Engineered Intermediate-Mass Black Hole Systems: Infrastructure Constraints, Observable Residue, and a Multi-Messenger Adjudication Framework

1. Introduction

1.1 The missing quadrant

Four papers in this series have examined the hypothesis that computation-optimizing civilizations migrate toward rapidly spinning massive black holes in dense old stellar systems. Paper A argued the motive: the thermodynamic advantages of Kerr accretion efficiency, horizon entropy disposal, and Bekenstein-bounded information storage. Paper B surveyed the candidates: the six-member family of inward-migration proposals and their falsifiability grades. Paper C planned the search: eight instrument-matched programs targeting Omega Centauri (NGC 5139), the nearest strong IMBH candidate. Paper D priced the move: closed-form crossover conditions between migration, densification, and seeding strategies.

Two questions remain unaddressed, in this series and, so far as we can determine, in the literature. First: could the infrastructure such a civilization requires actually persist at the destination? The destination is a demanding one. A cluster-core IMBH sits in a stellar environment three to four orders of magnitude denser than the solar neighborhood, threaded by gravitationally focused stellar traffic, and (if fueled) surrounded by a relativistic accretion flow. Black-hole computing proposals treat the hole as an idealized resource and the environment as empty. Megastructure stability analyses treat single stars in isolation. The intersection, engineered structures bound to a Kerr hole inside a live cluster core, is unexamined.

Second: if the campaign of Paper C ever records an anomaly, by what procedure would the community decide what it had seen? Radio SETI has mature per-signal verification practice and Bayesian population inference, and multi-messenger astrophysics has quantitative coincidence formalisms, but no published framework adjudicates a candidate technosignature across multiple messengers against an explicit menu of astrophysical alternatives. Paper C's rule that "every anomaly is adjudicated by at least two independent messengers" was stated as policy; here we give it mathematical content.

The two questions are coupled, and that coupling is the reason they share a paper. A feasibility envelope is a prior: it tells the adjudicator where in parameter space an engineered system could sit, and therefore which anomalies deserve elevated scrutiny and which are excluded on engineering grounds regardless of how anomalous they appear. Forward-modeled residue channels are likelihoods: they specify what an engineered system would look like, channel by channel, so that Bayes factors can be computed rather than gestured at. The paper thus runs in one direction: constraints (Section 2) produce observables (Section 3) which feed an adjudication engine (Section 4) that we exercise on real data (Section 5).

1.2 Conditionality and register

We work throughout with the fiducial system of Papers A and C: a black hole of mass M = 2×10⁴ M_☉ (bracketed by the 8.2×10³ M_☉ kinematic lower bound and the ~5×10⁴ M_☉ N-body preferred value), spin left free, embedded in the ω Cen core: central density ρ₀ ≃ 3×10³ M_☉ pc⁻³, core radius r_c ≃ 3.6 pc, line-of-sight velocity dispersion σ ≃ 20–23 km s⁻¹ near the center. Where a single value is needed we adopt σ = 21 km s⁻¹, the midpoint of that central range and consistent with the oMEGACat 3D kinematic analysis, and use it consistently in the influence radius of Section 2.3, the flyby velocities of Section 2.4, and Appendix A. The series' public calculators default instead to the cluster-averaged 18.2 km s⁻¹ of the shared measurement compilation, which is a global rather than central-region figure; the difference propagates as a factor (21/18.2)² ≃ 1.3 in r_infl and is within the parameter variations of Appendix A. The gravitational radius is

so the dynamic range between horizon scale and cluster scale is eight orders of magnitude in radius. The feasibility question is where in that range infrastructure can live.

2. The feasibility envelope

2.1 Constraint inventory

We consider a one-parameter family of architectures indexed by compactness: at one end, a dense swarm of independent elements on near-circular orbits at r ~ 10²–10⁴ r_g (the configuration favored by the thermodynamic argument of Paper A, since proximity to the horizon minimizes entropy-transport losses); at the other, an extended bubble or shell of elements at r ~ 10–10³ AU supported partly by radiation pressure or tether stress, the black-hole analogue of recently stability-verified Dyson bubbles. Intermediate cases (rings, nested tori) inherit constraints from both ends. We deliberately do not commit to an architecture; the envelope is the set of (r, architecture) pairs surviving all constraints simultaneously.

Six constraints bound the envelope: (i) relativistic orbit stability (inner); (ii) tidal stress on extended elements (inner); (iii) accretion-flow radiation and magnetic environment, if the hole is fueled (inner); (iv) cluster tidal truncation of bound orbits (outer); (v) stellar flyby perturbation and collision hazard (outer); (vi) thermal rejection capacity, which couples to the residue analysis of Section 3.

