The Law of Large Numbers in Complex Systems: From Statistical Convergence to Compressed Models

This paper, originally written in 2021 and substantially reworked in 2026, demonstrates that complex adaptive systems — biological, cognitive, and social — systematically violate the conditions required by the law of large numbers (independence, identical distribution, finite expectation), and that these violations are structurally entailed by the requirements of persistence rather than incidental. We introduce agent replaceability R(a) as a formal measure of the degree to which a system component's structural contribution can be recovered after loss, and show that decreasing replaceability forces a transition from population-level statistical averaging (the "LLN-strategy") to individual-level Bayesian inference with strong priors (the "model-strategy"). A scale-dependent compressibility criterion based on Kolmogorov complexity provides a non-circular characterization of this transition: the model-strategy becomes necessary when the environment contains compressible structure at the agent's operational scale that statistical averaging would destroy. The central theoretical contribution is the encapsulation principle: the two strategies are not alternatives but hierarchical levels of a single architecture. Every internal model is a compressed encoding of statistical regularities accumulated through an LLN-process at a longer timescale — many cheap trials → compression into model parameters → one costly action. The law of large numbers is never abandoned; it descends to a lower floor of the system's organization. This nesting is demonstrated to be recursive. Cognitive biases are shown to be Bayesian-optimal under strong priors and asymmetric loss functions, not deviations from rationality. Concrete case studies include bacterial chemotaxis (transitional case), feline ballistic motor control (full model-strategy with cerebellar forward/inverse models), microgravity experiments revealing the hierarchical failure modes of internal models under distributional shift, and neural-network robotics providing substrate-independent experimental confirmation — robots trained solely for viability spontaneously invented gait transitions. Six testable predictions are derived, including a maladaptation prediction from the encapsulation principle. The framework unifies probability theory, non-equilibrium thermodynamics, Kolmogorov complexity, predictive processing, and evolutionary biology within a single formal architecture. Part of the Unified Structural Theory of Complex Systems research program. Keywords: law of large numbers, Bayesian inference, Kolmogorov complexity, encapsulation principle, replaceability, persistence, predictive processing, cognitive heuristics, motor control, microgravity, robotics, non-equilibrium thermodynamics License: Creative Commons Attribution 4.0 International (CC BY 4.0) Author – Boris Kriger¹²¹ Information Physics Institute, Gosport, Hampshire, United Kingdom boris.kriger@informationphysicsinstitute.net² Institute of Integrative and Interdisciplinary Research, Toronto, Canadaboriskriger@interdisciplinary-institute.org

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