  We study the structure and elastic energy of the ground states of crystalline
caps conforming to a spherical surface. These ground states consist of positive
disclination defects in structures spanning from flat and weakly curved
crystals to closed shells. We compare two different continuum theories and one
discrete-lattice approach for the elastic energy of defective caps. We first
investigate the transition between defect-free caps to single-disclination
ground states and show that it is characterized by continuous symmetry-breaking
transition. Further, we show that ground states with icosahedral subgroup
symmetries in caps arise across a range of curvatures, even far from the
closure point of complete shells. Thus, while superficially similar to other
models of 2D `jellium' (e.g. superconducting disks and 2D Wigner crystals), the
interplay between the free edge of crystalline caps and the non-Euclidean
geometry of its embedding leads to non trivial ground state behaviors, without
counterpart in planar jellium models.
