\section{Cells and fixed offsets}
\label{sec:cells}

For an odd prime $p$, use coordinates
\begin{equation*}
D_{2p}=\Fp\rtimes\Z_2,
\qquad
(x,\epsilon)(y,\delta)
=(x+(-1)^\epsilon y,\epsilon+\delta).
\end{equation*}
Let $A\subseteq D_{2p}\setminus\{e\}$ be nonempty.
Write $R\subseteq\Fp^\times$ and $F\subseteq\Fp$ for its rotation and reflection coordinate sets, and put $r=|R|$, $f=|F|$ and $n=r+f$.
Coordinates within either pool are distinct; the two pools may intersect.
For $s\in\Fp$ and $(c,\epsilon)\in\Fp\times\Z_2$, write $s+(c,\epsilon)=(s+c,\epsilon)$.

\begin{lemma}
\label{lem:cells}
There is a random arrangement of $A$ into $m$ consecutive cells, each beginning with a variable, such that $n/4\le m\le n$ and every cell has length at most four.
All variables have the same type and distinct coordinates.
Conditional on the nonvariable entries and the variable coordinate set $X$, their coordinates $x_1,\ldots,x_m$ form a uniform permutation of $X$.
There are fixed sets $\D_j\subseteq\Fp\times\Z_2$, $|\D_j|\le4$, such that, for every assignment of $X$ to the variables, the nonempty prefixes ending in cell $j$ are exactly
\begin{equation}
\label{eq:cell-prefix}
s_j+\D_j,
\qquad s_j=\sum_{h=1}^j x_h.
\end{equation}
The prefixes within each cell are pairwise distinct for every assignment.
\end{lemma}

\begin{proof}
In the \emph{rotation-majority case} $r\ge f$, take $m=r$ cells, each starting with a variable rotation.
For $f=2t$, append two reflection slots to each of $t$ distinct cells.
For $f=2t+1$, reserve the last cell for one reflection and append two reflections to each of $t$ earlier cells.
Any such choice of cells is allowed and is fixed before sampling.
The reservation is feasible since $2t+1\le m$ implies $t\le m-1$.
The types are
\begin{equation*}
(\underline R),\qquad(\underline R,F,F),
\qquad\text{possibly a final }(\underline R,F).
\end{equation*}
Choose independent uniform assignments of $R$ and $F$ to their slots, with an empty permutation space interpreted as a singleton.
Conditioning on the reflections leaves a uniform permutation of $X=R$.
Here $m\ge n/2$, and each cell has length at most three.

In the \emph{reflection-majority case} $f>r$, start with $f$ successive reflection slots.
Choose arbitrary distinct reflection slots after which to place the rotations, and fix an arbitrary bijection from these rotation slots to $R$ before sampling.
Only the reflection assignment is random in this case.
Assign $F$ uniformly to its slots, pair successive reflections into cells and retain each following rotation in its reflection's cell.
The types are
\begin{equation*}
(\underline F,[R],F,[R]),
\qquad\text{possibly a final }(\underline F,[R]),
\end{equation*}
where brackets indicate optional entries and odd reflection slots are variable.
Then $m=\lceil f/2\rceil>n/4$, and cells have length at most four.
Conditioning on the even reflection slots leaves a uniform bijection from the odd slots to the remaining set $X$.

Every nonfinal cell has even reflection parity, so the prefix preceding each variable has parity zero and its coordinate has coefficient $+1$.
Permutations within fixed parity classes that preserve the total product in a semidirect product also occur in \citep[Section~2]{Costa2025}.
Let $\epsilon_*\in\Z_2$ be the common variable type.
If cell $j$ has $d_j$ fixed entries following its variable, let $(u_{j,h},v_{j,h})$ be the product of their first $h$ entries, including $(u_{j,0},v_{j,0})=(0,0)$.
Set
\begin{equation*}
C_0=0,\qquad C_j=C_{j-1}+(-1)^{\epsilon_*}u_{j,d_j},
\end{equation*}
\begin{equation*}
\D_j=\left\{
\bigl(C_{j-1}+(-1)^{\epsilon_*}u_{j,h},\epsilon_*+v_{j,h}\bigr)
:0\le h\le d_j\right\}.
\end{equation*}
The product before cell $j$ is $(s_{j-1}+C_{j-1},0)$, proving \eqref{eq:cell-prefix}.
The possible odd parity of the final cell occurs after its variable and causes no exception.

Subtracting $s_j+C_{j-1}$, a paired rotation cell with reflections $y,z$ has prefixes
\begin{equation*}
(0,0),\quad(y,1),\quad(y-z,0).
\end{equation*}
Its equal-parity pair differs by $y-z\ne0$.
A paired reflection cell with second reflection $y$ and optional rotations $a,b$ has prefixes
\begin{equation*}
(0,1),\quad(-a,1),\quad(-a-y,0),\quad(-a-y+b,0).
\end{equation*}
If $a$ is absent, set $a=0$ and omit the second point; if $b$ is absent, omit the fourth.
Every remaining equal-parity pair differs by a present nonzero rotation coordinate.
The shorter cells satisfy the same checks.
\end{proof}