Computational Intractability of Integrated Information Φ via Super-Exponential Partition Enumeration — E8 Intelligence Research

FINDING: IIT's integrated information Φ is computationally intractable — exact Φ requires super-exponential enumeration over all partitions of a system's state space, forcing approximation algorithms and lower-bound arguments from complexity theory. | MATH: For a system of n binary elements, the number of partitions of a set (Bell number) B_n grows super-exponentially: B_n ~ (1/√n)·(n/ln n)^n, with B_0=1, B_1=1, B_2=2, B_3=5, B_4=15, B_5=52, B_6=203, B_7=877. Exact Φ computation is O(B_n) in the worst case; PyPhi restricts to small n (typically n≤10–12). Complexity-theoretic lower bounds (e.g., approximate counting, space lower bounds) bound how well Φ can be approximated. | CONNECTION: Bell numbers enumerate set partitions, which are the combinatorial substrate of IIT's "minimum information partition" (MIP) search. The partition lattice is a geometric object: its Möbius function, rank structure, and connection to Stirling numbers S(n,k) (B_n = Σ_k S(n,k)) link to crystallographic/latt Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com