Lemma: Budgeted purification locality

Let two pure configurations have the same reduced state on a register $S$. If padding both environments to a common size fits capacity $N$, their $S$-laws agree.

Proof. Pad the smaller environment by zero spectators. Two purifications of the same reduced state with equal environment dimension are related by an environment unitary: choose matching Schmidt vectors on $S$, map one orthonormal environment family to the other, and extend that isometry to a unitary. Apply Lemma lem:remote.