In 2019, Paulsson and collaborators conjectured that stochastic biochemical control networks have fundamental limits on how much intrinsic noise can be simultaneously suppressed across multiple components. Ripsman, Kell, and Hilfinger recently proposed a formal proof strategy for unit-birth models based on a stationary information-theoretic decomposition. Here, we provide a rigorous mathematical justification for this argument. We consider continuous-time Markov chains on $\Z^N_{\ge 0}$ in which each component is degraded linearly and produced in unit births at a state-dependent rate depending on the other components but not on itself. Noise in component $i$ is measured by the Fano factor $F_{X_i}$, the ratio of stationary variance to mean, with Poisson value $1$ as baseline. Our first contribution is to isolate explicit hypotheses on moments, mean birth rates, and total-rate growth under which the formal information-flow identities can be rigorously justified. Following the proof outline of Ripsman, Kell, and Hilfinger, we then prove the conjecture. Our second contribution is to make these hypotheses checkable: a uniform positive lower bound on the birth rates and at-most-linear total growth dominated by the weakest degradation rate suffices, via Foster--Lyapunov methods. Our third contribution is a structural strengthening. Under a signed monotonicity condition on the rate functions, satisfied by structurally balanced signed interaction networks, we prove that the stationary distribution is associated with respect to the corresponding signed partial order. This upgrades the global tradeoff to the termwise bound $F_{X_i}\ge 1$ for every $i$. Hence, within the signed-monotone subclass, sub-Poissonian noise requires a frustrated interaction topology.