Joint Lyapunov Certificates for K-Agent Generative AI Governance: Stochastic Stability, Emergent Ensemble Risk, and Zero-Knowledge Governance Attestation

We develop a rigorous mathematical framework for the governance of systems of K self-adapting generative AI models under the principles of Model Risk Management (MRM). When multiple models share a meta-learning coupling through an interaction matrix, the per-agent Lyapunov analysis that underpins standard MRM is provably insufficient: individual agents can each satisfy their declared stability bounds while the joint system is in a regime of emergent ensemble-level drift. We formalize this gap through the Joint Lyapunov Proof (JLP)---a cryptographic and stochastic protocol that attests, without revealing proprietary weights, that the aggregate dynamics satisfy MRM Ongoing Monitoring standard at every validation epoch. Our main contributions are the following. We give a complete characterization of the infinitesimal generator of the joint quadratic Lyapunov function. We derive the exact critical coupling threshold above which the system loses mean-square stability. We prove a Noise-Floor Theorem and identify the correct target for zero-knowledge attestation. A per-epoch Succinct Non-Interactive Argument of Knowledge (SNARK) on the live weights is derived. All theoretical claims are validated against five numerical studies using a multi-agent softmax system.