Lemma: Timer arrivals are undirected first-passage times

For every positive timer realization, the flagging time of $x$ in Specification spec:cnot equals $T_G(0,x)$. The successful edges form a rooted tree. For independent continuous timers, ties occur with probability zero.

Proof. A successful edge $u\to v$ flags $v$ at time $S(u)+{\tau }_{\{u,v\}}$, so the successful ancestry chain gives a path of cost $S(v)$. Thus $T_G(0,v)\le S(v)$.

Conversely, suppose a vertex of minimum $T_G$ among those not yet flagged should have been reached earlier. On a minimizing path, let $\{u,v\}$ be the first edge crossing from flagged to unflagged vertices. Its timer began when $u$ was flagged. Since $S(u)\le T_G(0,u)$ at that stage of the induction, it expires no later than the path's proposed arrival at $v$. If the operation was cancelled, $v$ was already flagged earlier; otherwise it flags $v$. Both contradict the supposed delay. Induction over the finite graph proves equality.

Each non-root vertex is targeted exactly once, by an already flagged vertex. Arrival times strictly increase along successful edges, so there is no directed cycle. Continuous independent weights make distinct simple-path sums unequal almost surely: condition on an edge present in only one of the two paths. There are only finitely many simple paths in the finite graph.