Finite Certificates and Signed Structure in Ecological Coexistence

This paper develops structural criteria and exact certificates for stable coexistence in signed generalized Lotka–Volterra ecological networks. The framework fixes intrinsic growth signs, positive self-regulation, and the exact pattern of beneficial, harmful, and absent interactions. It asks whether some choice of interaction strengths consistent with those constraints admits a positive, locally asymptotically stable equilibrium. The central results identify a sharp boundary for a graphical coexistence criterion. A harmful interaction entering a beneficial cycle from outside provides a stable coexistence core. For communities in which every species has negative intrinsic growth, this cycle condition is also necessary whenever the beneficial interaction graph is strongly connected and becomes acyclic after deleting one vertex. The classification permits arbitrary branching, merging, harmful interactions, and prescribed zeros. An explicit five-species integer realization disproves the unrestricted converse. It supports locally stable coexistence with only one harmful interaction, although the harmful source belongs to every beneficial cycle through its target. The example is minimal in species count, harmful-interaction count, and beneficial feedback vertex number. Constructions extend this phenomenon to every community size of at least five species. The paper also establishes sharp interaction-count bounds. Among all-negative-growth networks with strongly connected beneficial support that fail the cycle condition, locally stable coexistence requires at least two more beneficial interactions than species and at least three more total off-diagonal interactions than species. Both bounds are attained at every eligible community size. A stability-preserving subdivision construction supplies the arbitrary-dimensional realizations. A complementary theorem characterizes the existence of globally attracting coexistence exactly: every species must be reachable through beneficial interactions from a species with positive intrinsic growth. Further necessary-and-sufficient local criteria cover dense single-benefactor networks, balanced interaction components, directed competitive rings with arbitrary nonzero growth signs, and specified beneficial cactus families. Component reduction and stable-core extension results organize these classifications for larger networks. A sharp universal augmentation theorem shows that every equilibrium-feasible signed pattern admits a locally stable extension obtained by adding three negative-growth species, with all interactions across the original–added boundary harmful in both directions. Three is the smallest universal number under these conditions. The construction preserves original signs and zeros while allowing numerical interaction strengths to be chosen anew. The analysis also characterizes cycle domination through universal positivity of normalized principal cofactors and proves that this property does not exclude local coexistence. Exact certificates classify the dense six- and seven-species non-extension sets relative to the cited lower-dimensional classification. An explicit integer-witness bound and rational polynomial refutations adapt classical real algebraic methods to provide a complete finite algebraic certificate theory for arbitrary fixed networks, without claiming practical efficiency. The accompanying materials include the manuscript, LaTeX source, and reproducible exact-arithmetic verification scripts. The results concern existence within prescribed sign classes; they do not assert stability for every compatible choice of interaction strengths. The unrestricted local structural classification remains open. The paper supplies exact classifications, minimal counterexamples, optimal sparsity bounds, and constructive tools that delimit and advance that remaining problem.