Bonnet et al., Lemma 8, for a partition given by a labeling $V → Fin p$: a refined labeling with at most $p · 2^p$ labels and one toggle on it whose radius-$r$ balls lie in the radius-$6r$ balls of the flip metric. Label the parts of $𝒫$ by $Fin |𝒫|$. The fibers of the refined labeling form the partition $𝒬$, and flips for a labeling with a symmetric toggle are exactly the flips for its fibers.