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arXiv:2306.03080 · The European Physical Journal Plus

Quantization of counterexamples to Dirac’s conjecture

Mauricio Valenzuela

Published Oct 23, 2023• 3 citations
arXivHTMLPDF
DOI
10.1140/epjp/s13360-023-04565-2

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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.