Finite-amplitude establishment beyond local invasion in cooperative chemical networks
Local kinetic exponents and next-generation radii diagnose infinitesimal invasion, but they need not determine whether a nonlinear organized state is accessible. Motivated by the kinetic component of Gate~7 in a twelve-gate nonequilibrium framework, we study cooperative networks whose productive linearization contains no closed cycle. We prove that an acyclic productive Jacobian has $R=0$ and $λ_{\rm kin}=-\min_i d_i<0$ for every drive, while a nonlinear return path can create a stable finite-amplitude branch through a saddle-node. In a canonical catalyst--assembly model, the saddle's stable manifold---not a threshold in either concentration---is the deterministic establishment boundary, so equal-abundance seeds of different composition can have opposite outcomes. At finite copy number, establishment becomes a committor and first-passage problem. For a benchmark protocol with $Γ=2$, $Ω=16$, $η=0.02$, and horizon $T=1000$, density-Hill and pair-count generators give first-entrance probabilities $0.633$ and $0.339$; their interpolated half-success drives are $1.900$ and $2.148$, an approximately $13.1\%$ shift. The associated constant-source deterministic system retains a stable low state, so these transitions are fluctuation- and discrete-arrival-enabled. The finite-amplitude regime is not confined to exact acyclicity: a weak linear return leaves a finite interval with $0<R<1$, $λ_{\rm kin}<0$, and bistability. A general cooperative-return criterion separates the superlinear local condition from the global return-profile structure required for a fold. The resulting hierarchy of branch existence, basin accessibility, and finite-time stochastic establishment replaces a single onset threshold by a protocol-dependent kinetic criterion.