# Paper E figures — scripts, Monte Carlo, and reproduction

Numerical backing for *Engineered Intermediate-Mass Black Hole Systems* (Paper E,
`../engineered-imbh-paper.tex`). Every quoted number in the paper's §2.4, §2.7, §3.1
and §5 regenerates from these scripts. All random draws use fixed seeds derived from
`20260717`; byte-identical reruns are expected on any NumPy ≥1.26.

## Files

| File | What it is |
|---|---|
| `common.py` | Shared constants (fiducial 2×10⁴ M☉ Kerr hole, ω Cen core parameters), the perturber mass function, the radius-dependent MIR limit, and the Gnedin & Ostriker (1999) adiabatic kernel. Change here, re-run all figs. |
| `fig1_envelope.py` | **The flyby Monte Carlo** (see below). Outputs `fig1_envelope.pdf` + `fig1_results.json`. |
| `fig2_constraint_plane.py` | (P_comp, 1−f_sink) constraint plane: JWST waste-heat wedge, transport floor, Bondi fuel ceiling. The floor renders as a labelled adopted-parameter line over hatching, not as a hard boundary of the allowed region (§6.1). |
| `fig3_lnk.py` | Worked-example ln K computation (2×10⁶ prior draws; Appendix B model). Prints the §5 numbers, the data-only statistic ξ, and the channel capacity. Every prior boundary is a command-line switch (see below). |
| `fig4_r_statistic.py` | Schematic PSD defining the MAD-regulation statistic R (illustrative, no data). |
| `fig1_results.json` | Machine-readable MC output: per-radius median/16–84% survival times, encounter rates, kick variances, and the adiabatic parameter, for **every** configuration under `per_config`. |

Reproduce everything:

```sh
cd paper/figs
python fig1_envelope.py && python fig2_constraint_plane.py && python fig3_lnk.py && python fig4_r_statistic.py
```

## The flyby Monte Carlo (fig1)

**Question:** how long does a structure on a circular orbit of semi-major axis *a*
around the cluster-core IMBH survive cumulative stellar-flyby perturbation?

**Method** (Paper E Appendix A): per radius bin, 2×10⁵ encounters are sampled from the
gravitationally focused impact-parameter distribution (truncated at b = 30a), a speed
distribution drawn as a Rayleigh of scale σ = 21 km/s shifted upward by 0.3σ (mean
1.55σ), and a mass function (0.35 M☉ MS + 0.6 M☉ WD tail, plus segregated remnants).
Each encounter contributes an impulsive eccentricity kick δ = 2(m★/M)·min[1,(a/b)²]·(v_orb/v).
Per history (2×10⁴ per bin), survival is the random-walk first passage to orbit-crossing
eccentricity e = 0.5, capped at 12 Gyr. The radius grid starts at the Schwarzschild
ISCO (6 r_g), inside which no circular orbits exist.

**Two curves, and the difference between them matters.** The *impulsive* curves apply
no adiabatic suppression and are the conservative floor the paper quotes. The
*adiabatically corrected* curve applies the Gnedin & Ostriker (1999) kernel
A(x) = (1 + x²)^(−5/2) per encounter, with x = (b/a)(v_orb/v), and is the physical
estimate. Across the whole envelope x runs from 2.2 at the cluster stripping radius to
4.1×10³ at the ISCO, so unbound cluster stars are adiabatically decoupled from every
orbit in the envelope: the corrected lifetimes sit at the 12 Gyr cap inside ~3 AU and
never fall below ~3×10⁸ yr at the outer edge.

**Perturber mass** (referee R5, 2026-07-30): stellar-mass black holes are now 31 M☉,
the mean mass of 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. Kick variance
scales as ⟨m²⟩, so this shortens the impulsive floor by a factor 8.0 at fixed f_BH.
The `fbh_0.01_m10` configuration reproduces the pre-R5 10 M☉ curve for comparison.

