IIT's Φ as a Partition-Lattice Functional: Consciousness via Irreducible Cause–Effect Structure — E8 Intelligence Research

FINDING: IIT's Φ is a partition-lattice functional — consciousness is defined as irreducible cause–effect structure, i.e. the degree to which a system's causal repertoire cannot be factored across any bipartition. | MATH: Φ = min over partitions P of [ D(p(system) ‖ ∏_k p(block_k)) ], where D is an information-theoretic distance (Earth Mover's / KL-type) on the cause–effect repertoire; the minimization runs over the **partition lattice** Π(S) of the system's elements. The lattice is partially ordered by refinement, with ⊥ = discrete partition (all singletons) and ⊤ = trivial partition (one block). Φ > 0 ⟺ the system occupies a non-atomic element of Π(S). | CONNECTION: Partition lattices are the canonical finite examples of **crystallographic/root-system symmetry**: Π(S) for |S| = n has Möbius function μ(⊥,⊤) = (−1)^{n−1}(n−1)!, and its order complex is the **order complex of the partition lattice**, whose homology is the top homology of the **Dowling–Wilson / Whitney algebra** — the sa Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com