The Phase Transition in 5 Point Energy Minimization

Let R_s(r)=sign(s)/r^s be the Riesz s-energy potential. (This is the usual power-law potential.) This monograph proves the existence of a computable number S=15.048... such that the triangular bi-pyramid is the unique minimizer with respect to R_s, amongst all 5-point configurations on the sphere, if and only if s lies in (-2,0) or (0,S). This establishes the existence of the long-conjectured phase transition constant in 5-point energy minimization.