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: \( EI(X; MIP) = minₚₐᵣₜᵢₜᵢₒₙ [ H(X) - ∑_i H(X_i | mechanism) ] \). Tononi's original: \(Φ = minₚₐᵣₜᵢₜᵢₒₙ 1/2 ∑_i [ H(X^i) - H(X^i | complement) ]\). Tegmark's contribution: quantum IIT, \(Φ_Q\) via density matrices and von Neumann entropy \(S(ρ)\), with \(Φ_Q = minₚₐᵣₜᵢₜᵢₒₙ S(ρ || ⊗_i ρ_i)\) (relative entropy). Algorithmic variant (arXiv:1405.0126): \(Φ_A = K(X) - minₚₐᵣₜᵢₜᵢₒₙ ∑_i K(X_i | rest)\) using Kolmogorov complexity \(K(·)\), avoiding lossy integration. | CONNECT Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com