2.2 Inner boundary

Orbit stability. Circular equatorial orbits around a Kerr hole are stable outside the innermost stable circular orbit, r_ISCO = 6 r_g (Schwarzschild) shrinking to r_g (prograde, extremal). For M = 2×10⁴ M_☉ this is ~10⁻³ AU: dynamically, matter can orbit extraordinarily deep. Orbital periods are P ≃ 2π(r³/GM)^1/2 ≃ 9 s at the Schwarzschild ISCO and minutes at 100 r_g (619 s), and light-crossing latency across the swarm is seconds at the inner edge of the fueled envelope (r/c = 9.8 s at 100 r_g), part of the computational attraction (Paper A, §4).

Tidal stress. A rigid element of size ℓ at radius r experiences differential acceleration ~2GMℓ/r³. Requiring internal stress below a material strength S for an element of density ρ_el gives

where 10 GPa is the tensile strength of present-day carbon composites, with ρ_el = 10³ kg m⁻³, r = 10² r_g, and M = 2×10⁴ M_☉. Tides therefore never forbid the swarm architecture (elements of km scale are unconstrained down to ~10 r_g); they forbid monolithic structures below ~10³ r_g and thereby select swarms at small radii, a conclusion that parallels the single-star case (Wright 2020).

Radiation and magnetic environment. If the hole is fueled at the level required for the Blandford–Znajek power budget of Paper A, the inner ~10² r_g contains a magnetically arrested flow with field strength B ~ 10²–10⁴ G near the horizon and a jet along the spin axis. Equatorial swarm orbits outside the disk body (r ≳ 10² r_g, inclined to avoid both disk and jet) survive; structures inside ~30 r_g face erosion by the flow itself. We adopt r_in ≃ 10² r_g as the practical inner boundary of a fueled configuration, and r_ISCO for a dormant one.

Formation coherence under relativistic precession. A constraint with no single-star analogue: swarm elements at different radii and inclinations precess differentially. Pericenter advance and Lense–Thirring nodal precession scale as (r_g/r) and a_★(r_g/r)^3/2 per orbit. At r = 10² r_g the orbital period is 2π(r³/GM)^1/2 ≃ 619 s ≃ 10 min, and a formation with a 2 per cent radial spread accumulates a differential precession phase of 2π in 1.7×10³ orbits, about 12 days. Precession is not what binds, however: the same 2 per cent spread decoheres by ordinary Keplerian shear (n ∝ r^−3/2) through 2π in 33 orbits, about 6 hours, a factor 50 faster at the same threshold. Keplerian shear carries no radial dependence in units of the orbit, so it applies across the whole envelope rather than only inside 10³ r_g. The relativistic terms are what remain once a formation is built co-radially, which is the configuration that defeats shear by construction, and it is for that configuration that the numbered result below is stated. Because this exclusion recurs throughout the residue analysis, we state it as the paper's first numbered result:

Rigid geometric formations (the flat rings and discs whose stability the single-star literature analyzes around stars) are therefore excluded deep in the envelope. This is the Kerr-specific amendment to the single-star stability literature: the deep envelope selects swarm over monolith (tides, above), and within swarms, symmetric or phase-agnostic designs over shaped formations.

2.3 Outer boundary

Cluster truncation. The hole dominates the cluster potential inside its influence radius

Orbits bound to the hole are progressively stripped by the fluctuating cluster field as r → r_infl; by analogy with tidal truncation of binaries in clusters, we adopt r ≲ 0.1 r_infl ~ 4×10³ AU as a stipulated prior edge for secure orbits, not a derived one. Binary truncation scales with the Jacobi radius, which for 2×10⁴ M_☉ in the adopted uniform core (ρ₀ = 3×10³ M_☉ pc⁻³) is r_J = (M/4πρ₀)^1/3 = 0.81 pc, or 4.15 r_infl; the stipulated edge is then 0.024 r_J, and 0.1 r_J would be 1.7×10⁴ AU. The edge is a lever (Appendix C tabulates the downward direction), and the operative outer constraint below is stellar traffic rather than truncation.

