Living Intelligence: Information Constrained Stochastic Control with State Dependent Diffusion, Adaptive Foraging, and the Lyapunov Information Equivalence Principle

This paper introduces Living Intelligence as a control-theoretic framework for continuously adaptive intelligent systems, formalized as continuously adaptive stochastic controllers (CASCs). The framework establishes stochastic stability guarantees via Lyapunov analysis, characterizes a compact invariant set with explicit exponential convergence rates, and proves stability-preserving exploration under a Fisher information threshold switching policy. The unifying result is the Lyapunov-Information Equivalence Principle (LIEP): a necessary and sufficient condition linking Lyapunov boundedness and information-regularized optimality in a single inequality. Connections to rate-distortion theory and maximum-entropy reinforcement learning are established. No phenomenological or biological equivalence claims are made.