We propose a population-level description of cross-time memory effects in macroscopic quantum tunneling of a current-biased Josephson junction subject to a dynamically suppressed interval \([t_1,t_2]\) . Starting from the Caldeira–Leggett influence functional, we identify the controlled approximations under which bath-mediated correlations between instantons can be coarse-grained into a Volterra equation for the survival probability S(t). The resulting cross-time kernel couples pre-window times \(t'<t_1\) to post-window times \(t>t_2\) with an amplitude proportional to \(e^{-\lambda _c\Delta t_b}M_\textrm{pre}\) , where \(\Delta t_b=t_2-t_1\) and \(M_\textrm{pre}\) is a history-weighted pre-window survival functional. The switching distribution \(P(t)=-\dot{S}(t)\) then contains a Markovian clustering peak at \(t_2\) and a non-Markovian post-window tail. We clarify that the kernel is a controlled phenomenological reduction, not an exact microscopic identity, and we state explicit positivity, validity, and thermal-crossover conditions. Numerical diagnostics are reformulated in terms of S(t) and show how the full switching distribution changes with barrier duration and pre-barrier history. The analysis indicates that ordinary resistively shunted junctions have too short a memory time for nanosecond barriers, whereas engineered high-Q resonator environments provide a plausible route to testing the predicted peak-plus-tail signatures.
