reality the other SRFs are not negligible in this system; in particular εTide ≃10−13 turns out to be the dominant effect. This example is thus a controlled demonstration, not a fully realistic system.

[Figure: Figure 9. As in Figure 6, but for the Sun–Jupiter-like system described in §3.3 where the only SRF included is due to a rotational bulge.]

12 Klein & Hamilton

[Figure: Figure 10. As in Figures 4 and 7, but for the example from §3.3 with the rotational bulge providing the only SRF.]

• HR24 focused on quantifying the jumps in the space of approximate adiabatic invariants over a single eccentricity peak. They did not explore the consequences of these jumps for long-term dynamical evolution. Here we have demonstrated that jumps alter the maximum eccentricity emax and secular period PK, sometimes by orders of magnitude (e.g., Figure 3) and can drastically change the time spent above a given eccentricity T(> e) (e.g., Figure 4).
set of approximate adiabatic invariants CK, jz. This means that the inner binaries of hierarchical triples may explore phase space far more efficiently than previously recognized. Nonadiabatic jumps were already reported by HR24 (see also Rasskazov & Rafikov 2024) for one specific SRF, namely 1pN GR apsidal precession, and they called the effect ‘relativistic phase space diffusion’. Our findings extend/differ from those of HR24 as follows:
• HR24 focused on GR, but we have shown that GR is not the only SRF that can drive nonadiabatic jumps: any apsidal precession effect that is sufficiently strong at high eccentricity (see equation (1)) can do the same.
The jumps in jz deserve further comment. They are inevitable because the allowed range of CK decreases with |jz| (§2.1); thus a jump that drives CK toward the edge of its range often cannot be accommodated at fixed jz, so |jz| tends to decrease on average. To our knowledge this is the first known quadrupole-order, three-body mechanism that drives long term evolution of jz. Previously, long-term evolution of jz was understood to require either octupole-order coupling (the EKL orbit-flip mechanism; e.g., Lithwick & Naoz 2011; Katz et al. 2011; Naoz 2016; Sidorenko 2018; Lei 2022; Klein & Katz 2024b), the addition of a fourth body in a quadruple configuration (e.g., Pejcha et al. 2013; Hamers & Lai 2017; Fang et al. 2018; Klein & Katz 2023, 2024c, 2025), or an aspherical cluster potential perturbing the triple (Petrovich & Antonini 2017). Such evolution is important because jz is the main determinant of the eccentricity maximum emax. Thus, on average nonadiabatic jumps drive the binary towards ever more extreme orbital dynamics. In the past, SA dynamics was sometimes treated as a minor correction to DA dynamics, via approximations such as equations (19)–(20) (Grishin et al. 2018; Hamilton & Rafikov 2019). Nonadiabatic jumps invalidate this approach: they drive behavior that, to our knowledge,
• HR24 referred to the effect of jumps as ‘diffusion’, implying that individual jumps are in some sense small. However, we have found that a single jump can change CK by O(1). This is not small (recall the maximum dynamic range is CK ∈[−3/2, 1]), so the word ‘diffusion’ may be misleading.
• HR24 believed that jz did not undergo nonadiabatic jump behavior. We have shown that this is not true. In fact, as we discuss further below, jumps in jz are both inevitable and dynamically important.
• HR24 showed that nonadiabatic jumps arise for binaries subject to a range of external potentials. We specialized here to a genuine hierarchical triple (in which the outer tertiary is a point mass, i.e., a Keplerian potential), but we do not anticipate that anything would change qualitatively if we were to relax this.

SRF Catalysis of Extreme Orbital Evolution 13
is not well described — even statistically — by any analytic or semi-analytic theory.

Such variability means that the questions posed in the Introduction — namely, (i) what is emax? and (ii) what is PK? — cease to be meaningful for systems residing in the ‘fourth quadrant’ of Figure 1.