Theorem: Local directional inclusion

Suppose each scheduled communication has a specified continuous route with velocity in $K_{\mathrm{sched}}(x,t)$, and

at every point visited by those routes. Include gate, encoding, decoding, and storage dependencies in the same path contract. Then the full schedule is physically contained, and Theorem thm:physical applies.

Proof. Every scheduled route is an admissible physical path. Concatenated routes remain admissible piecewise almost everywhere. The physical relation of each dependency edge is therefore certified.