As artificial intelligence (AI) is increasingly deployed in high-stakes public decision-making (from resource allocation to welfare distribution), public trust in these systems has become a critical determinant of their legitimacy and sustainability. Yet existing AI governance research remains largely qualitative, lacking formal mathematical frameworks to characterize the precise conditions under which public trust collapses. This paper addresses that gap by proposing a rigorous coupled dynamics model that integrates a discrete-time Hawkes process -- capturing the self-exciting generation of AI controversy events such as perceived algorithmic unfairness or accountability failures -- with a Friedkin-Johnsen opinion dynamics model that governs the evolution of institutional trust across social networks. A key innovation is the bidirectional feedback mechanism: declining trust amplifies the intensity of subsequent controversy events, which in turn further erode trust, forming a self-reinforcing collapse loop. We derive closed-form equilibrium solutions and perform formal stability analysis, establishing the critical spectral condition rho(J_{2nt}) < 1 that delineates the boundary between trust resilience and systemic collapse. Numerical experiments further reveal how echo chamber network structures and media amplification accelerate governance failure. Our core contribution to the AI governance field is a baseline collapse model: a formal stability analysis framework demonstrating that, absent strong institutional intervention, even minor algorithmic biases can propagate through social networks to trigger irreversible trust breakdown in AI governance systems.