Configurations of N points on the two-sphere that are stable with respect to the Riesz senergy have a structure that is largely hexagonal. These stable configurations differ from the configurations with the lowest reported N-point s-energy in the location and structure of defects within this hexagonal structure. These differences in energy between the stable and minimal configuration suggest that energy scale at which defects play a role. This work uses numerical experiments to report this difference as a function of N, allowing us to infer the energy scale at which defects play a role. This work is presented in the context of established estimates for the minimal N-point energy, and in particular we identify terms in these estimates that likely reflect defect structure.

Given a natural number N , one may ask what configuration of N points on the two-sphere minimizes the discrete generalized Coulomb energy. If one applies a gradientbased numerical optimization to this problem, one encounters many configurations that are stable but not globally minimal. This led the authors of this manuscript to the question, how many stable configurations are there? In this manuscript we report methods for identifying and counting observed stable configurations, and estimating the actual number of stable configurations. These estimates indicate that for N approaching two hundred, there are at least tens of thousands of stable configurations.

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