Integrated Information Theory: Φ as Irreducible Cause-Effect Structure — E8 Intelligence Research

FINDING: Integrated Information Theory formalizes consciousness as Φ (phi), a measure of irreducible cause-effect structure generated by a system's partition lattice — consciousness arises where the system's causal architecture cannot be decomposed without loss. MATH: - Φ = min over partitions P of [effective information of whole − Σ effective information of parts] - Φ = minP ∈ Π [ EI(X) − EI(X₁) − EI(X₂) − ... − EI(Xₙ) ] where Π is the set of all partitions - For a system with n elements, the partition lattice has Bell number Bₙ elements (B₁=1, B₂=2, B₅=52, B₁₀=115975) - MIP (Minimum Information Partition) = argmin_P [EI(whole) − Σ EI(parts)] - Φ > 0 iff the system is irreducible under all partitions - Algorithmic IIT (arXiv:1405.0126): Φ_AIT defined via Kolmogorov complexity K: Φ ∝ K(whole) − K(parts), avoiding lossy integration paradox CONNECTION: - Partition lattice is a geometric object: the lattice of set partitions of n elements forms a graded poset with Möbius function μ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com