Lemma: Complete destructive test

The map $\mathcal{M}$ is CPTP, including arbitrary untouched references. Discarding its classical record gives ${\mathrm{Tr}}_{S}X$. Its output contains no quantum $S$ factor.

Proof. Its Kraus operators are $|r{⟩}_R\otimes ⟨r{|}_S\otimes I_T$, whose squared moduli sum to the identity. Summing diagonal blocks gives the partial trace. The codomain states exactly which factors survive.