Emergent Continuity and Long-Time Dynamics in Finite Nonlinear Networks

A foundational tension exists between the discrete, finite nature of physical microstates and the continuous formalisms used to describe macroscopic dynamics. In this theoretical letter, we utilize the framework of iterative topological condensation to propose a physically motivated resolution. We model the physical substrate as a globally energy-conserving network of N discrete, finite nonlinear oscillators. We demonstrate that continuous descriptions can emerge as effective coarse-grained representations via Kuramoto phase-locking and relational entropy reduction. Furthermore, while the network possesses a finite structural capacity, the strictly conserved kinetic energy drives periodic cycles of structural dissolution and subsequent re-condensation. This avalanche dynamic acts as a temporal engine, allowing a spatially and materially finite system to ergodically explore a potentially unbounded sequence of macroscopic states over time. This framework provides a possible dynamical mechanism bridging discrete microstates and the emergence of continuous macroscopic variables in complex systems.