K智能体生成式AI治理的联合李雅普诺夫证明：随机稳定性、涌现集成风险与零知识治理证明
Joint Lyapunov Certificates for K-Agent Generative AI Governance: Stochastic Stability, Emergent Ensemble Risk, and Zero-Knowledge Governance Attestation
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中文总结 AI 辅助
该研究针对K个耦合自适应生成式AI模型系统，提出联合李雅普诺夫证明（JLP）框架，推导临界耦合阈值等理论结果，结合SNARK实现零知识治理证明，经多智能体softmax系统数值研究验证。
中文摘要 AI 辅助
我们基于模型风险管理（MRM）原则，为K个自适应生成式AI模型组成的系统开发了严格的数学治理框架。当多个模型通过交互矩阵共享元学习耦合时，支撑标准MRM的单智能体李雅普诺夫分析被证明是不足的：各智能体可分别满足其声明的稳定性边界，但联合系统却处于涌现集成层面漂移的状态。我们通过联合李雅普诺夫证明（JLP）——一种密码学与随机协议，在不泄露专有权重的情况下，证明聚合动力学在每个验证周期都符合MRM持续监控标准，以此明确这一缺口。我们的主要贡献包括：完整刻画了联合二次李雅普诺夫函数的无穷小生成元；推导了系统失去均方稳定性的精确临界耦合阈值；证明了噪声底定理并确定了零知识证明的正确目标；推导了针对实时权重的周期简洁非交互式知识论证（SNARK）。所有理论结论均通过多智能体softmax系统的五项数值研究得到验证。
英文摘要
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.
