Living Intelligence: Information-Constrained Stochastic Control with State-Dependent Diffusion, Adaptive Foraging, and the Lyapunov–Information Equivalence Principle
Year: 2026
Type: article
Abstract: ABSTRACT I introduce Living Intelligence as a rigorous control-theoretic framework for continuously adaptive intelligent systems—formalized as continuously adaptive stochastic controllers (CASCs)—operating under persistent stochastic perturbations and strict information-rate constraints. A CASC is modeled as a controlled Itô diffusion with state-dependent diffusion coefficient Σ(x) ∈ ℝⁿˣⁿ and an information-regularized quadratic objective that jointly penalizes state-control cost and mutual information consumption. Three principal analytical results are established. Theorem 1 proves stochastic ultimate boundedness via generator-based Lyapunov analysis with explicit, computable constants (α, c). Theorem 2 characterizes a compact stochastic invariant ellipsoid and provides the explicit exponential convergence bound 𝔼[V(Xₜ)] ≤ e^{−αt} V(x₀) + c/α. Theorem 3 establishes stability-preserving exploration under a Fisher-information threshold switching policy, with a quantitative rate condition ensuring no loss of stochastic boundedness. The unifying result is the Lyapunov–Information Equivalence Principle (LIEP, Principle 1): a CASC simultaneously achieves Lyapunov boundedness and information-regularized optimality if and only if there exist α > 0 and c ≥ 0 such that ℒV(x) + λ (d/dt) I(Xₜ ; Uₜ) ≤ −α V(x) + c holds along all closed-loop trajectories. This single inequality encodes Lyapunov stability, information-theoretic optimality, and the quantitative trade-off between them. Explicit connections are established to rate–distortion theory, input-to-state stability, and maximum-entropy reinforcement learning. Three falsifiable quantitative predictions are derived. No phenomenological, biological, or consciousness claims are made. I have introduced Living Intelligence as a rigorous mathematical paradigm for continuously adaptive intelligent systems, formalized as continuously adaptive stochastic controllers (CASCs) and anchored by the Lyapunov–Information Equivalence Principle (LIEP). The LIEP is both necessary and sufficient for a CASC to simultaneously achieve stochastic ultimate boundedness, information-regularized optimality, and stability-preserving exploration. Three theorems establish these guarantees with explicit, computable constants; three falsifiable predictions make the framework experimentally accessible. What distinguishes this work from prior visionary treatments of adaptive or "living" machine intelligence is not ambition but mathematical discipline: every claim is supported by a theorem, every theorem has verifiable conditions, every constant is explicit, and no phenomenological, biological, or consciousness claims are made at any point. This discipline is the prerequisite—not the obstacle—for building a foundation that others can rigorously extend, instantiate, and cite. The LIEP signature inequality connects adaptive control, information theory, and continuous-time machine learning through a single expression. Its rate–distortion interpretation [8,12], ISS generalization [18], and maximum-entropy RL connection [14,23] position it as a cross-disciplinary citation anchor. Numerical validation, nonlinear extensions, and mean-field multi-agent generalizations are the natural and immediate next steps.
Author: LOOI, KWOK WAH
Language: English
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7155007533
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