Radial Hyperbolic Measures: Shell Geometry, Pyramid Limits, and Gaussian Phase Transitions

In high-dimensional hyperbolic space, concentration of a radial measure near a shell need not determine the pyramid limit: angular concentration and hyperbolic expansion alter separation. An effective radius is the scale on which positive-mass sets actually separate, which the raw radius need not give. With vanishing rescaling and radial fluctuations, radius convergence gives weak convergence to the pyramid of all metric measure spaces with the resulting diameter bound. At modal radial Gibbs shells, exponential decay of intrinsic shell curvature and dimension-normalized tangential Bakry-\'Emery Ricci curvature recovers the radius. The Gaussian intrinsic to hyperbolic volume and that obtained by wrapping a Euclidean Gaussian have different critical orders. They are L\'evy below those orders, infinitely dissipate above them, and at criticality converge to the corresponding diameter-bounded pyramids.