A Finite-Size Crossover in OllivierRicci Curvature of Random Regular Graphs

We study the sign of mean OllivierRicci curvature (ORC) on random k-regular graphs as a function of the density parameter η = k 2 /N. Exact linear programming computation of the Wasserstein-1 distance across N ∈ {50, 100, 200, 500, 1000} reveals a nite-size crossover at η c (N) = 3.75 − 14.62/ √ N (bootstrap 95% CI: η ∞ c ∈ [3.61, 3.84]; R 2 = 0.995). The transition slope scales as N −0.20 , indicating a smooth crossover rather than a sharp phase transition. Erd®sRényi and BarabásiAlbert graphs exhibit the same phenomenon at shifted η c , with the ordering η c (BA) < η c (ER) < η c (reg) explained by degree heterogeneity. A semi-analytical framework based on the Hehl matching formula reproduces the crossover with MAE = 0.009 and predicts a clustering-dependent phase boundary η c (N, C). Sphere-embedded ORC on the CayleyDickson tower (S 3 through S 127) saturates at a positive asymptote κ∞ > 0 with power-law exponent β = 0.283 ± 0.034, below the Johnson Lindenstrauss bound. A Lean 4 formalization provides machine-checked proofs of regime exclusivity (η < η c ⇒ κ < 0; η > η c ⇒ κ > 0). We validate the crossover on 6 real-world semantic networks spanning 4 languages, achieving 6/6 correct phase classication.