\section{Two-pass repair}
\label{sec:repair}

Fix $k\ge2$, put $M=4k$ and suppose $m\ge k$.
Let $x_1,\ldots,x_m\in\Fp$, set $s_t=\sum_{j\le t}x_j$, $s_0=0$, and fix $\D_j\subseteq\Fp\times\Z_2$ with $|\D_j|\le B$.
All indices refer to the original ordering, and decorations remain attached to positions.
For $0\le t\le m$, define
\begin{equation}
\label{eq:potential-targets}
\mathcal V_t=\left\{
s_t+\sum_{j\in J}x_j+d:
\begin{array}{l}
J\subseteq\{t+1,\ldots,\min(m,t+M)\},\\
d\in\D_h,\quad h\in[1,m]\cap[t,t+M]
\end{array}\right\}.
\end{equation}
Then $|\mathcal V_t|\le B(M+1)2^M$ and $\mathcal V_m=s_m+\D_m$.

A source consists of a window $W=\{b+1,\ldots,b+w\}\subseteq[m]$ of length $w\le M$, disjoint sets $U,Q\subseteq W$ with $k\le|Q|\le2k$, and a cyclic order of $Q$.
A rotation is safe if
\begin{equation}
\label{eq:safe-rotation}
s_b+\sum_{j\in U}x_j+\sum_{j\in I}x_j+d\notin\mathcal V_t
\end{equation}
for every nonempty proper prefix $I$ of the rotation, every $d\in\bigcup_{h\in[1,m]\cap[b,b+M]}\D_h$ and every $t\in\{0,\ldots,m\}$ with $|t-b|>2M$.
A prefix denotes its set of original labels when used in a sum.
The window $W$ bounds original labels; the rotating list may occupy different positions.
Since the safety test includes every decoration in the source interval, it protects the list's proper prefixes whenever they end at positions in $[1,m]\cap[b,b+M]$ and the list has starting sum $s_b+\sum_{j\in U}x_j$.

Partition $[m]$ into consecutive intervals $Q_i=\{b_i+1,\ldots,b_i+n_i\}$, $k\le n_i\le2k$, $1\le i\le a$.
Such a partition exists by using lengths $k+u,k,\ldots,k$ when $m=ak+u$, $0\le u<k$.
For $c\in Q_i$, let $T_i(c)$ be the original cyclic order of $Q_i$ starting after $c$ and omitting $c$.
Before sampling coordinates, list the following sources:
\begin{enumerate}
\item For each $i$, use $b=b_i$, $W=Q_i$, $U=\varnothing$, $Q=Q_i$ and its original cyclic order.
\item For each $i<a$ and $(c,d)\in Q_i\times Q_{i+1}$, use $b=b_i$, $W=Q_i\cup Q_{i+1}$, $U=\{c\}$, $Q=(Q_i\setminus\{c\})\cup\{d\}$ and cyclic order $(T_i(c),d)$.
\end{enumerate}
Let $\mathcal R_{\mathrm{menu}}$ assert that every listed source has a safe rotation.
The list has at most
\begin{equation}
\label{eq:menu-size}
a+\sum_{i<a}n_i n_{i+1}\le(1+4k^2)m/k
\end{equation}
items, all satisfying the source restrictions.

\begin{proposition}
\label{prop:repair}
If $\mathcal R_{\mathrm{menu}}$ holds, there is a permutation $\sigma$ of $[m]$ with $|\sigma(j)-j|\le4k-2<M$ such that, on setting $u_j=\sum_{h\le j}x_{\sigma(h)}$, equality between a point of $u_i+\D_i$ and a point of $u_j+\D_j$, $i<j$, implies $j-i\le4M$.
\end{proposition}

\begin{proof}
For each source of the first family, choose a safe rotation, let $c_i$ be its first label, and write the block as $(c_i,T_i(c_i))$.
After this first pass, completed blocks retain their sums, so every proper nonempty prefix inside block $i$ is $s_{b_i}+\sum_{h\in I}x_h$.
With its position's decoration, this lies in $\mathcal V_{b_i}$ and avoids $\mathcal V_t$ for $|t-b_i|>2M$.
Put a cut at $h_i=b_i+1$.
Since $n_i\ge2$, its prefix is proper and has sum
\begin{equation}
\label{eq:cut-sum}
s_{b_i}+x_{c_i}.
\end{equation}

