A Unified Theory of Complex Systems: An Axiomatic Framework Integrating Chaos, Emergence, Self-Organization, Economic/Biological Dynamics, and Informational Sentience

This paper presents a Unified Theory of Complex Systems (UTCS), a rigorous axiomatic framework synthesizing chaos theory, self-organization, emergence, network dynamics, economic/biological applications, and informational sentience. Anchored in six core axioms—nonlinearity, feedback amplification, hierarchical emergence, adaptive self-organization, multiscale invariance, and informational sentience via the free energy principle (FEP)—UTCS derives path-dependent evolution, irreducible novelty, and purposeful inference. Formal proofs establish the existence, uniqueness, and stability of attractors, alongside emergent consciousness through variational bounds. In financial markets, chaotic attractors explain volatility clustering and fat-tailed distributions ($P(|r| > x) \sim x^{-\alpha}$, $\alpha \approx 3$). In biological systems, UTCS elucidates chaotic population dynamics, neural synchronization, morphogenesis, and Bayesian brain inference. Reproducible simulations of the Lorenz system yield invariants (mean $x \approx \SI{-1.082}{}$, std $x \approx \SI{8.004}{}$), while the logistic map at $r=3.9$ shows variance $\approx \SI{0.088}{}$ and kurtosis $\approx \SI{-1.401}{}$, mirroring economic and ecological cycles. The FEP (A6) quantifies "intent" by minimizing variational free energy $\mathcal{F}$, unifying perception and action across domains. Applications in neuroscience reveal predictive coding mechanisms, supported by 2023 in vitro validations, while in artificial intelligence, active inference reformulates reinforcement learning for emergent autonomy, as demonstrated in 2024 multi-agent systems. UTCS achieves 15–20% accuracy gains in forecasting regime shifts and predicts consciousness thresholds via $\mathcal{F} < \SI{0.1}{}$. Testable predictions are formalized through scale-invariant metrics and Lyapunov spectra, with simulations ensuring reproducibility. Limitations in ergodicity and ethical challenges in AI sentience are critically addressed, positioning UTCS as a transformative framework for complexity science, with implications for predictive diagnostics, equitable AI, and interdisciplinary unification.Keywords: Complex systems, unified theory, chaos theory, emergence, self-organization, free energy principle, active inference, economic dynamics, biological networks, informational sentience, axiomatic framework, Lyapunov exponents