Abstract. We present an empirically supported two-parameter model for predicting the geometry of semantic networks. Using exact linear programming to compute Ollivier–Ricci curvature on 11 networks across 7 languages, we show that the density parameter η = ⟨k⟩²/N drives a curvature sign change at a critical threshold ηc(N) = 3.75 − 14.62/√N (R² = 0.995). A second parameter, the clustering coefficient C, separates hyperbolic (C > 0.10) from Euclidean (C < 0.02) regimes. Dutch SWOW (η = 7.56) is the first real semantic network observed to cross the phase boundary into spherical geometry (κ̄ = +0.099), directly confirming the density-driven sign change. Sphere-embedded ORC across the Cayley–Dickson tower (S³, S⁷, S¹⁵) eliminates negative curvature for all 11 networks by d = 8, demonstrating that semantic hyperbolicity is entirely metric-dependent. Degree-matched null models reveal that semantic organization makes networks less hyperbolic than random graphs with the same degree—the opposite of naive expectation. The depression symptom network shows the largest effect: clinical structure eliminates 75% of the curvature. A Lean 4 formalization (25 modules, 8097 lines, 0 sorry in 7 core modules) provides machine-checked proofs of Wasserstein non-negativity, curvature bounds, and regime exclusivity. An independent implementation in the Sounio language cross-validates all results (33/33 sign agreement). 13 pages • 8 figures • 6 tables • 16 references
Demetrios C. Agourakis (Sat,) studied this question.