Stellar flybys. The operative outer constraint is sharper than truncation: stellar traffic. The rate at which cluster stars pass within impact parameter b of the hole, including gravitational focusing, is

with n_★ ≃ 10⁴ pc⁻³ (mean stellar mass 0.3 M_☉ at the quoted mass density) and v_rel ≃ √2 σ ≃ 30 km s⁻¹, the velocity used in the focusing term throughout. At b = 10³ AU the focusing term is ~40 (it would be ~80 if evaluated at v = σ; we use v_rel) and Eq. 4 gives one passage per ~10³ yr; at b = 10² AU, one per ~10⁴ yr; at b = 10 AU, one per ~10⁵ yr (focusing-dominated regime, Γ ∝ b). Each passage tidally perturbs swarm orbits at the impulsive level δ ~ 2(m_★/M)(a/b)²(v_orb/v_rel) for structure semi-major axis a < b, saturating at ~2(m_★/M)(v_orb/v_rel) for penetrating passages. The mass ratio controls everything: at m_★/M ~ 10⁻⁵ even a penetrating passage delivers δ ≲ 10⁻², so no single flyby disrupts a hole-bound orbit, and the hazard is cumulative, a random walk in eccentricity. Two further protections apply. Direct star–element scattering requires approach within a fraction ≲10⁻⁷ of the orbital cross-section per penetrating passage: negligible. And deep in the envelope the encounter duration b/v_rel exceeds the orbital period by orders of magnitude, so the impulsive estimate above is an overestimate: adiabatic invariance suppresses the coupling of slow perturbations to fast orbits exponentially.

Secular cluster-tide forcing. Flybys are the granular part of the cluster field; the smooth part forces eccentricity secularly, the cluster analogue of Lidov–Kozai oscillations analyzed for binaries in cluster tides by Hamilton & Rafikov (2019). The secular timescale for a hole-bound orbit of semi-major axis a is of order t_sec ~ P_orb M/M_cl(a), where M_cl(a) = (4π/3)ρ₀a³ is the smooth cluster mass enclosed by the orbit. Because the hole outweighs the enclosed cluster mass by ≳10⁵ everywhere in the envelope, the forcing is weak: at a = 10³ AU, M_cl ~ 10⁻³ M_☉ against 2×10⁴ M_☉ and t_sec ~ 3×10⁹ yr, falling as a^−3/2 to ~4×10⁸ yr at the 4×10³ AU stripping radius. The estimate assumes the tide's full quadrupole is available; ω Cen's measured near-spherical shape (global ellipticity ~0.1, rounder still in the core) suppresses the non-axisymmetric component that drives the largest eccentricity excursions, so these are upper limits on the forcing rate. Secular tides therefore do not bind interior to the stripping radius: they become comparable to the flyby-diffusion floor of Section 2.4 only at the envelope's outermost edge, where stripping already terminates it.

2.4 Monte Carlo of flyby histories

We quantify the cumulative hazard with a Monte Carlo over encounter statistics (Figure 1; method and parameters in the Appendix; code in the paper repository, paper/figs/fig1_envelope.py): impact parameters from the focused distribution of Eq. 4 within b < 30a, relative speeds drawn as described in the Appendix, impulsive kicks with the saturated form above, and dynamical survival defined as eccentricity random-walking to orbit-crossing (e ~ 0.5). The perturber mass function has two stellar components (main sequence plus white-dwarf tail) and, because kick variance scales as m_★² and mass segregation concentrates heavy remnants inside the influence radius, a segregated remnant extension: neutron stars (1.4 M_☉, 2 per cent number fraction) and stellar-mass black holes at a fiducial local number fraction of 1 per cent, bracketed by 0.1 and 3 per cent. We take the black-hole mass to be 31 M_☉, the mean mass of holes inspiraling into the IMBH in the González Prieto et al. (2025) ω Cen models, which identify 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. Two caveats attach to this unbound-field comparison bracket, which retains the stipulated 0.1–3 per cent fraction: the inspiraling set is segregation-biased toward the heavy end, so the population mean is lower than 31 M_☉; and a bottom-heavy initial mass function of the kind those models require (α₃ = 2.5) produces fewer black holes. The population that actually operates, the bound cusp channel below, is not assembled from separate figures; it is anchored jointly on the extended dark mass and the mean remnant mass directly. At 31 M_☉ per object, the 2–3×10⁵ M_☉ extended dark component favored by the pulsar timing is ~10⁴ black holes, a global number fraction of 0.1–0.15 per cent against ~7×10⁶ cluster stars, consistent with the same bottom-heavy IMF and reaching a central, segregation-enhanced value of 1.15 per cent, close to the unbound bracket's own fiducial. The dataset that drives the H_q–H_sub contest also pins the Monte Carlo's dominant parameter.

The result is structurally simpler than the constraint inventory led us to expect: flybys never set the outer boundary, but the remnant tail controls the lifetime normalization. The median impulsive-diffusion lifetime is roughly flat from the ISCO to 4×10³ AU, because the kick variance per encounter and the encounter rate scale inversely (⟨δ²⟩ ∝ a⁻¹ from the saturated penetrating passages that dominate it, Γ ∝ a in the focused regime), so their product is scale-free. Its value is ~4×10⁹ yr for the stars-plus-WDs function alone, dropping to ~1×10⁹, ~2×10⁸, and ~9×10⁷ yr at black-hole fractions of 0.1, 1, and 3 per cent. The perturber mass matters more than the fraction bracket: at fixed f_BH = 1 per cent, moving from 10 to 31 M_☉ raises ⟨m_★²⟩ by a factor 8.0 and shortens the floor from ~9×10⁸ to ~2×10⁸ yr, a larger displacement than the whole 0.1–3 per cent bracket produces at fixed mass.

