An exact SWAP protocol

Let $X$ be the set of distinct route sites. Prepare (eq:encoder). At tick $k$:

• if $p_k=p_{k-1}$, apply the identity to all sites;

• otherwise, use the block $\{p_{k-1},p_k\}$, apply its SWAP channel, and apply identity singleton channels elsewhere.

This is one partition layer. The block diameter is at most $a_j$, and the schedule is determined by the target, not by the payload.