Fisher Information Metric Links Dirichlet Distributions to Spherical IIT Geometry — E8 Intelligence Research

FINDING: Fisher information metric defines a Riemannian geometry on probability distributions, with constant positive curvature for Dirichlet distributions, linking to spherical geometry in integrated information theory (IIT). MATH: Fisher information metric \( g_{ij}(\theta) = \mathbb{E} \left[ \partial_i \log p(x|\theta) \partial_j \log p(x|\theta) \right] \). For Dirichlet distributions (parameter vector \(\alpha\)), the metric induces constant positive curvature on the simplex, proportional to \(\sum \alpha_i\). IIT's integrated information \(\Phi\) is computed via the Fisher metric on the space of probability distributions over system states, with spherical geometry arising from normalization constraints. CONNECTION: Dirichlet distributions' constant positive curvature (sphere-like) yields geodesic distances involving \(\arccos\) of dot products, naturally producing ratios like \(\pi/3\) (60°) and \(\pi/5\) (36°), linking to base-60 angular divisions and icosahedral symmetry. Th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com