Sharp Relaxed Triangle Inequality for Rényi Divergence of Order 1/2 Between Univariate Gaussians

For Kullback–Leibler divergence between multivariate Gaussians, a sharp relaxed triangle inequality was recently obtained by Xiao et al. (arXiv:2602.02577v1, 2026). This paper develops an analogous extremal program for the Rényi divergence of order α = 1/2 in the one-dimensional Gaussian family. The main result is a sharp closed-form supremum S(Δ, Δ) = 2 sinh(2s*), where s* solves an explicit transcendental equation, achieved by a mirror-symmetric configuration. The proof is fully analytic and proceeds by sector decomposition: a Cauchy–Schwarz bound for the positive sector, a Schur-concavity argument for the negative sector, and a contrapositive elimination for the mixed sector. We also establish that the supremum cannot be achieved at zero means (a structural difference from the KL case), provide a budget-decomposition functional for general Rényi order α, and prove a tensorization reduction under commuting covariances in higher dimensions. The package contains the main paper and supporting documents: a self-contained Schur-concavity proof, and three addenda covering the mixed-sector boundary analysis, edge bounds with symbolic verification, and the analytic closure of the mixed-sector interior.