Super-Exponential Φ Computation via Bell-Number Partition Lattice — E8 Intelligence Research

FINDING: IIT's Φ (integrated information) is computed over the **partition lattice** of a system's cause-effect structure; the number of partitions grows as the **Bell number B_n**, making exact Φ computation super-exponential (NP-hard / #P-hard), which is why PyPhi uses heuristics and why "higher-arity" causal analysis is computationally bounded. MATH: - Partition lattice: partitions of an n-element set form a lattice Π_n ordered by refinement; |Π_n| = B_n (Bell number). - Bell numbers: B_n = Σ_{k=0}^{n} S(n,k), where S(n,k) are Stirling numbers of the second kind. - Recurrence: B_{n+1} = Σ_{k=0}^{n} C(n,k) B_k. - Asymptotic growth: B_n ~ (1/√(n·r)) · (n/r)^n · e^{n(r−1/r)−1}, with r = W(n) (Lambert W). Super-exponential: log B_n ~ n log n − n log log n. - MIP (minimum information partition) search: Φ = min over partitions P of the "unpartitioned" cause-effect repertoire distance (earth-mover / KL-type). Exact MIP is NP-hard (Oizumi et al.; Barrett & Seth). - Toniann Pitassi's "deter Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com