Stencil size and permission data

The permission stencil obeys

Indeed, the integer cube with coordinate bound $\lfloor j/\kappa \rfloor$ lies inside the Euclidean radius-$j$ ball, and that ball lies inside the coordinate radius-$j$ cube. Consequently

Permission is tested by the integer inequality

The model may use this fixed membership rule instead of storing a fully enumerated stencil. If the stencil is explicitly materialized, its entry count is $\left|S_j\right|$, and its displacement-bit storage is $O_d(j^d\log (j+1))$. These documentary permission choices are not additional quantum registers.

Only the $q$ route sites are initialized in a given realization. The theorem does not require a quantum register at every site in a bounded continuum region, nor does it count a full infinite lattice as an initialized finite apparatus.