+ p_2(t) \ket{\tilde{P}_2} \to \ket{P_{\rm ss}} .$$ The evolution for $t \gg 1/|\text{Re}(\lambda_3)|$, prior to reaching the stationary state, is within the one-dimensional MM, as it corresponds to the evolution of $p_{1,2}(t) \geq 0$ in the linear combination above, with $p_1(t)+p_2(t)=1$, and is described by the effective generator .

We now add resets to this metastable dynamics, focusing for simplicity on the one-dimensional metastable manifold case. First, we note that we can rewrite Eq.  as $$\ket{P^\Gamma_\text{ss}}=\ket{P_\text{ss}}-\sum_{i=2}^{D}\frac{\left\langle l_i|C_0\right\rangle}{\frac{\lambda_i}{\Gamma}-1}\ket{r_i}.$$ As a result, we see that, if we consider $\Gamma \approx |\lambda_2|\ll|\lambda_3|$, then the coefficients in the sum for the terms $i\geq3$ are small compared to the coefficient for $i=2$, so we can truncate to only the first two terms: $$\label{MetaTruncation}
\ket{P^\Gamma_\text{ss}}\approx\ket{P_\text{ss}}-\frac{\Gamma\left\langle l_2|C_0\right\rangle}{\lambda_2-\Gamma}\ket{r_2}.$$ Since the spectrum of the process with resets is simply a real shift by $\Gamma$, values on this scale preserve the gap in the spectrum required for metastability, as for $\Gamma\approx|\lambda_2|$ we still have $\text{Re}(\lambda_2)-\Gamma\ll \text{Re}(\lambda_3)-\Gamma$. Physically, this regime corresponds to the average time between resets being comparable to the lifetime of the metastable phases, in which the short timescale dynamical modes represented by $i\geq3$ are averaged out and make negligible contribution to the stationary state.

Using the definitions of 