Adiabatic decoupling of the unbound channel. That floor is an impulsive estimate, and the impulsive approximation does not hold anywhere in the envelope. For the penetrating encounters that carry the kick variance, the adiabaticity parameter is x = ω_orbτ_enc = (b/a)(v_orb/v), which at b ≃ a reduces to v_orb/v_rel: 2.2 at the cluster stripping radius, 4.5 at 10³ AU, 14 at 10² AU, and 4×10³ at the ISCO. Impulsive behaviour requires x ≪ 1, which for this system means a ≳ GM/v_rel² ≃ 2×10⁴ AU, five times outside the envelope. Unbound cluster stars are therefore adiabatically decoupled from every orbit an installation could occupy, and suppressing that correction is conservative in an unbounded way rather than by a stated factor. Applying the Gnedin & Ostriker (1999) kernel A(x) = (1+x²)^−5/2 per encounter, which is calibrated against N-body results and is the less suppressing of the standard choices, lifts the fiducial median to the 12-Gyr cluster-age cap within ~3 AU and to ≃7.6×10⁸ yr at the envelope edge, where the suppression is weakest. We carry both curves in Figure 1 and quote the impulsive one as the floor. The scale-free flatness is a property of the impulsive kernel, not of the physical problem: the corrected curve rises inward.

The Bahcall–Wolf cusp. The Monte Carlo uses the core-average perturber density, and a relaxed bound cusp around the hole raises it locally. The cusp exists. The González Prieto et al. (2025) realizations, which we cite elsewhere for growth history, report a density cusp inside the influence radius forming after 300 Myr, with the stellar population settling at γ ≃ 1.3 and the black-hole population steepening to γ ≃ 2.0: the multi-mass structure predicted by Bahcall & Wolf (1976, 1977), heavy species steeper than light. (The single-mass benchmark γ ≃ 1.75, reproducing the classic Bahcall–Wolf result inside the same ω Cen comparison, is Baumgardt (2004)'s N-body value rather than a separate output of the multi-mass ω Cen realizations; we cite it for context but do not use it as an input.) The timescale argument agrees. ω Cen's catalogued relaxation times are ~4.0 Gyr in the core and ~12 Gyr at the half-mass radius (Harris 1996); the influence radius lies deep inside the core radius, so the core value governs, and it is roughly a third of the cluster age. (A local Spitzer relaxation time evaluated at ρ₀ and σ with ⟨m⟩ = 0.3 M_☉ gives ~13 Gyr instead, the convention difference residing in ⟨m⟩ and lnΛ; we use the catalogued core value and flag the discrepancy.) Either way the segregation clock settles it: cusp formation for a species of mass m proceeds on ~(⟨m⟩/m) t_relax, which for a 31 M_☉ remnant is 0.03–0.6 Gyr on either convention.

The cusp case is therefore the fiducial, and it interacts with the adiabatic correction in the direction that matters. Cusp members are bound, moving at the local Keplerian speed rather than at v_rel, so x ~ 1 for them and they are not adiabatically protected. That is the regime where the impulsive treatment is least controlled, not most: the impulsive limit wants x ≪ 1 and the adiabatic limit wants x ≫ 1, and the bound members sit between them. We apply the same Gnedin & Ostriker kernel there because it interpolates and is calibrated against N-body, but the correct statement is that the bound channel is the one where the kernel choice matters and the unbound channel is the one where the impulsive treatment is simply wrong: running the bound channel with no kernel at all drops the conditional edge floor from 4.3×10⁷ to 5.9×10⁶ yr, a factor 7.4, carried as the lower bracket on this channel rather than the kerneled number alone. Resonant relaxation is a further omission: for a near-Keplerian bound population it can dominate two-body relaxation in eccentricity, it is not in this model at all, and its absence pushes in the direction of longer lifetimes. Because N(<r) ∝ r^3−γ = r at γ = 2, only ~0.4 black holes are expected inside the stripping radius and ~4 inside r_infl: the population is granular, not a smooth density, usually absent at the edge, and the Monte Carlo is rebuilt around it rather than estimated analytically. Each species' cusp density follows N_s(<r) ∝ r^3−γ_s (γ_★ = 1.3, γ_NS = 1.5, γ_BH = 2.0; González Prieto et al. 2025), normalized against the core-average density at r_infl; the perturber count for each history is Poisson-drawn from this profile rather than treated as a rate. At the envelope edge a bound black hole is present in 25 per cent of histories; when one is present the diffusion floor is ~4×10⁷ yr, and when none is present the channel reaches the 1.2×10¹⁰-yr cluster-age cap, which is what the median history does. Three normalizations control those two numbers: the stipulated segregation enhancement of Appendix A (across 3–30 the presence probability spans 20–90 per cent), the joint dark-mass anchoring (Appendix A), and the choice to normalize the core number density on the observed mass density rather than a fixed number density, worth a factor 2.7 in the perturber count on its own. Under the retired fixed-number convention (Appendix G), presence at the edge is 53 per cent and the median history is the one that grinds; the conditional floor itself barely moves between the two conventions (4.3×10⁷ yr either way), since the normalization sets how often a bound remnant is there, not how fast it works when it is (Figure 1, panel b; Appendix A). That bimodal bound-member diffusion channel, not an analytic estimate, is the binding clock at the envelope edge: flybys never bind anywhere in the envelope, and the deep envelope is limited by maintenance rather than by dynamics.

