Some notation: let $P$ be an interval $[a,b]$. We say $P$ is a \emph{patch} when the $\Z{i}$ event holds for all $i \in [a,b]$ and $\NZ{a-1}$ and $\NZ{b+1}$ holds. We denote this event by $P\in\mathcal{P}$, so

        \begin{align*}

            P\in\mathcal{P} \equiv \NZ{a-1} \cap \Z{a} \cap \Z{a+1} \cap \cdots \cap \Z{b-1} \cap \Z{b} \cap \NZ{b+1} .

        \end{align*}

        Note that we have the following partition of the event $\Z{v}$ for any vertex $v\in[n]$:

        \begin{align*}

            \Z{v} = \dot\bigcup_{P : v\in P} (P\in\mathcal{P})

        \end{align*}

            R^{(n)}

            &= \frac{1}{n}\sum_{v\in[n]}\E^{(n)}(\text{\# resamples of } v) 

            \tag{by definition}\\

		    &= \frac{1}{n}\sum_{v\in[n]}\sum_{t=1}^{\infty}t\cdot \P^{(n)}(v \text{ is resampled }t\text{ times}) \\

            &= \frac{1}{n}\sum_{v\in[n]}\sum_{t=1}^{\infty}\sum_{P\text{ patch}}t\cdot\P^{(n)}(v \text{ is resampled }t\text{ times and }v\in P\text{ and } P\in\mathcal{P})

            \tag{partition}\\

            &= \frac{1}{n}\sum_{v\in[n]}\sum_{t=1}^{\infty}\sum_{P\text{ patch}}t\cdot\P^{(n)}(v \text{ is resampled }t\text{ times and }v\in P | P\in\mathcal{P}) \; \P^{(n)}(P\in\mathcal{P})\\

		    &= \frac{1}{n}\sum_{P\text{ patch}}\E^{(n)}(\# \text{ resamples in }P|P\in \mathcal{P}) \; \P^{(n)}(P\in\mathcal{P})\\

            &= \sum_{s=1}^{n-1}\E^{(n)}(\# \text{ resamples in }[s] \;|\; [s]\in \mathcal{P}) \; \P([s]\in\mathcal{P}) +\mathcal{O}(p^{n})

            \tag{by translation symmetry}\\

            &= ???? \\