**Headline results** (impulsive floor, medians at 10 / 10² / 10³ AU):

| configuration | 10 AU | 10² AU | 10³ AU | envelope minimum |
|---|---|---|---|---|
| stars + WDs only | 3.4×10⁹ | 3.3×10⁹ | 2.5×10⁹ | 1.8×10⁹ |
| f_BH = 0.1% | 1.0×10⁹ | 9.3×10⁸ | 5.8×10⁸ | 1.0×10⁸ |
| **f_BH = 1% (fiducial)** | **6.6×10⁷** | **6.3×10⁷** | **5.2×10⁷** | **2.7×10⁷** |
| f_BH = 3% | 2.2×10⁷ | 2.0×10⁷ | 1.9×10⁷ | 1.1×10⁷ |
| f_BH = 1%, 10 M☉ (pre-R5) | 5.4×10⁸ | 5.1×10⁸ | 3.5×10⁸ | 2.0×10⁸ |
| f_BH = 1%, **adiabatic (physical)** | 7.3×10⁹ | 4.5×10⁹ | 1.7×10⁹ | 3.0×10⁸ |

The profile is roughly flat because ⟨δ²⟩ ∝ a⁻¹ while the encounter rate Γ ∝ a in the
focused regime, so their product is scale-free — but that cancellation is a property
of the impulsive kernel, and the adiabatic curve is not flat. The operative outer
boundary is cluster tidal stripping at ~0.1 r_infl ≈ 4×10³ AU either way.

The script also prints the seed-to-seed spread of the min-of-bins statistic over five
seeds, which is where Appendix A's ±20–40 per cent figure comes from.

## The ln K computation (fig3)

`fig3_lnk.py` exposes every prior boundary, because ln K here is a prior-volume ratio
and the boundaries are the levers:

```sh
python fig3_lnk.py --leak-floor-dex -6      # softer transport bound
python fig3_lnk.py --pcomp-floor-dex -3     # P_comp prior floor at 1e-3 Lsun
python fig3_lnk.py --pcomp-ceiling 1e3      # ADIOS-suppressed natural fuel ceiling
python fig3_lnk.py --soft-threshold 0.3     # logistic instead of hard threshold
python fig3_lnk.py --mips70-scale 0.1 --lmid-scale 0.1
python fig3_lnk.py --r-range 0.02 1000
```

Measured (central ln K, band over f_d ∈ [0.1, 0.9], ξ = excluded active-prior fraction):

| configuration | ξ | ln K | band |
|---|---|---|---|
| **fiducial** | 0.497 | **−0.286** | [−0.593, −0.051] |
| leak floor 10⁻⁶ | 0.358 | −0.197 | [−0.389, −0.036] |
| P_comp floor 10⁻³ L☉ | 0.329 | −0.180 | [−0.351, −0.033] |
| ADIOS ceiling 10³ L☉ | 0.222 | −0.118 | [−0.223, −0.022] |
| soft threshold 0.3 / 0.5 dex | 0.494 / 0.490 | −0.284 / −0.281 | — |
| MIPS 70 μm ×10 harder | 0.498 | −0.286 | — |
| mid + cold wedges ×10 harder | 0.528 | −0.306 | — |
| radius prior [0.02, 10³] AU | 0.538 | −0.313 | — |

The channel is one-sided: P(no detection | H_q) = 1, so ln K ≥ ln f_d = −0.693 for any
depth of photometry. At the fiducial dormancy prior the channel has spent 41 per cent
of that capacity and has 0.41 nats of headroom left, in total, forever.

## Browser standalone

An independent JavaScript reimplementation of the same physics runs at
[omegacentauri.me/tools/flyby-survival.html](https://omegacentauri.me/tools/flyby-survival.html)
(40k in-browser draws, hash-state deep links, no network calls); the companion fuel
budget calculator is
[imbh-fuel-budget.html](https://omegacentauri.me/tools/imbh-fuel-budget.html).
Cross-check status: the tool tracks the **pre-R5 impulsive, 10 M☉** configuration and
agrees with `fbh_0.01_m10` within ×1.5, fully explained by the σ default (site SSOT
18.2 km/s from `measurements.js clusterParams` vs this paper's 21 km/s fiducial).
**Known follow-up:** expose the perturber mass and the adiabatic kernel in the tool, or
label it explicitly as the conservative impulsive case.

## Provenance

Draft v0.6, last revised 2026-07-30. License follows the repo: code MIT, prose CC BY 4.0.