Exponential Population Tails and Geometric Ergodicity in a Three-Species Stochastic Reaction Network

We prove rate-universal positive recurrence, exponential stationary molecule-count moments, and weighted geometric ergodicity for a restricted weakly reversible three-species stochastic mass-action network: reversible creation and annihilation of an A+C pair, reversible 2A↔2B exchange, and any strongly connected unary-conversion graph. An explicit exponential Foster–Lyapunov function with a quartic imbalance penalty establishes a uniformly negative drift outside a finite set; exact symbolic and independently sampled generator checks accompany the proof. The network has deficiency one, is not double-full, and generic rates are non-complex-balanced. The theorem does not settle the unrestricted weakly reversible positive-recurrence conjecture, and historical originality and peer review remain unverified.