IIT 4.0's Phi: Wasserstein Metric and the Integration-Information Tradeoff — E8 Intelligence Research

FINDING: Integrated Information Theory (IIT) formalizes consciousness as a quantity Φ (phi) — the irreducible cause-effect power of a system's state — but current formalizations face a fundamental tension between information loss and integration, with algorithmic information theory offering a lossless alternative. | MATH: IIT 4.0 defines Φ via the *Earth mover's distance* (Wasserstein metric) between the cause-effect repertoire of a mechanism and its partitioned counterpart: Φ = min_partition W( p(mechanism_state) || p(partitioned_state) ). Tononi's original (2008) formulation used effective information: EI(X→Y) = H(Y) − H(Y|X) summed over minimal partitions. The algorithmic alternative (arXiv:1405.0126) replaces Shannon entropy with Kolmogorov complexity: Φ_AIT = K( system_state ) − Σ K( subsystem_states ), requiring no lossy compression. | CONNECTION: The Wasserstein metric is a *transportation* distance — its optimal solution on discrete state spaces corresponds to a linear programm

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com