Route description and compilation boundary

For a lattice destination $b$ with ${\Vert b\Vert }_2\le L$, every route coordinate in lattice units has magnitude at most $\lceil L/h_j\rceil +1$. A signed coordinate can therefore be stored with

bits. An explicit route list uses at most

coordinate bits, together with $O(\log (N+2))$ bits for its length and an execution index. For fixed $d,L,\ell$, this is $O(j\log (j+1))$ classical storage.

Given an exact lattice destination, the route is finitely computable: $N$ is determined by comparisons of algebraic numbers, and (eq:route-points) uses rational-coordinate rounding. No optimization oracle is needed.

For an arbitrary real continuum target, ${\mathsf{Q}}_{j}x$ specifies a mathematical family. Effective compilation additionally requires a target representation sufficient to certify the relevant rounding choices. No algorithmic access to arbitrary noncomputable real coordinates is asserted.