Generative Closure in Dissipative Systems: Spectral Thresholds and Structural Laws

Many complex systems must reproduce the conditions required for their continued operation despite persistent dissipation. This paper develops a structural theory of generative closure in dissipative cooperative systems, identifying the network architectures capable of sustaining self-reproducing dynamics and the quantitative laws that govern them. The framework rests on a spectral characterization: a dissipative cooperative system exhibits generative closure — i.e., admits a non-trivial endogenously sustained equilibrium — if and only if the spectral bound s\left(M\right) of the net reinforcement operator M=A-Δ is positive. In irreducible systems this equilibrium is strictly positive in all components; in reducible systems we distinguish strictly positive from non-trivial equilibria supported on the downstream closure of generative cores, through a constructive proof across topological strata. Generativity is structurally localized in strongly connected components — generative cores — whose internal reinforcement exceeds dissipative losses. Building on this foundation, we develop three quantitative results. A structural sensitivity theorem yields closed-form expressions for the response of the spectral bound to perturbations, identifying critical edges and vulnerable nodes through left and right Perron eigenvectors. A coupling amplification theorem proves that bidirectional coupling between generative cores strictly amplifies the global spectral bound — linearly at exact spectral resonance and quadratically for any fixed non-zero spectral gap — with a corollary on the rescue of subcritical components. Under a separate smooth common-order hypothesis, a critical exponents theorem characterizes the generative threshold as a threshold bifurcation with explicit exponents — universal in the relaxation time, p-dependent in the amplitude — providing an early-warning signature for approach to the zero operational state. A worked example illustrates all results.