Lemma: Class anchoring

Let $A\subseteq {\bigcup }_j{K}_j$, with $A\cap K_j\ne \varnothing$ exactly when $j\in J$. On a tower state,

In particular, pinching one member of each selected class is sufficient. The equality need not hold on an arbitrary operator outside the tower-supported subspace.

Proof. On a member $q\in K_j$, a branch cross term carries $|r_j⟩⟨{r^{\prime }}_j|$. Pinching that member preserves the term precisely when $r_j={r^{\prime }}_j$. The number and choice of members pinched within the class are therefore irrelevant on the branch expansion. The condition $A\subseteq {\bigcup }_j{K}_j$ excludes additional pinching of residual degrees of freedom.