Theorem: Resource-controlled operational null-boundary approximation

In the model of Section sec:model, the following hold.

1. Exact locality ceiling. For every admissible $n$-tick realization, payload influence from $0$ to $b$ requires

   $${\Vert b\Vert }_2\le cn{\delta }_j.$$

2. Uniform lattice transport. For $j>\kappa$ and every $b\in {\Lambda }_j$, the explicit route implements ${\mathrm{id}}_D$, including every untouched reference, at time ${\tau }_j(b)$, with

   $$0\le {\tau }_j(b)-\frac{{\Vert b\Vert }_2}{c}\le \frac{\kappa {\Vert b\Vert }_2}{c(j-\kappa )}+{\delta }_j.$$

   It uses $N_j(b)$ ticks, $m_j(b)\le N_j(b)$ SWAPs, and $q_j(b)=m_j(b)+1\le N_j(b)+1$ initialized sites.

3. Uniform null-target approximation. For every $L>0$, every null target ${\Vert x\Vert }_2=cT\le L$, and every $j>\kappa$, there is a finite payload-independent admissible protocol with destination $x_j$, arrival time $T_j$, and logical channel ${\Phi }_j(T,x)$, such that

   $${\Vert x_j-x\Vert }_2\le {\epsilon }_j,0\le T_j-T\le B_j(L),{\Phi }_j(T,x)={\mathrm{id}}_D.$$

   It uses at most $H_j(L)$ ticks and $H_j(L)$ initialized sites.

4. Conditional implementation convergence. Suppose the encoding and decoding errors of these protocols are at most ${\alpha }_j$ and ${\beta }_j$, uniformly over the target family, and every complete tick map has error at most $e_j$. Then

   $$\underset{{\Vert x\Vert }_2=cT\le L}{\sup }{d}_{\diamond }({\tilde{\Phi }}_j(T,x),{\mathrm{id}}_D)\le \min \{1,{\alpha }_j+H_j(L)e_j+{\beta }_j\}.$$

   In particular, ${\alpha }_j\to 0$, ${\beta }_j\to 0$, and $j{e}_j\to 0$ are sufficient for uniform complete-channel convergence.

For a fixed nonzero null target $r={\Vert x\Vert }_2>0$, the constructed tick and initialized-site counts additionally satisfy

The site count has the optimal leading coefficient in the declared static-site model.

Proof. Part (i) is Corollary cor:speed. Part (ii) combines Lemmas lem:routing, lem:repetitions, and lem:identity. Terminal identity waits preserve the complete transport identity, so Lemma lem:endpoint gives part (iii). Lemma lem:telescoping and the bound on the total number of ticks give part (iv).

For the last assertion, set $r_j={\Vert x_j\Vert }_2$. The route has $m_j$ nonzero increments, so

For nonzero $x_j$,

Since $r_j\to r>0$, these inequalities imply

There is at most one additional wait, and $q_j=m_j+1$. This proves (eq:leading-resources).

For any other admissible static-site protocol that influences the same rounded destination, Proposition prop:memory-lower gives

Therefore its asymptotic site count cannot have a smaller leading coefficient.