  Combining a brilliant obserbation of A. Tumanov with our computational
approach to Thomson's 5-electron problem, we prove that the triangular
bi-pyramid is the unique global minimizer for the Rieze potential R_s(r) =
sign(s) r^{-s} amongst all configurations of 5 points on the unit sphere,
provided that s in (-2,0) or s in (0,13]. The lower bound is sharp and the
upper bound is pretty close to the presumed sharp cutoff of about 15.040908. I
hope to reach the sharp cutoff in a sequel paper. The discussion section of
this paper has a brief sketch of the approach I will take in the sequel to deal
with exponents in the range [13,15.04.0809...].