The same remnant, held at r_p = 4×10³ AU, also forces the orbit secularly on t_sec(a) = (M/m_pert)(r_p/a)³P_orb(a) ≃ 1.2×10⁷ yr at a ≃ 9×10² 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; a competing quench from apsidal precession forced by the light cusp population is comparable near 10³ AU and evaluated per history. Folding in the 77.5 per cent mutual-inclination window for the resonance, this forcing is coherent and oscillatory rather than diffusive, so it sets a station-keeping cadence over its 4×10²–4×10³ AU operating range rather than a survival limit, and is absent below it.

The survival criterion itself needs a second clock. The e = 0.5 threshold marks dynamical death, when swarm orbits cross and membership is lost; an unmaintained swarm dies operationally much earlier, because differential kicks pump internal velocity dispersion, and once neighboring orbits cross at their much smaller spacing (Δa/a ~ 10⁻³ for a dense swarm) the collisional cascade of Section 3.3 begins. Scaling the random walk to that threshold gives a grinding-onset time of (e_swarm/e_cross)² ≃ 4×10⁻⁶ of the diffusion floor, ≈3×10² yr against the impulsive floor near 1 AU and longer once the adiabatic correction is applied: of order 10²–10³ yr, consistent with the abandonment timescale derived independently in Section 3.3. The two clocks are reconciled by maintenance: the station-keeping needed to null the accumulated differential drift is a small trim budget (the per-encounter kicks are Δv ≲ cm s⁻¹ against orbital velocities of ~70 km s⁻¹ at the 4×10³ AU stripping radius and ~4×10³ km s⁻¹ at 1 AU), but it is strictly required. A passively bound swarm is not thereby functional; every long-lived swarm in this paper is a maintained swarm. The operative outer boundary remains the cluster stripping radius of Section 2.3, ~0.1 r_infl ~ 4×10³ AU, with the flyby Monte Carlo setting the maintenance-free lifetime interior to it.

2.5 The envelope

Combining Sections 2.2–2.4: for the fiducial system, engineered swarms (symmetric or phase-agnostic, per the precession constraint) persist passively from r ≃ 10² r_g (2×10⁻² AU, fueled) or r_ISCO (dormant) out to the cluster stripping radius ~4×10³ AU. The unbound field is adiabatically decoupled everywhere in that range; the operative floor is the bound-member diffusion channel, ~4×10⁷ yr at the envelope edge in the 25 per cent of realizations that draw a bound heavy remnant, and the cluster-age cap in the other three quarters. Beyond the stripping radius, orbits bound to the hole are not durable and only cluster-orbiting architectures (outside this paper's scope) remain.

The envelope is the paper's first result (Figure 1). Two corollaries matter downstream. First, the thermodynamically favored location (deep, near the flow) is also the dynamically safest, doubly so once adiabatic protection is counted: the environment does not penalize the architecture Paper A's physics prefers, which is a nontrivial consistency check the hypothesis could have failed. Second, the envelope concentrates any engineered mass within ~4×10³ AU ~ 10⁻¹ r_infl of the hole, with the thermodynamic gradient pushing occupation deep into the arcsecond-unresolvable interior: observable only through its energetic and dynamical residue. That is the subject of Section 3, after two remaining environmental questions: hazards other than flybys, and fuel.

2.6 Hazards beyond flybys

Three further environmental processes deserve quantitative treatment. Each is a candidate defeater; none defeats, though the first comes closest.

