The Regulated Entropy Injection Theorum: A Living Information Theory Extension on Persistence, Noise and Play

This paper introduces the Regulated Entropy Injection Theorem (REIT), a structural result in stochastic dynamical systems that formalizes how adaptive systems persist under environmental volatility. Classical stochastic theory treats noise as either destabilizing or externally imposed. REIT demonstrates that in switching attractor landscapes with bounded viability constraints, persistence is maximized not by minimizing noise nor by maintaining constant stochasticity, but by scheduling entropy exposure at an interior rate. The theorem emerges from the opposition of two monotonic processes derived from large deviation theory: escape hazard increases with noise amplitude, while relocation efficiency following environmental switching also increases with noise. Their composite hazard therefore admits an interior minimum, yielding a persistence-maximizing entropy schedule. Computational validation in switching double-well systems confirms: Existence of interior entropy optima Superiority of temporally structured noise over constant noise Volatility-dependent phase transitions in optimal scheduling regimes Robustness of the effect under parameter perturbation and dimensional extension Within Living Information Theory, REIT formalizes regulated entropy injection as an operational mechanism of persistence under perturbation. Geometry stabilizes attractors; regulated stochasticity enables traversal between them. Together, they produce a unified dynamical account of adaptive systems facing environmental change. The theorem provides a mechanistic interpretation of biological play, neural variability, regenerative destabilization, and structured exploration. More broadly, it proposes a general design principle for adaptive biological and artificial systems: exploration must be temporally regulated within viability constraints to maximize persistence under switching environments. This work bridges nonlinear dynamics, large deviation theory, information geometry, and biological theory, and generates empirically testable predictions for both living and engineered systems.