Odd-Cycle Frustration in Competitive Networks: An Exact Topology Characterization

This release gives an exact support-topology characterization for a class of competitive generalized Lotka–Volterra communities whose coexistence equilibrium is permanent and globally asymptotically stable, yet cannot be made evolutionarily stable by any positive projective scaling of the associated replicator game. The main theorem states that a finite simple support graph admits a relatively open family of such communities if and only if the graph is nonbipartite, equivalently, if and only if it contains an odd cycle. For bipartite support, a diagonal signature switches the interaction matrix to a positive-stable Z-matrix; nonsingular M-matrix diagonal stability then supplies an ESS-equivalent Volterra–Lyapunov representative. The converse is constructive. The article begins with an exact rational three-species seed possessing a one-way edge and a rank-one obstruction to every positive scaling. A directed two-subdivision operation replaces that edge by a three-edge path through two fast logistic mediators. The Schur reduction of the enlarged competitive system is exactly the original GLV field. A scaled invariant-measure boundary-growth certificate proves robust permanence, triangular pullback estimates give order-epsilon tracking on compact invariant sets, and a compactness-plus-local-basin argument upgrades the singular reduction to genuine global asymptotic stability for every sufficiently small finite epsilon. The new middle edge is again one-way and its obstruction coordinate is zero, so the operation iterates to every odd cycle. A split-free weak-attachment lemma then extends the construction to every graph containing an odd cycle. The accompanying verifier uses exact SymPy arithmetic to check the local odd-cycle characteristic identity and root slope for sampled orders 3, 5, 7,and 9, the rational seed, its complete boundary ledger, the all-scalings obstruction, the symbolic Schur identity, the critical graph, and the first five-species subdivision. No floating-point inequality is used to certify an algebraic claim. The analytic openness step is stated separately as a robust global-attraction lemma: strict robust permanence gives a common compact interior absorbing set for nearby systems, a fixed quadratic Lyapunov estimate gives a uniform local basin, and a converse Lyapunov function forces every positive orbit into that basin. The split-Lyapunov and limiting invariant-measure arguments are written out explicitly. The accompanying algebraic verifier does not certify permanence,finite-epsilon global attraction, relative openness, novelty or priority, or a numerical admissible upper bound for the singular parameter epsilon. Those are analytic or literature-review claims in the manuscript. The proof gives an existential interval of admissible epsilon values, containing rational choices, but this release does not extract a certified numerical endpoint. This manuscript is a companion to Dimensional Thresholds and Exact Obstructions for Evolutionary Stability under Replicator Equivalences (Kevin Fathi, working paper, 2026), which establishes the all-scalings obstruction framework and the dimensional thresholds. This is a research working paper. The proof and exact computations have undergone adversarial internal checking, but the new finite-parameter subdivision theorem and the priority claim remain subject to independent specialist and peer review.