Proposition: Exact relay attainment

On ${\mathbb{Z}}^D$ with the ${\ell }^1$ metric, for every $\tau \ge 1$ there is an $x$-perturbation pair with $d(x,A)=r\tau$ and readout laws ${\delta }_0,{\delta }_1$. Consequently $c_{\mathrm{lat}}=r$.

Proof. Take $A=0$, $x=r\tau e_1$, and place a classical bit $\beta \in \{0,1\}$ at $x$. At step $j=0,\dots ,\tau -1$, choose the phase containing the interval

Relocate the record from its right endpoint to its left endpoint. All other updates may be identities.

After $\tau$ ticks, the record is at $A$. Reading it gives ${\delta }_{\beta }$. The two runs differ only in the initial bit at $x$ and use identical rules. The lower bound $r$ combines with Proposition prop:upper.