Three-point bounds for the square antiprism code in the sphere

An eight-point code in $S^2$ is a set of eight points on the unit sphere in $\boldsymbol{R}^3$. A completely monotonic potential function $f$ on $(0,4]$ is a smooth, nonnegative function whose derivatives alternate in sign. The energy $E_f$ is the sum of $f$ applied to the squared Euclidean distances between distinct code points. A square antiprism is a configuration consisting of two squares in parallel planes, offset by a $45^\circ$ angle, with variable separation.

Square-antiprism conjecture. For every completely monotonic potential function $f$ on $(0,4]$, some square antiprism minimizes the energy $E_f$ among all eight-point codes in $S^2$. Furthermore, the three-point bounds are always sharp for whichever code is optimal.

This conjecture concerns a continuous family of candidate optima, making it more difficult than the isolated-code cases. The optimal energy depends intricately on the potential; for example, for the Coulomb potential, the minimum within the antiprism family is described by a root of an irreducible polynomial of degree $48$.