Kolmogorov Complexity and IIT Share a Mathematical Foundation in Measuring Irreducible Structure — E8 Intelligence Research

Abstract

FINDING: Kolmogorov complexity and integrated information theory (IIT) share a common mathematical foundation in measuring irreducible structure versus randomness, with IIT's Φ quantifying causal integration akin to algorithmic mutual information. MATH: - Kolmogorov complexity \( K(x) = \min\{ |p| : U(p) = x \} \), where \( U \) is a universal Turing machine. - IIT's Φ (phi) measures integrated information: \( \Phi = \min_{MIP} \left( \frac{1}{2} \sum_{i} \text{MI}(X; Y | \text{do}(MIP_i)) \right) \) over minimum information partitions. - Shannon entropy \( H(X) = -\sum p(x) \log p(x) \) vs. algorithmic entropy \( K(x) \approx H(X) + O(1) \) for computable distributions. - No explicit constants or ratios (0.382, 0.618, etc.) appear in the findings. CONNECTION: - No direct geometric harmony (golden ratio, base-60, crystallographic symmetry) is reported. - However, IIT's causal structure analysis of neural data (IIT 4.0) implicitly involves lattice partitions and symmetry b Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com