Tidal disruption events. A cluster-core IMBH tidally disrupts stars scattered into its loss cone. Rates for IMBHs in evolved globular clusters, computed with loss-cone methodology of the kind developed for the supermassive case (Stone & Metzger 2016), are ~5×10⁻⁸–5×10⁻⁷ yr⁻¹ per cluster for main-sequence stars (our evaluation of Fragione et al. (2018)'s Eq. 27 empty-loss-cone rate over the half-mass density bracket of its Eq. 8, 5×10⁻⁶–5×10⁻⁵ yr⁻¹ per IMBH-hosting cluster, reduced by an assumed IMBH occupation fraction of 0.01; the WD-suppression factor below is Fragione et al. (2018)'s stated result), with white-dwarf disruptions rarer by a factor of ~30 at z ≲ 0.5, the epoch relevant here (the factor approaches ~100 only at z = 2) (Fragione et al. 2018); the ω Cen-specific value from the González Prieto et al. (2025) models is ~5×10⁻⁸ yr⁻¹, and we use that number where a target-specific rate is needed. One reciprocal flag for Paper A: this is also the natural loss-cone supply rate, and the staged architecture of Paper A's Appendix A requires ~1–2×10⁻³ stars yr⁻¹ during spin-up (6.8×10⁴ stars over 4.2–6.6×10⁷ yr, Paper A's own basis for the spin-up duration, which is a different quantity from the 3.5×10⁷ yr Salpeter-time estimate of this paper's Section 5), so natural refilling falls short by four to five orders of magnitude and essentially the whole fuel budget of that phase must be delivered by engineered orbit-shaping. During a disruption the accretion luminosity approaches Eddington, L ~ 3×10⁴² erg s⁻¹ for the fiducial mass, sustained for months to years of fallback: the flux at r = 1 AU is ~10⁹ times the solar constant, and no plausible material hardening survives it in place. Feasibility therefore requires that a TDE be survivable by response rather than endurance: loss-cone stars are in principle identifiable and trackable long before disruption, and the months-long fallback rise gives further warning, so temporary evacuation outward along the envelope, or shadowing at large inclination, is an engineering requirement we impose on any architecture rather than a reason to exclude one. The expected recurrence time of ≳10⁷ yr sets the natural amortization horizon of the installation: a 10⁸–10⁹-yr occupation must plan for several such events. The TDE duty cycle also reconciles engineered occupation with the present silence at no extra cost: post-TDE fallback outshines any steady configuration for ~10²–10³ yr per ~10⁷-yr recurrence, a duty cycle of ≲10⁻⁴, so catching ω Cen mid-flare was never likely under any hypothesis. And TDEs cut the other way, as an observational gift: any future TDE flare in ω Cen both confirms the hole and activates the variability channel of Section 3.2 at high signal-to-noise, and the framework of Section 4 treats a TDE with anomalous light-curve regulation as one of the few single-epoch events that can move ln K substantially.

Black-hole wander. The hole is a Brownian particle in the stellar bath: N-body characterizations give r.m.s. displacements of order 10⁻²–10⁻¹ pc for 10³–10⁴ M_☉ holes in ω Cen-like cores, extrapolating de Vita et al. (2018)'s fitted wander-amplitude scaling law (directly simulated over 75–150 M_☉) to this mass range using its stated inverse-mass-power dependence. For the fiducial 2×10⁴ M_☉ hole the expected wander is ≲10⁻² pc ~ 2×10³ AU: comparable to the 4×10³ AU envelope. That figure is a lower bound on two counts. The equilibrium amplitude in a multi-mass bath is set by the mass-weighted mean m_eff = ⟨m²⟩/⟨m⟩ rather than by ⟨m⟩, since the velocity diffusion scales with n⟨m²⟩ while the drag scales with n⟨m⟩ (Chatterjee, Hernquist & Loeb 2002; Merritt, Berczik & Laun 2007); for the segregated mass function of Section 2.4 that is m_eff = 13.1 M_☉ against ⟨m⟩ = 0.75 M_☉ at the fiducial f_BH = 1 per cent and 31 M_☉ perturbers, so an equal-mass estimate understates the amplitude by √(m_eff/⟨m⟩) = 4.2. Measured instead against a remnant-free bath (m_eff = 0.46 M_☉), which is closer to the mass function the N-body results carried, the understatement is 5.4. And the quoted N-body results predate the retained black-hole populations now inferred for ω Cen, so the mass function they carried is lighter than the one Section 2.4 adopts. Paper C's wander program (§5.2 there) inherits both corrections, in the direction that makes the measurement easier. Structures bound to the hole simply ride along (the swarm orbits the hole, and hole plus swarm wander together through the cluster), so wander does not threaten the installation; what it threatens is the observer's astrometry, since the kinematic center and the hole need not coincide at the 10⁻²-pc level. Paper C's astrometric programs already marginalize over center position; we note here that the wander amplitude is itself mass-dependent and therefore carries independent information about M in long-baseline proper-motion data.

