Quantization of Ricci Curvature in Information Geometry

TL;DR

This paper proves a long-standing conjecture that the volume-averaged Ricci scalar in certain information geometries is quantized to half-integers for tree and complete graphs, but not in general, and explores curvature properties in Gaussian networks.

Contribution

It resolves the 20-year-old conjecture on Ricci scalar quantization in information geometry and extends the analysis to Gaussian DAG networks with curvature dichotomy.

Findings

Proves Ricci scalar quantization for tree and complete-graph bitnets.

Disproves quantization in general by counterexamples with loops.

Identifies positive curvature in discrete bitnets and negative in Gaussian networks.

Abstract

In 2004, while studying the information geometry of binary Bayesian networks (bitnets), the author conjectured that the volume-averaged Ricci scalar <R> computed with respect to the Fisher information metric is universally quantized to positive half-integers: <R> in (1/2)Z. This paper resolves the conjecture after 20 years. We prove it for tree-structured and complete-graph bitnets via a universal Beta function cancellation mechanism, and disprove it in general by exhibiting explicit loop counterexamples. We extend the program to Gaussian DAG networks, where a sign dichotomy holds: discrete bitnets have positive curvature, while Gaussian networks form solvable Lie groups with negative curvature.