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.

We must therefore conclude that, if both SA fluctuations and SRFs are important in the dynamics of a hierarchical triple, those dynamics cannot be approximated by any currently-known DA or modified-DA theory.

It follows that in the fourth quadrant, only brute-force integration of the SA (or N-body) equations of motion can be relied upon.

This conclusion also sharpens how the DA-validity criterion (17) should be applied in practice.

In the compact-object–merger literature the DA criterion is conventionally written as tLK(1 −e2 max)1/2 ≳Pout (e.g., Liu & Lai 2018; Hamilton & Rafikov 2021), where tLK stands in for the secular timescale; with our normalization tLK = Psec, so this reads Pout/Psec ≲jDA min.

Thus, DA is frequently assumed reliable whenever Pout/Psec is merely somewhat smaller than jDA min.

A rigorous asymptotic statement of DA validity, however, calls for the strong inequality Pout/Psec ≪jDA min (Eq. (17)) rather than the marginal ≲.

This might seem a pedantic distinction, but in fact it is crucially important.

For instance, Liu & Lai (2018) partitioned their perturber mass–semimajor-axis plane of merging compact-object triples into adjacent ‘DA’ and ‘SA’ regions whose dividing curve is exactly Pout/Psec = jDA min (their Figure 2).

Our GR-only example (§3.1) would be deemed safe for DA according to this partitioning (see Eqs. (23) and (25)), but Figure 2 makes clear that in fact, O(1) jumps in CK, and secular evolution of jz, both occur.

While DA/SA discrepancies have been reported in ZLK calculations including SRFs (Liu & Lai 2018; Huang et al. 2026), they have been attributed to the usual small fluctuations δe (equation (20)) along the lines of Ivanov et al. (2005); Grishin et al. (2018).

The phenomenon we have described here is a distinct one, in which the SRF precession itself drives discrete jumps in the DA constants from one secular cycle to the next.

This may have significant astrophysical consequences, because nonadiabatic jumps lead to qualitatively different statistics for extreme-eccentricity passages than would have been predicted in their absence.

In real systems, dissipative processes such as tides or GW emission can turn sufficiently high-eccentricity passages into migration or merger channels (Fabrycky & Tremaine 2007; Wen 2003; Miller & Hamilton 2002; Liu & Lai 2017, 2018; Liu et al. 2019).

Thus, if we get those high-eccentricity statistics wrong, we might badly misestimate the observable rate of LIGO/Virgo sources,
hot Jupiters, etc. Population synthesis calculations of such processes that relied on formulae like (20)-(19) may therefore need to be revisited.

Our mechanism should not be confused with the relativistic ‘precession resonance’ of Kuntz (2022), in which the GR apsidal precession of the inner orbit becomes commensurate with the outer orbital frequency and the eccentricity is resonantly excited to a bounded ceiling.

The two effects differ fundamentally.

In Kuntz (2022)’s resonance, the rapid GR precession occurs at low eccentricity, because the binary radiates gravitational waves and thus shrinks its semimajor axis.

In this regime SRFs are strong enough to dominate the secular dynamics (εGR ∼πPsec/Pout ≫1, so ZLK cycles are fully suppressed).

By contrast, our jumps are purely conservative — the inner semimajor axis is fixed and no dissipation enters — and our SRFs have εi ≪1, so ZLK cycles operate.

While our systems do reach very fast precession rates that become commensurate with the outer orbital period, they do so only transiently during a singly-averaged fluctuation at high e, and for the rest of their evolution the apsidal precession is much slower than the outer orbital frequency.