Gas drag and erosion. ω Cen retains a measurable intracluster medium: pulsar dispersion measures in comparable clusters give central ionized densities n_e ~ 0.07–0.3 cm⁻³ (Freire et al. 2001; Abbate et al. 2018); that range is the 47 Tucanae analogy of Section 2.7. (oMEGACat VII's own ω Cen figure, 1.94 cm⁻³, is flagged by its authors as a foreground-dominated upper limit and is not used here or anywhere in this paper.) Ram-pressure drag on a swarm element of areal density Σ ~ 1 g cm⁻² at orbital velocity of a few ×10³ km s⁻¹ (the deep envelope; 4×10³ km s⁻¹ at the 1 AU fiducial) removes a fractional momentum ~ρ_gasvt/Σ per time t: at the measured densities this is ≲10⁻⁵ per kyr, running 1.9, 6.1, and 8.0×10⁻⁶ per kyr at n_e = 0.07, the fiducial 0.23, and 0.3 cm⁻³, and scaling as n_ev/Σ. Drag never disrupts an orbit outright, but the term is not negligible over the envelope's own lifetimes: unopposed, the fiducial figure removes the full orbital momentum of a 1 g cm⁻² element in 1.6×10⁸ yr, the same order as the bound-member diffusion floor of Section 2.4. Drag therefore belongs in the maintenance budget rather than in the list of defeated processes, and an abandoned installation loses to it on a timescale the relic window of Section 3.3 already assumes. Sputtering and dust impacts are similarly small in an old cluster with no star formation and depleted debris populations. Gas matters for fuel, however, which is the next subsection.

2.7 The fuel budget

The Blandford–Znajek architecture of Paper A needs mass supply, and the measured medium bounds it. ω Cen has no direct ionized-gas density measurement of its own; we adopt n_e ≃ 0.23 cm⁻³ by analogy with the comparable cluster 47 Tucanae (Abbate et al. 2018), an explicit ambient-value choice rather than an ω Cen-specific measurement. (oMEGACat VII's own ω Cen-specific figure, n_e = 1.94 cm⁻³, is ~8× higher and is flagged by its authors as an upper limit likely dominated by foreground, non-intracluster ionized gas rather than the cluster's own medium; we do not use it here.) Bondi accretion from gas at this density and relative velocity ~σ gives

about 10⁻⁴ of the Eddington rate. The figure is bounded from both sides. It is an upper bound insofar as the density at the hole's actual location may be below the cluster mean: the accretion non-detections are consistent with the hole occupying a locally evacuated region, and the Bondi radius (GM/(σ² + c_s²) ≃ 3×10⁴ AU) samples gas the surveys average over. It is a lower bound insofar as it ignores harvesting, though by far less than earlier versions of this paper and of Paper F claimed (Appendix G). Those versions put ~10³ giants inside the influence radius; that figure counted the stars there rather than the ones currently on the giant branch. Recounting with the Kroupa turnoff weight (0.035) and the fraction of a turnoff star's main-sequence life spent above 10³ L_☉ (0.032), the expected standing giant population inside this paper's own r_infl = 0.20 pc is 0.18. At 10⁻⁸ M_☉ yr⁻¹ each, the aggregate wind supply is a few per cent of Ṁ_B on this paper's own numbers. Local wind harvesting is therefore a factor-of-order-unity term, not an orders-of-magnitude one; the "orders of magnitude" route is deliberate mass import alone, which the waste-heat ceiling of Section 3.1 already prices. At MAD-plus-spin effective efficiencies of order unity, realized only at high spin (η_jet ≃ 1.4 at a_★ ≃ 0.99, falling roughly as a_★² to a factor ~5 lower at a_★ = 0.5; Paper A), the ambient rate gives extractable power P ~ Ṁ_Bc² ≃ 3×10³² W ≃ 7.6×10⁵ L_☉ at a_★ ≃ 0.99 and ≈9.5×10² to 3.8×10³ L_☉ at the MAD equilibrium spin a_★ = 0.035–0.07 of Section 3.4, with the fuel ceiling carrying the same η(a_★) dependence: on the thin-disc branch the ambient medium alone, with no harvesting of stellar winds or imported fuel, funds a computational budget two orders of magnitude above the waste-heat ceiling derived in Section 3.1, and on the MAD branch it does not (Section 3.4).

