Renormalization-Group Geometry of Homeostatically Regulated Reentry Networks

Abstract

Reentrant computation-recursive self-coupling in which a network continuously reinjects and reinterprets its own internal state-plays a central role in biological cognition but remains poorly characterized in neural network architectures. We introduce a minimal continuous-time formulation of a homeostatically regulated reentrant network (FHRN) and show that its population dynamics admit an exact reduction to a one-dimensional radial flow. This reduction reveals a dynamically fixed threshold for sustained reflective activity and enables a complete renormalization-group (RG) analysis of the reentry-homeostasis interaction. We derive a closed RG system for the parameters governing structural gain, homeostatic stiffness, and reentrant amplification, and show that all trajectories are attracted to a critical surface defined by $\gamma\rho=1$, where intrinsic leak and reentrant drive exactly balance. The resulting phase structure comprises quenched, reactive, and reflective regimes and exhibits a mean-field critical onset with universal scaling. Our results provide an RG-theoretic characterization of reflective computation and demonstrate how homeostatic fields stabilize deep reentrant transformations through scale-dependent self-regulation.