On the Dynamics of Sustained Adaptive Trajectories in Embedded Systems

Adaptive dynamics is commonly framed in terms of the stability of subsystems under non-equilibrium conditions. However, this state-based perspective may not fully account for why some environments appear capable of supporting extended adaptive progression, while others yield only short-lived stable configurations. To address this challenge, we propose a coupled dynamical description in which adaptive states are defined on the coupled system–environment embedding, where dynamics are jointly determined by informational structure and physical configuration. Within this formulation, viable adaptive transitions are confined to intermediate scales at which variation is both probabilistically accessible and adaptively consequential, giving rise to a scale-dependent transition structure. Analysis of the resulting dynamics identifies distinct adaptive regimes governed by the joint balance of informational accessibility and thermodynamic support. In the sustained regime, progressive transitions statistically increase the persistence of successive adaptive states while recursively reorganizing the transition structure governing subsequent evolution. As a consequence, adaptive trajectories need not follow the cumulative probabilistic suppression expected from independent stochastic transitions over extended timescales. The resulting framework yields a substrate-independent mesoscopic description of adaptive dynamics in which sustained progression emerges as a conditional property of the coupled informational and physical structure of interactive system–environment embeddings. The resulting formulation provides a general framework for analysing the probabilistic structure of adaptive states and transitions, with potential applications to early chemical organization, adaptive dynamics in non-equilibrium systems, and other physical processes within the framework of sustained stochastic variation, informational organization of the interactive environment and thermodynamic support.