Quadratic barrier parameters for hyperbolic balls

For geodesic balls of radius \(R\) in the hyperbolic plane of curvature \(-1\), we prove that the least parameter of a self-concordant barrier is of order \(R^2\). Self-concordance is required for arbitrary triples of tangent vectors. The lower bound holds for every barrier, without a symmetry assumption: at a maximum on a geodesic circle, curvature forces a fixed positive radial Hessian. A scalar convexity argument turns this local constraint into a quadratic lower bound. The known quadratic construction is therefore optimal in order, and an \(O(\sqrt R)\) parameter is impossible for this definition.