Two examples of orbit evolution in the (1 − e, a) plane. The blue line shows the formation of an EMRI that is missed by our standard definition, and requires the correction described in the text to be rightfully counted. The pink line instead shows a regular reinitialisation. We also plot the evolution line of an orbit that passes through the minimum of the t_GW = 10⁻³t_rlx curve following Eq. (32) (dashed brown line). It can be seen that all orbits that cross into the excluded region while being in the GW-dominated region fall below this evolution line, meaning they will reach the solid green curve, as long as the stochastic kicks due to two-body relaxation are small (see Fig. 1 for the meaning of the other elements). Here we set M_• = 4 × 10⁶ M_⊙.