Kolmogorov Complexity and Integrated Information Share a Core Measure of Irreducible Structure — E8 Intelligence Research

FINDING: Kolmogorov complexity and integrated information theory (IIT) share a common mathematical core: the measure of irreducible structure, linking algorithmic randomness to causal geometry. MATH: - Kolmogorov complexity \( K(x) = \min\{ |p| : U(p) = x \} \), where \( U \) is a universal Turing machine. - IIT's central quantity: \( \Phi \) (integrated information), defined via the Earth Mover's Distance (Wasserstein metric) between cause-effect repertoires of a system and its partitioned parts. - Shannon entropy \( H(X) = -\sum p(x) \log p(x) \) vs. Kolmogorov complexity: \( K(x) \approx H(X) + \text{constant} \) for computable distributions, but diverges for non-computable sequences. - No explicit constants (0.382, 0.618, etc.) appear in the provided sources; the link is structural. CONNECTION: - IIT 4.0 uses a lattice of partitions (the "cause-effect structure") to compute \( \Phi \). This lattice is isomorphic to the Boolean lattice of subsets, which underlies root sy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com