Theorem: Finite-time offset and limiting velocities

Let $q=|V(Q)|$, $D_{\max }={\max }_e|{\Delta }_e|$, and $R_Q=(q-1)D_{\max }$. Every path completed within elapsed time $t$ has displacement

From any specified initial quotient vertex, the set of subsequential limiting displacement/time ratios, for unbounded elapsed times, is exactly $K_Q$.

Proof. Remove closed subwalks until the remaining walk has no repeated vertex. Decompose removed closed walks into simple cycles. Their total displacement is a time-weighted sum of cycle velocities, with total cycle time at most $t$. The zero vector accounts for unused time. The remaining simple walk has at most $q-1$ edges and displacement norm at most $R_Q$, proving (eq:offset).

Divide by $t$ to obtain one inclusion for limiting ratios. Conversely, take any convex combination of finitely many cycle velocities and zero. Repeat the cycles in integer counts whose durations approximate the chosen proportions on a scale tending to infinity. Strong connectivity supplies fixed connecting walks from the chosen start and between cycles. Their total cost is bounded for this fixed combination. Waiting realizes its zero component. The normalized displacement tends to the desired combination.