The list between cuts $h_i$ and $h_{i+1}$ is $P_i=(T_i(c_i),c_{i+1})$, of length $n_i$.
Rotate it safely using the second-family source indexed by $(i,c_i,c_{i+1})$.
These position intervals are disjoint.
Their preserved sums, together with the fixed first label $c_1$, preserve every cut sum \eqref{eq:cut-sum}, even if its label moves.
Thus $s_{b_i}+x_{c_i}$ remains the starting sum for $P_i$, even though the sum through the original boundary $b_i$ may change.
For $h_i<j<h_{i+1}$, the final prefix is
\begin{equation*}
u_j=s_{b_i}+x_{c_i}+\sum_{h\in I}x_h,
\end{equation*}
where $I$ is a proper nonempty prefix of the chosen rotation of $P_i$.
The labels in $\{c_i\}\cup I$ lie within $M$ original positions after $b_i$, and $j\in[b_i,b_i+M]$.
Thus \eqref{eq:potential-targets} and \eqref{eq:safe-rotation} give membership and protection at the original anchor $b_i$.

Leave the final tail $T_a(c_a)$ unchanged.
Its preceding cut sum is preserved, so its prefixes retain their first-pass values.
The cuts, interiors between cuts and final tail cover every position $j<m$.
For each such $j$, the corresponding original boundary $a_j=b_i$ satisfies
\begin{equation}
\label{eq:protected-clocks}
\begin{gathered}
|j-a_j|\le M,\qquad u_j+\D_j\subseteq\mathcal V_{a_j},\\
(u_j+\D_j)\cap\mathcal V_t=\varnothing
\quad\text{if }|t-a_j|>2M.
\end{gathered}
\end{equation}
When there is one block, the first pass alone gives these conclusions.
At the final position, $u_m=s_m$ and $u_m+\D_m=\mathcal V_m$; set $a_m=m$.

Each pass moves labels within intervals of length at most $2k$, so their composition has displacement at most $4k-2$.
If decorated points at $i<j$ coincide, protection at $i<m$ and membership at $j$ force $|a_i-a_j|\le2M$.
Hence $j-i\le|i-a_i|+|a_i-a_j|+|a_j-j|\le4M$.
\end{proof}

\begin{lemma}
\label{lem:replay}
Suppose a cell realisation satisfies $\mathcal L_K$ for $K\ge6M+1$.
After any variable permutation $\sigma$ with $|\sigma(h)-h|\le M$, no equal nonempty prefixes can end in distinct cells $i<j$ with $j-i\le4M$.
\end{lemma}

\begin{proof}
Equal prefixes give an identity subword by left cancellation.
It begins after the selected prefix in cell $i$, which already contains that cell's variable, and ends in cell $j$.
The subword therefore starts strictly after the variable of cell $i$ and uses exactly the variable slots $i+1,\ldots,j$; its other entries remain fixed.
The interval
\begin{equation*}
W=[\max(1,i-M),\min(m,j+M)]\cap\Z
\end{equation*}
contains the subword's cells and every original variable label $\sigma(h)$ it uses, and $|W|\le j-i+2M+1\le6M+1$.
The prescriptions $h\mapsto\sigma(h)$ for $i+1\le h\le j$ form a partial injection of $W$ into itself.
Extend it to a permutation of $W$ by matching unused domain and range points.
That local permutation reproduces the identity subword in the original realisation, contradicting $\mathcal L_K$.
\end{proof}