Proposition: Fine-graining anchoring

If $C_{\mathsf{FG}}(s,\vec{m})\le N$, every $\mathsf{S}\in {\mathcal{A}}_N$ has the target law on any capacity-$N$ purification whose measured reduction is $\mathrm{diag}(\vec{m}/M)$.

Proof. On the certificate, Lemma lem:uniformity makes the $E^{\prime }$-law uniform on its $M$ labels. Nullity restricts joint $S,E^{\prime }$ outcomes to the prescribed pairs $(k(r),r)$. Composability therefore gives weight $1/M$ to each pair and marginal $m_k/M$ on $S$.

For another purification, pad the smaller environment to the larger of its size and the certificate size. Both fit $N$. Lemma lem:locality transfers the law.