Stability of AI Governance Systems: A Coupled Dynamics Model of Public Trust and Social Disruptions

AI systems are increasingly entrenched in public governance, yet scholarship lacks formal tools to determine when deviations of public trust in algorithmic institutions dissipate and when they grow into collapse. Stability refers here to asymptotic recovery from finite state perturbations under fixed structural parameters. We address this gap by developing a mathematical framework for institutional trust stability that couples a Friedkin-Johnsen opinion dynamics process with a Hawkes-inspired intensity process for AI controversies. Motivated by the Computers-Are-Social-Actors literature and recent studies of trust in large language models, this bidirectional coupling reveals that governance stability depends on the structural architecture of the information environment rather than absolute trust levels. We derive an exact spectral stability criterion delineating resilience from collapse, demonstrating how event self-excitation and memory persistence systematically narrow the stable parameter regime. Our structural analysis yields four counterintuitive structural implications: high-trust systems can be structurally fragile, low-trust environments can be structurally stable, dynamical stability neither measures nor guarantees algorithmic fairness or legitimacy, and network topology reshapes equilibrium heterogeneity while its effect on spectral stability is uniformly bounded in an explicit memory-dominated regime. Governance assessment should therefore pair normative evaluation of harms and fairness with structural analysis of recoverability, rather than treating either as a proxy for the other.