Ollivier Ricci Curvature as a Geometric Biomarker for Biomedical Networks: From Ontology to Comorbidity Aging Trajectories

Network geometry offers a principled lens for understanding the structure of biomedical knowledge. We apply exact Ollivier-- Ricci curvature (ORC) -- a discrete analogue of Riemannian curvature computed via optimal transport -- to medical ontologies, disease comorbidity networks, biological interaction networks, and brain functional connectivity graphs. Three main results emerge. First, within a single database (the Human Phenotype Ontology), the formal IS-A taxonomy is hyperbolic ([Formula], tree-like), while the disease co-occurrence network is spherical ([Formula], clique-rich) -- a six-order-of-magnitude gap in the density parameter that the curvature phase transition framework predicts without free parameters. Second, age-stratified disease comorbidity networks from 8.9 million Austrian hospital patients reveal a geometric aging trajectory: mean ORC increases monotonically from [Formula] (age 20-30) to [Formula] (age 80+), driven by rising clustering and density that encode the accumulation of multimorbidity. Third, sedenion ([R]16) Mandel-brot orbit features -- exploiting the zero-divisor structure of the Cayley-Dickson tower -- discriminate ASD-like from ADHD-like brain network topology (AUROC = 0.990, sedenion-only), providing complementary geometric information to ORC. Canonical biological networks (C. elegans neural, E. coli gene regulatory, protein-protein interaction) are uniformly spherical, suggesting that evolved biological networks universally favour redundant, triangle-rich connectivity. All core mathematical claims are machine-verified in Lean 4 (0 sorry in 7 core modules). These results establish ORC as a quantitative geometric biomarker for biomedical network analysis and demonstrate that the same phase transition framework governing semantic networks extends to clinical and biological domains.