  We show that the late time Hartle-Hawking wave function for a free massless
scalar in a fixed de Sitter background encodes a sharp ultrametric structure
for the standard Euclidean distance on the space of field configurations. This
implies a hierarchical, tree-like organization of the state space, reflecting
its genesis as a branched diffusion process. An equivalent mathematical
structure organizes the state space of the Sherrington-Kirkpatrick model of a
spin glass.
