Theorem: Explicit finite-volume FPP comparison

On a graph with $M$ edges, couple $S_q$ to monotone infection $I_{\mathrm{P}}$ driven by the same Poisson clocks. At an event touching an infected site, the monotone process sets both endpoints occupied. Then

The law of $I_{\mathrm{P}}$ is the law of the infected set for i.i.d.\ undirected $\mathrm{Exp}(\lambda )$ FPP weights.

Proof. While the two supports agree, events touching no occupied endpoint do nothing in both processes. At any other event, the Clifford support agrees with the monotone update with probability $(q^2-1)/(q^2+1)$. Couple these outcomes directly. The probability of first disagreement is bounded by $2/(q^2+1)$ times the expected number of edge events, which is $\lambda Mt$.

For the second assertion, the timer construction of Lemma lem:timers has, for exponential timers, independent exponential residual clocks on its active boundary edges. This follows inductively from memorylessness after the next minimum clock and from independence of newly armed edges. Its jump rate to an uninfected vertex is $\lambda$ times the number of infected neighbours. The Poisson construction has the same finite-state generator and the same initial state. Thus the two laws agree.