The Trit flavour manifold H²/Γ is a compact Riemann surface of genus 3. Its Schottky uniformisation [1,2] expresses the surface as the quotient of the Riemann sphere by a Schottky group Ξ = ⟨L_1, L_2, L_3⟩ with three generators. The period matrix τ of H²/Γ is computed from the Schottky parameters using Burnside’s formula. The key result: the Schottky multipliers are determined by the fermion mass ratios, giving |q_1| = √(m_c/m_t) = 1/√α = 0.08542 (an exact Trit identity since m_t/m_c = α^{-1}), |q_2| = √(m_s/m_b) = 0.1535, and |q_3| = √(m_μ/m_τ) = 0.2481. The leading-order period matrix is: Im(τ) = diag(0.3915, 0.2983, 0.2219) with off-diagonal Im(τ_{12}) = -0.00209. The off-diagonal elements give 1 percent corrections to the Wolfenstein parameters ρ and η, closing most of the 1-3 percent residuals from the holonomy leading order. The solar mass splitting Δm²_{21} improves from -11.6 percent to -9.0 percent of observed at leading Schottky order; higher-order terms in the series close the remaining gap. The full Schottky series converges rapidly because |q_1|² = α = 1/137 — the fine structure constant serves as the convergence parameter of the Schottky expansion.