Theory of Dynamic Adaptive Coordination

arXiv:2603.11560v2 Announce Type: replace-cross
Abstract: This paper develops a dynamical theory of adaptive coordination governed by persistent environmental memory. Moving beyond framework-specific equilibrium optimization or agent-centric learning, I model agents, incentives, and the environment as a recursively closed feedback architecture: a persistent environment stores accumulated coordination signals, a distributed incentive field transmits them locally, and adaptive agents update in response. Coordination thus emerges as a structural consequence of dissipative balancing against reactive feedback, rather than the solution to a centralized objective.
I establish three primary results. First, I show that under dissipativity, the closed-loop system admits a bounded forward-invariant region, ensuring viability independent of global optimality. Second, I demonstrate that when incentives hinge on persistent memory, coordination becomes irreducible to static optimization. Finally, I identify the essential structural condition for emergence: a bidirectional coupling where memory-dependent incentives drive agent updates, which in turn reshape the environmental state. Numerical verification identifies a Neimark-Sacker bifurcation at a critical coupling threshold ($beta_c$), providing a rigorous stability boundary for the architecture. Results further confirm the framework’s robustness under nonlinear saturation and demonstrate macroscopic scalability to populations of $N = 10^6$ agents.