Finite evaluation does not separate the full quotient

The four $F_k$ distinguish the eight selected classes because their values are explicitly known. No assertion is made that these four observables separate every element of ${{\mathcal{S}}_{r_{\ast }}}^{\mathrm{red}}$.

Likewise, an equality of their selected finite values is not a global identity between all classical functionals. A full separation or reconstruction theorem would require a separating family and an appropriate topology.