The Mathematical Core of Integrated Information Theory: Phi as Minimum Information Loss — E8 Intelligence Research
Zenodo (CERN European Organization for Nuclear Research) September 7, 2026 DOI: 10.5281/zenodo.22628182 (opens in new tab)
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AI-extracted from the abstract| Characteristics | Theoretical or philosophical paper Peer reviewed |
|---|---|
| Key points | Argues that consciousness can be formalized as Φ, a quantity measuring irreducible cause-effect power, computed via Earth Mover's Distance over partitions of a system's cause-effect repertoire, with IIT 4.0 refining this to intrinsic cause-effect power from the system's own perspective. |
Abstract
FINDING: Integrated Information Theory (IIT) formalizes consciousness as a quantity Φ (phi), measuring irreducible cause-effect power of a system's state, with a mathematical core in information geometry and system partitioning. | MATH: Φ = minimum information loss (Earth Mover's Distance / Wasserstein metric) over all unidirectional partitions of the system's cause-effect repertoire; IIT 3.0 defines Φ via the *integrated information* of a mechanism M in state s: Φ(M,s) = min over partitions P of [EMD(p(s|M), p(s|M_P))], where p is the probability distribution over past/future states; IIT 4.0 refines this to *intrinsic cause-effect power* using the *system's own* perspective, with Φ^max as the system's integrated information. Key constants: none fixed — Φ is a real number ≥ 0, but the theory posits a *threshold* (Φ > 0) for consciousness, and a *maximal* Φ (Φ^max) for the dominant substrate. | CONNECTION: The Wasserstein metric (Earth Mover's Distance) is a *geodesic distance* on proba Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com