Method (Paper E Appendix A, browser-reduced sample counts). For each log-spaced radius bin a ∈ [6 r
g, stripping radius] AU (the grid starts at the Schwarzschild ISCO, inside which no circular orbits exist): draw encounters with impact parameter from the gravitationally focused distribution truncated at b
max = 30a, speed from a Rayleigh of scale σ shifted up by 0.3σ, and mass from the perturber mass function; compute the impulsive eccentricity-kick variance ⟨δ²⟩; convert to a survival time via the random-walk first-passage estimate n
★ = e
cross²/⟨δ²⟩ divided by the encounter rate, with CLT scatter giving the 16–84% band; cap at 12 Gyr.
Perturber mass function, perturber mass, and the fBH control (Paper E v0.6). The background is two-component stellar (0.35 M☉ weight 0.70, 0.60 M☉ white-dwarf tail weight 0.30). A mass-segregated heavy-remnant extension adds neutron stars (1.4 M☉, fixed number fraction f
NS = 2%) and stellar-mass black holes at the perturber mass set by the control, segregated fraction
f_BH (fiducial 1%, grid 0.1/1/3%). The heavy tail is the controlling perturber, because kick variance scales as ⟨m²⟩, so both the fraction and the mass matter and the mass matters more. The default perturber mass is now 31 M☉, the mean mass of black holes inspiraling into the IMBH in the González Prieto et al. (2025) ω Cen models, which name 10 M☉ as the assumption they are correcting; the metal-poor progenitors of a cluster at [Fe/H] ~ −1.5 leave heavier remnants than the solar-metallicity default. Moving from 10 to 31 M☉ at fixed f
BH = 1% raises ⟨m²⟩ by a factor 8.0 and shortens the impulsive floor from ~5×10⁸ to ~6×10⁷ yr, a larger displacement than the whole 0.1–3% fraction bracket produces at fixed mass. Set the control back to 10 M☉ to reproduce the pre-2026-07-30 curves; set f
BH = 0 to recover the stars-only baseline (~1.8×10⁹ yr).
The adiabatic correction, and why the impulsive floor was only a floor. The impulsive kernel above is valid only when the encounter is short compared with the orbit, x = ω
orbτ
enc = (b/a)(v
orb/v) ≪ 1. For this system x is 2.2 at the cluster stripping radius and ~4×10³ at the ISCO, so unbound cluster stars are adiabatically decoupled from every orbit an installation could occupy: the impulsive numbers above are the conservative floor for a channel that does not operate, not a physical estimate. Ticking the adiabatic-correction control applies the Gnedin & Ostriker (1999) kernel A(x) = (1+x²)
−5/2 to δ² per encounter, the less-suppressing of the standard choices and calibrated against N-body results. With it, the fiducial median reaches the 12 Gyr cluster-age cap inside ~3 AU and stays above ~3×10⁸ yr at the envelope edge. The flat, scale-free profile is a property of the impulsive kernel; the corrected curve rises inward.
The bound-cusp channel (Paper E D1-MODEL, 2026-07-30). What actually binds at the envelope edge is the bound cusp population: members move at the local Keplerian speed, so x ~ 1 for them (not ≪ 1), the impulsive kernel with the same adiabatic correction applies, and they are granular rather than diffuse. Each species follows N
s(<r) ∝ r
3−γs (γ
★ = 1.3, γ
NS = 1.5, γ
BH = 2.0, González Prieto et al. 2025); at γ
BH = 2 the expected black-hole count is ≲ 1 everywhere in the envelope, so each history draws a Poisson perturber count from the local shell rather than using a smooth rate, and the resulting lifetime distribution is bimodal: a bound black hole is present in 53% of histories at the envelope edge, giving a diffusion floor of ~4×10⁷ yr (16th percentile ~2×10⁷ yr); when none is drawn the channel reaches the cluster-age cap. The perturber population is now anchored jointly rather than assembled from separate figures: f
BH,central = 10 × (M
dark/m
BH)/(7×10⁶ stars), fiducial 1.15% at M
dark = 2.5×10⁵ M☉ and m
BH = 31 M☉ (Bañares-Hernández et al. 2025 pulsar timing), narrowing the old 0.1–3% fraction bracket (factor 30 in variance) to a factor 3.1 in the M
dark/m
BH bracket. The same bound remnant also forces the orbit secularly (Kozai–Lidov): t
sec(a) = (M/m
pert)(r
p/a)³P
orb(a) with the outermost bound black hole fixed at r
p = 4×10³ AU, but relativistic apsidal precession quenches the resonance (t
sec/t
GR ∝ a
−4) inward of a ≈ 4×10² AU, where t
sec ≈ t
GR ≈ 3.7×10⁷ yr. That forcing is coherent and oscillatory, not diffusive, so it is a station-keeping cadence over its 4×10²–4×10³ AU operating window, not a third survival curve. Tick
overlay bound-cusp channel above (with paper reference mode) to plot both against the unbound impulsive/adiabatic curves; this channel is not yet live-sampled in the browser, only the published fiducial run is shown.
Paper reference mode. Ticking
paper reference plots the exact fiducial arrays from
fig1_results.json verbatim (2×10⁵ encounter draws × 2×10⁴ histories × 40 bins, fixed seed 20260717, σ = 21 km/s,
fig1_envelope.py, full bundle at
/papers/source/): no live sampling, so it reproduces the published numbers rather than approximating them. The live MC trades sample count for interactivity; expect its median to sit inside the reference band, not land on it. The
paper MC floor summary card always shows the published floor for the grid f
BH nearest your setting, for side-by-side comparison. σ default from
measurements.js clusterParams (18.2 km/s, cluster-averaged); Paper E adopts σ = 21 km/s, the midpoint of the 20–23 km/s central-region range of Baumgardt & Vasiliev (2021) and Nitschai et al. (2023). The difference lengthens survival times by tens of per cent and is inside the paper’s stated parameter variations.