MIP-Based Φ Decomposition via Lattice Möbius Inversion in IIT — E8 Intelligence Research

FINDING: IIT formalizes consciousness as integrated information Φ, computed via the **minimum information partition (MIP)** — the weakest cut of a system's cause-effect structure — with lattice-based Möbius inversion providing the algebraic backbone for decomposing Φ into irreducible higher-order interactions. MATH: - **Φ (phi)** = min over partitions P of the effective information (EI) lost when the system is cut: Φ = min_P [ EI(system) − EI(P) ] where EI is measured via Kullback–Leibler divergence between forward and backward cause-effect repertoires. - **MIP** = argmin_P of the above; the partition that least damages integrated information. - **Möbius inversion on the lattice of partitions**: For a system's cause-effect structure, Φ can be expressed as a sum over the partition lattice: Φ = Σ_{π ∈ Π} μ(π, 1̂) · φ(π) where μ is the Möbius function on the partition lattice Π, and 1̂ is the top element (the full system). This isolates irreducible (holistic) contribution Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com