Positive Recurrence and Moment Convergence for a Stochastic Biochemical Mean–Noise Controller

The combined biochemical mean–noise controller introduced by Lim and coauthors prescribes stationary output moments through exact signed-memory balance identities. These identities alone do not establish the existence of an invariant probability distribution, the integrability required to justify stationary compensation, or convergence of output moments at deterministic times. This preprint analyzes the original six-species, eleven-reaction stochastic controller at the published factorial-moment reference s = 181/20, for each fixed plant-production gain k in [2, 16]. Conditional on the published native antithetic-controller Lyapunov construction of Briat, Gupta, and Khammash, the analysis establishes nonexplosion, irreducible positive recurrence, uniqueness of the invariant probability distribution, finiteness of all polynomial stationary moments, convergence in total variation, and convergence of expectations of polynomial-growth observables. The resulting stationary output mean is 3, the variance is 61/20, and the Fano factor is 61/60. The proof combines positive- and negative-memory stochastic excursion estimates, global first-entry control, completed returns to a finite set, and regenerative renewal. The contribution concerns the long-run stochastic properties of the original published controller, not a new biochemical circuit or experimental realization. It does not establish convergence rates, stability for other noise-reference values, or behavior under indefinite gain switching. The accompanying technical supplement provides the detailed proofs and complete exact rational certificates required for the stated pointwise inequalities. No experimental validation is claimed.