Proposition: A static-site footprint bound

Suppose a realizing network uses $q_n$ fixed spatial sites. All quantum and classical communication edges between distinct sites have length at most $a_n$. If a source–receiver dependency path exists and their displacement is $x_{n,\zeta }$, then

If, additionally, the resource model keeps a $d$-dimensional register at every used site live simultaneously, its live quantum dimension is at least $d^{q_n}$.

Proof. The communication path lies in a connected finite graph on the used sites. Erasing loops gives a simple spatial graph path with at most $q_n-1$ edges. The triangle inequality gives $\Vert x_{n,\zeta }{\Vert }_2\le (q_n-1)a_n$, proving (eq:sites). The simultaneous-register clause gives the product dimension.