Lemma: Endpoint, waiting, and uniform resource bounds

For $j>\kappa$ and every null target with ${\Vert x\Vert }_2\le L$, the construction (eq:null-construction) satisfies

At most one extra terminal waiting tick is needed. The total tick count and the number $q_j$ of initialized sites obey

For $j\ge 2\kappa$,

Proof. The spatial estimate is (eq:rounding-error). Put $r={\Vert x\Vert }_2=cT$ and $r_j={\Vert x_j\Vert }_2$. Then

By Lemma lem:routing,

and

The upward-rounded target time has delay less than ${\delta }_j$. Taking the maximum of these two tick-boundary times proves (eq:time-bound).

If ${\tau }_j(x_j)<T$, then

Since ${\tau }_j(x_j)$ is already a tick-boundary time, one additional tick suffices. Otherwise no additional tick is needed.

The route uses at most

ticks and one more site than its number of nonzero increments. Including the possible terminal wait gives (eq:resource-H). The zero target uses one source register and zero ticks and also satisfies these estimates.

For $j\ge 2\kappa$, substitute $j-\kappa \ge j/2$ and ${\epsilon }_j=\kappa \ell /(2j^2)$ into (eq:B). This gives (eq:B-expanded). Similarly,