this system using multiple different methods. The red dashed and black dashed
lines reproduce the SA and DA results from Figure 2 respectively. The blue dashed line shows the SA integration without the test-particle approximation (i.e., including the back-reaction onto the outer orbit), and the thick gray line shows the results of a direct N-body integration (no averaging at all) using SpaceHub (Wang et al. 2021). We see that the two SA integrations and SRF Catalysis of Extreme Orbital Evolution 19 the N-body integrations produce results that are indistinguishable, giving us confidence in our approach.

[Figure: Figure 11. The first eccentricity peak from Figure 2, integrated with multiple methods. The red dashed and black dashed lines reproduce the SA and DA results from Figure 2 respectively. The blue dashed line shows the SA integration without the test-particle approximation (i.e., including the back-reaction onto the outer orbit), and the thick gray line shows the results of a direct N-body integration (SpaceHub).]

However, we are concerned not only with the dynamics of a single eccentricity peak, but also with the statistics of long term evolution over many secular cycles. As is well-known in planetary dynamics (Laskar 1989; Sussman & Wisdom 1992), tiny changes in the approximations used (and even in numerical choices like the exact timestep) can, over long enough timescales, creep in to the dynamics and produce, e.g., phase offsets between the results of different integrations. And since nonadiabatic jumps are extraordinarily sensitive to orbital phases, these tiny offsets can drive altogether different orbital behavior. Thus we should not rely upon any integration method to be exact — all one can ultimately rely upon is the statistics that these methods produce. To study this, in Figure 12 we recreate the residence time plot for the BBH–SMBH triple of §3.1 (cf. Figure 4) using four different dynamical treatments. The curves in panel (c) are identical to those in Figure 4, showing the result for SA dynamics without any back-reaction onto the outer orbit. Panel (b) re-integrates the same 50 realizations with the back-reaction restored (i.e., without the test-particle approximation). Panel (d) shows the realization-independent DA reference curves from Figure 4. Panel (a) re-integrates them with the N-body code. The residence time diagnostic is useful because it combines two kinds of averaging: it integrates over

[Figure: Figure 12. Cumulative residence time T(> e)/Ttot versus log10(1 −e) for the BBH–SMBH triple of §3.1 (Figures 2– 4), computed using the same four different methods used for Figure 11. Colors are the same as in Figure 4. The initial conditions are those of the 30+30 M⊙BBH listed in Table 1.]

the full time evolution of each trajectory and aggregates over the ensemble of outer mean-anomaly realizations M2. Three points are immediate. First, panels (a), (b) and (c) agree at the qualitative level over the entire eccentricity range: the SRFs produce the same characteristic enhancement of the high-e residence tail, the +SRF (red) median crosses to the left of the no-SRF (blue) median at the same log10(1 −e) ≃−3, and the realization-torealization spread covers a comparable few-decade band. Secondly, the close agreement between panels (b) and 20 Klein & Hamilton (c)

shows that the inner-on-outer back-reaction has negligible effect on the residence statistics for this system, validating the test-particle approximation. Thirdly, the DA panel (d) is qualitatively distinct from the other three: the residence curves terminate abruptly at the analytic DA emax values and have the +SRF branch to the right of the no-SRF branch (the opposite of the SA/N-body ordering). The main-text story that SRFs catalyze, rather than suppress, excursions to extreme eccentricity is therefore an SA feature that survives both the back-reaction and the full N-body integration, but is absent in the DA limit.