A Large-Diameter Fundamental-Gap Lower Bound for Horoconvex Domains

We prove a large-diameter fundamental-gap lower bound for compact horoconvex domains in real hyperbolic space of curvature $-1$. The geometric part reduces large horoconvex domains to a fixed-width radial-height problem in all dimensions. The analytic part proves the needed radial-height theorem by comparing the low-energy Dirichlet form with a limiting angular operator on the sphere, while the radial complement is separated by a one-dimensional branch gap and endpoint Green estimates. The result gives the polynomial $D^{-3}$ scale matching the Nguyen–Stancu–Wei large-diameter upper bound.