-permutation matrix pencils, and a new one-defect normalization. The two signed-permutation arguments also yield forbidden families that apply in arbitrarily large ecological communities. As an archival strengthening, the accompanying computation regenerates exact integer witnesses for all 53 additional five-species non-extension cases previously reported as possible without retained sample matrices. It also supplies exact witnesses for the 18 possible four-species non-extensions outside the companion eleven-case impossibility closure. Taking the official isomorphism enumeration and constructive extension theorem of Meng and collaborators as inputs, the results complete the five-species census. Among 248,436 nontrivial five-species ecological topologies, exactly 24 are impossible and 248,412 are possible. The reproducibility package contains the manuscript source, compiled paper, official candidate records, exact impossibility certificates, integer feasible-stable witnesses, independent witness sets, reconstruction scripts, verification reports, and a standard-library Python checker. All theorem-facing finite calculations are performed with exact integer arithmetic.