Interaction-Embedded Internal Time

This work develops a mathematically precise notion of internal time for multi-agent self-modifying systems, where time is not given by an external clock but is constructed from self-change and interaction. Each agent carries a local state, exchanges messages, and updates an internal time process via an additive functional whose increments depend on a metric of self-modification and an axiomatized interaction intensity. On top of this dynamical layer, the paper introduces value–meaning representations and semantically realized potentials, so that internal time becomes a value-regulated quantity rather than arbitrary bookkeeping. Within a global Markov framework, the internal time of each agent is treated as an additive functional of the augmented state process. Under Foster–Lyapunov conditions ensuring positive Harris recurrence, the paper proves the existence of almost-sure internal-time velocities (“social proper-time velocities”) and derives natural-law inequalities that couple these velocities to long-run averages of self-change, interaction intensity, and semantic drift. Symmetry and value-alignment assumptions yield rigorous synchronization results, while heterogeneous coupling and value structures generically lead to persistent de-synchronization. Non-ergodic regimes are characterized in terms of ergodic components, making clear that internal-time velocities are random variables indexed by long-run behavioral classes. The framework is related to time-changed Markov processes, classical models of subjective time, Age-of-Information metrics, and active-inference style semantic functionals. Several toy models—including a noisy consensus system and a three-agent threshold model—illustrate how internal-time synchronization can undergo phase-like transitions as coupling parameters vary. Potential applications include distributed and federated learning, human–AI collaboration, and social simulation, where internal time can serve as a local, semantically grounded tempo for scheduling computation, communication, and interaction.