Wasserstein Distance as a Geometry-Aware Metric Linking Optimal Transport to IIT 4.0 Consciousness Measure — E8 Intelligence Research

FINDING: Wasserstein distance provides a geometry-aware metric between probability distributions that remains well-defined when KL divergence and other classical divergences break down, and it is being connected to Integrated Information Theory (IIT) 4.0's measure of consciousness (Φ) via optimal transport formulations. | MATH: p-Wasserstein distance W_p(μ,ν) = (inf_{γ∈Γ(μ,ν)} ∫ d(x,y)^p dγ(x,y))^{1/p}, where Γ(μ,ν) is the set of couplings (joint distributions with marginals μ,ν). For p=1: W_1(μ,ν) = ∫|F_μ(x) − F_ν(x)|dx (1D). For p=2: W_2²(μ,ν) = ∫|F_μ⁻¹(t) − F_ν⁻¹(t)|²dt. KL divergence D_KL(P‖Q) = Σ P(x)log(P(x)/Q(x)) is undefined when supports don't overlap; Wasserstein remains finite and metrizes weak convergence. IIT 4.0 defines Φ-structure via cause-effect repertoires; the distance between repertoires can be measured by W_p, giving a metric on the space of integrated information structures. | CONNECTION: The Wasserstein space (P_p(X), W_p) is a geodesic space with a formal Rieman Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com