IIT's Phi: Formalizing Consciousness as Irreducible Integrated Information — E8 Intelligence Research

FINDING: Integrated Information Theory (IIT) formalizes consciousness as a quantity Φ (phi), measuring irreducible cause-effect power of a system's state, with recent work linking it to algorithmic information theory and lossless integration. | MATH: Core IIT quantity: Φ = minimum information partition (MIP) — the *irreducible* integrated information, computed via effective information: \( EI(X; MIP) = \min_{\text{partition}} \left[ H(X) - \sum_i H(X_i | \text{mechanism}) \right] \). Tononi's original: \(\Phi = \min_{\text{partition}} \frac{1}{2} \sum_i \left[ H(X^i) - H(X^i | \text{complement}) \right]\). Tegmark's contribution: quantum IIT, \(\Phi_Q\) via density matrices and von Neumann entropy \(S(\rho)\), with \(\Phi_Q = \min_{\text{partition}} S(\rho || \otimes_i \rho_i)\) (relative entropy). Algorithmic variant (arXiv:1405.0126): \(\Phi_A = K(X) - \min_{\text{partition}} \sum_i K(X_i | \text{rest})\) using Kolmogorov complexity \(K(\cdot)\), avoiding lossy integration. | CONNECT Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com