\section{Robust local separation}
\label{sec:local}

Fix an integer $K\ge1$ and consider the full probability space of Lemma~\ref{lem:cells}.
In the rotation-majority case this includes both independent permutation factors; in the reflection-majority case only the reflection assignment is random.
Let $\mathcal L_K$ be the event that, in every interval of at most $K$ cells, every permutation of its variables leaves every nonempty contiguous subword in that interval with nonidentity product.
For each realisation, these permutations move only variable coordinates among variable slots in the chosen window.
All nonvariable entries, including sampled nonvariable reflections, remain fixed during these permutations.

\begin{lemma}
\label{lem:local}
There are $m_0(K)$ and $c_K>0$, depending only on $K$, such that $m\ge m_0(K)$ implies
\begin{equation*}
\Pr(\mathcal L_K)\ge c_K
\end{equation*}
in either cell construction, uniformly over $p,A$, the permitted placements and the ratio of the two pool sizes.
\end{lemma}

The proof, given in Appendix~\ref{app:local}, applies the quantitative permutation local lemma to canonical assignments.
Its probability is unconditional in the cell construction.
In particular, no lower bound on each fixed-decoration fibre is asserted.
The condition excludes short identity subwords beginning at the first entry as well as those between two nonempty prefixes; the final conclusion still concerns nonempty prefixes only.