Spectral Stability and Amplification Propagation in Interacting Agent Networks: (CAIT Part=4)

This paper develops a spectral operator theory for network amplification fields arising from the CAIT Judgment Exposure Subsystem. The framework extends canonical single agent dynamics to graph-coupled populations of interacting agents and models amplification propagation as a nonlinear spectral process evolving over network topology through spillover operators. Interacting Network Systems capture multiple networks, each containing interacting agents both AI agents or/and Human agents. The resulting coupled system defines a network amplification field on R5N governed jointly by intrinsic instability mechanisms, topological interaction structure, and exposure mediated amplification transport. The dynamics are formulated through the global amplification operatorFλ(X;G,Θ) = 0, where X ∈ R5N denotes the global amplification state, G is the graph interaction operator, Θcollects intrinsic amplification parameters, and λ is the spillover coupling intensity. Under standard smoothness, boundedness, and irreducibility conditions on the interaction topology and local amplification laws, existence, uniqueness, and persistence of homogeneous network equilibria are established. Linearization about equilibrium yields the Kronecker structured network Jacobian Jnet = IN ⊗Jlocal +λG⊗C,providing an exact spectral decomposition separating intrinsic local dynamics from topology induced amplification transport.A unified spectral amplification theorem is established showing that the spectral radius ρ(Jnet) is the governing invariant controlling equilibrium persistence, synchronization onset, bifurcation transitions, and systemic instability propagation. In particular, topology dependent critical coupling thresholds emerge according to λc ∼ ρ(G)−1, N →∞,demonstrating asymptotic collapse of local stability margins under increasing topological amplification density.The paper further introduces the inequality curvature functional K(X∗) = λmax∇2Φ(X∗) , and proves that amplification asymmetry arises as a spectral geometric instability whose dominant curvature modes align with the principal eigenspaces of the interaction topology. Consequently, instability propagation, inequality formation, synchronization structure, and amplification bifurcation emerge as manifestations of a single spectral mechanism. Finally, it is shown that the governing spectral structure is invariant under a well defined class of smooth local amplification perturbations, thereby defining a universality class for network coupled amplification systems. Collectively, the results establish a rigorous mathematical theoryof amplification driven instability, topology induced bifurcation, spectral synchronization, and inequality curvature formation in nonlinear network amplification fields. Networked dynamical systems, spectral graph theory, Jacobian analysis, operator theory, non linear stability, bifurcation theory, Kronecker structure, spectral abscissa, coupled systems, amplification dynamics, inequality propagation, complex networks, fixed point theory. Utility and Forward LinkageThe analytical results developed in this paper including the global operator model, Kronecker structured Jacobian, spectral stability law, critical coupling thresholds, fixed point equilibrium structure, heterogeneity propagation mechanisms, topology driven bifurcation criteria, and inequal ity curvature formulation provide the mathematical foundation for analyzing systemic behavior in AI–human networks. While the present paper focuses exclusively on the formal spectral and operator theoretic structure, the managerial, organizational, and governance implications of these results are developed separately in Paper 5, where the spectral mechanisms identified here are translated into decision relevant frameworks for risk management, organizational design, and AI governance.