Three-point bound certification of the snub cube as an optimal spherical code in R3

Show that the Bachoc–Vallentin three-point semidefinite programming bounds certify that the configuration of the 24 vertices of a snub cube is an optimal spherical code on S^2, matching the known geometric proof of optimality.

Background

The authors obtain a three-point bound proof for the 9-point optimal code in R3 and conjecture the same approach should verify the 24-point snub cube, which is already known to be optimal by geometric arguments due to Robinson.

A semidefinite programming proof would provide an independent certification and illustrate further sharpness of three-point bounds in low-dimensional cases.