Phi, Compression, and Consciousness: Unifying IIT with Algorithmic Complexity — E8 Intelligence Research

FINDING: Integrated Information Theory (IIT) formalizes consciousness as a quantity Φ (phi), measuring irreducible causal integration in a system; Tegmark's analysis links Φ to physical complexity, while algorithmic information theory (AIT) reframes Φ via lossless compression. | MATH: Core IIT: Φ = minimum information partition (MIP) distance — Φ = min over partitions P of (H(X) − Σ H(X_i|P)) / H(X) (normalized), or in Tononi's 2008 form: Φ = Σ p(mechanism) · EI(mechanism) where EI = effective information = H(X) − H(X|mechanism). Tegmark's contribution: Φ scales with system size N and interaction strength J — approximate bound Φ ≤ N·log₂(k) for k-state units; AIT variant (arXiv:1405.0126): Φ_AIT = K(X) − K(X|mechanism) using Kolmogorov complexity K, avoiding lossy integration — requires K(X) ≈ K(X|mechanism) + log₂(1/ε) for ε-recovery. | CONNECTION: Φ's partition-minimization mirrors spectral gap in graph Laplacians — Φ ≈ λ₂ (algebraic connectivity) for symmetric networks, linking to r Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com