Remark: What has been evaluated

The certificate specifies the entire input set, all admissible Boolean functions through exhaustive orbits, every kernel entry, every posterior value, and explicit integer comparisons. It does not rely on a floating-point search or an unprinted optimizer.

The general three-point joining optimizer has not been evaluated here. That is not the selected next theorem: the selected next theorem is the finite Hellinger certificate, which is now proved.