An explicit nonempty regular route

Choose an axial regular spatial segment of developed length $L>0$, compactly inside $B_R$. Place stations $A_0,\dots ,A_n$ at equal spacing $\ell =L/n$. Let the temporal laboratory window be long enough for the following finite schedule, fixed before the run.

At $A_0$, install:

• the one $m$-qubit clock and a $g$-bit catalogue;

• the table and clean work memory;

• a $p$-qubit unknown payload, $p\ge 1$;

• a probe for the optional normalized score in (eq:evaluation).

Every other station has a blank $p$-qubit payload register and a blank classical history buffer.

Carry out the finite clock–table instrument locally, retain all outcomes, and wait through a prescribed positive buffer interval. This waiting interval is an installed schedule, not a singular host event. Let the complete retained history have $b$ bits, including the clock code, catalogue index, sign outcome, and report. At each stage:

1. copy the classical history into the next outgoing buffer, using $b$ positive-duration local copying operations;

2. carry that buffer to the next station;

3. exchange the two adjacent payload contents by opposite carriage paths, not by an instantaneous remote gate.

Assign the carriage a duration $2\ell$ in developed time. Its speed is $1/2$. The complete route and its closure stay in the regular laboratory.