Theorem: Buffered null approximation in the declared library

For every target $v=(\tau ,x)\in \partial {\mathcal{L}}_4\setminus \{0\}$ and integer $L\ge 1$, the library of Specification spec:bus has a finite identity-transmission protocol with:

relative to the comparator-null travel time $\tau$. Its complete final channel is the input message at the receiver tensor blank states on all other declared registers.

A realization retaining all registers uses $2L+1$ qubits, $L$ carrier legs, and $2L$ SWAP gates. With gate duration ${\delta }_L=L^{-2}$, the largest declared gate-generator norm is $\pi \hslash L^2$.

Proof. Place $L+1$ memories along the straight spatial segment, with successive displacement $x/(sL)$. A carrier travels each segment in time $\tau /L$; its speed is comparator-null since $s\Vert x/(sL)\Vert =\tau /L$.

At each leg, SWAP the message from its current memory into a fresh blank carrier, propagate it, and SWAP it into the next blank memory. During both gates the carrier and memory are coincident and stationary. There are $2L$ dwell intervals of duration $L^{-2}$. Thus the total arrival time is $\tau +2/L$, with the required spatial endpoint.

A SWAP transfers the whole input operator, including every untouched reference. Induction over the legs puts the unknown input at the last memory and returns every earlier memory and every carrier blank. This proves the complete channel identity.

The stationary memories number $L+1$, and the carriers number $L$. The gate count is $2L$. For the SWAP operator $S$,

satisfies $e^{-i{\delta }_L{H}_{\mathrm{swap}}/\hslash }=S$ and has norm $\pi \hslash /{\delta }_L$. The claimed generator bound follows.