Optimality and uniqueness for the Thomson problems with n=9 and n=10

The Thomson problem asks how to place identical charges on a sphere to minimize the sum of the reciprocals of their pairwise distances. We prove that the minimizing arrangements for nine and ten charges are unique up to rotations, reflections, and relabeling. For nine charges, the arrangement has two aligned equilateral triangles at opposite heights and three equally spaced equatorial points facing the sides of the resulting prism. For ten charges, it has two opposite poles and two parallel squares at opposite heights, with one square rotated by 45 degrees relative to the other. In each case, a one-variable equation determines the optimal height. Polynomial energy bounds show that every arrangement with energy no greater than the candidate's must have each pairwise distance close to one occurring in the candidate. The fact that the points lie in three-dimensional space then determines how these distances fit together in one arrangement. After alignment and removal of rotational freedom, a uniform positive second-derivative bound forces the arrangement to equal the candidate exactly. The finite checks use exact arithmetic, exhaustive enumeration, and rigorous interval bounds. Verification code and certificates for the nine-point result: https://doi.org/10.5281/zenodo.23852043.Verification code and certificates for the ten-point result: https://doi.org/10.5281/zenodo.23845065.