Why a separate probe is needed

The instrument ${\mathcal{M}}_C$ supplies a label and surviving matter. It does not, by itself, make $m(\beta )$ or $n_{\ell }(\beta )$ an expectation value of that matter.

We now construct one explicit finite comparison probe. Its inputs are the already calculated table entries. The construction is deliberately a table realization, not a physical derivation of those entries from a spin-two interaction.