Governed Multi-Agent Dynamics: Stability, Stratification, and Complexity Bounds under Endogenous Constraint Fields

We study heterogeneous multi-agent systems (MAS) in which governance policies act as endogenous constraint fields on agent-state evolution, rather than as external supervisory signals. Each agent's structural configuration evolves on a Polish metric space under three coupled vector fields: intrinsic dynamics, environmental coupling, and a governance operator. Under a GENERIC (General Equation for Non-Equilibrium Reversible–Irreversible Coupling) decomposition, we show that systems satisfying four structural conditions admit a metric gradient-flow representation on a Riemannian manifold.

Building on this foundation, we establish three results with explicit quantitative guarantees.

Theorem I — Governance-Induced Stabilization Threshold. The governance intensity μ has a computable critical value μ_c separating unstable from stable regimes. For the two-agent system, μ_c = α − 1 in closed form: the system is unstable for μ < μ_c and asymptotically stable for μ > μ_c.

Theorem II — Potential-Induced Stratification. When governance assigns each agent a strongly convex potential from a finite set of k distinct potential classes, the resulting gradient flows produce k exponentially convergent form strata at rate μ_l per class, with inter-stratum separation δ_min > 0. The cluster partition is recoverable from the null eigenspaces of the governance graph Laplacian.

Theorem III — Form Complexity Bound. A Grönwall-based argument yields three formulations (time-varying, stochastic, worst-case) of a structural complexity ceiling B(t). The Complexity Diagnostic shows that persistent bound violation is a sufficient diagnostic of loss of governance-induced complexity dissipation.

We further establish empirical-measure convergence for each governance class and a stochastic extension with Lyapunov stability in expectation under bounded Hessian. Three minimal numerical illustrations demonstrate qualitative consistency with the analytical predictions.