That headroom assumes the Bondi rate reaches the horizon, and in natural hot, low-Eddington flows it does not: the inflow declines inward, Ṁ(R) ∝ R^s with s ≃ 0.3, as most of the captured gas is unbound into outflows (Blandford & Begelman 1999; Yuan & Narayan 2014). Over the eight decades from the Bondi radius (r_B = GM/(σ² + c_s²) ≈ 3.3×10⁴ AU) to r_g, the suppression is (r_B/r_g)^0.3 ≈ 290, giving P_fuel ≃ 2.6×10³ L_☉; adopting instead the empirical Sgr A* figure of ~1 per cent of the Bondi-radius rate gives ≃8×10³ L_☉. Either value lands at or below the warm-swarm waste-heat ceiling of Section 3.1, so for a natural flow the supply and concealment constraints bind together, and concealment alone binds only in the engineered case. The engineered case is available, and it unifies two of this paper's results: a system drawing more than the ADIOS-suppressed rate is a system that has suppressed its own outflow, and that is the same control action, regulating horizon magnetic flux and mass supply, that produces the low-R variability deficit of Section 3.2. The fuel budget and the R statistic are two faces of one control action. We therefore carry both cases through the paper: natural-suppressed (P_fuel ~ 10³–10⁴ L_☉) and engineered-unsuppressed (P_fuel ≃ 7.6×10⁵ L_☉), with the Appendix-C prior ceiling stated for the pair.

For an installation with outflow control, then, the binding constraint on a present-day ω Cen system is thermal (getting rid of entropy without detection), never supply; without it, the two constraints meet. The unsuppressed ordering is a nontrivial output: for stellar-mass holes in the same environment the two constraints reverse even in the engineered case, an independent reason the hypothesis family selects IMBHs. The same numbers close the loop on the accretion non-detection: the deep ATCA and JWST limits require the natural radiative efficiency of any Bondi-fed flow to be ≲4×10⁻³, uncomfortable for the quiescent-IMBH null at the upper end of the mass range and what an engineered-occupation flow predicts for output extracted as work rather than radiation; the inverse fundamental-plane treatment of the same candidate reaches the complementary conclusion, that its radio silence constrains gas supply, accretion efficiency, and compact-core production rather than excluding the hole (Abbassi et al. 2026). The adjudication in Section 5 prices this observation for both hypotheses rather than letting either claim it informally.

3. Observable residue

3.1 Waste-heat floor

Paper A argued that horizon entropy disposal permits computation with radiated waste heat far below the Dyson-sphere expectation. Here we make the floor quantitative, and we first fix the bookkeeping, because the mainstream position is an energy-conservation argument: Curtis et al. (2026) state that Dyson-scale computation must reradiate nearly all of the energy it absorbs, making mid-infrared thermal emission the robust technosignature. The horizon-sink architecture disputes that position at the level of energy conservation itself, not merely entropy accounting. Let P_comp be the power processed by the swarm and f_sink the fraction of the waste energy delivered across the horizon rather than radiated; the entropy rides with its carriers, so a single fraction serves for both. Energy beamed across the horizon is not reradiated at all: it adds to M, and since T_H ∝ M⁻¹ the disposal channel deepens as it is used, a weakly self-improving property. What must be radiated is only the missed fraction, L_waste = (1 − f_sink) P_comp, re-emitted thermally at the temperature set by the swarm radius. For single-sided radiators at unit covering fraction, T_eff ≃ (L_waste/4πr²σ_SB)^1/4: 1 L_☉ at r = 1 AU gives 394 K, 1 L_☉ at 10³ AU gives 12.5 K, and a 50 K swarm at 10³ AU corresponds to L_waste ≈ 260 L_☉. This T_eff(r) assumes unit covering fraction with single-sided radiators; a sparse swarm of covering fraction f_cov runs hotter by f_cov^−1/4, moving mass into the JWST wedge and shifting the composite ln K of Section 5 to −0.61 at f_cov = 10⁻², with the excluded active-prior fraction rising from 0.76 to 0.92 (Appendix C, --fcov).

The engineering limit on f_sink is transport: waste energy must be carried inward, against the swarm's own power draw, by mass flux or directed radiation into the horizon. The capture-cone and étendue accounting of Appendix A.3 bounds 1 − f_sink ≳ 10⁻⁴ for swarm architectures within the envelope, set by the fraction of beamed power unavoidably intercepted and re-thermalized by swarm elements along the transport path. The sign of its influence is worth stating: a lower true floor (better beaming than the A.3 accounting allows) weakens the mid-infrared charge against engineered occupation and moves ln K toward zero, while a higher floor tightens the P_comp ceiling and strengthens it (Section 6.1). The floor is therefore

Converting that floor into a P_comp bound requires a temperature axis, because which instrument limits L_waste depends on where the thermal emission peaks. The deep JWST limits at the kinematic center constrain warm sources: a swarm at r ≲ a few AU re-radiates at T_eff ≳ 150 K, peaks within MIRI coverage, and inherits the sub-L_☉ point-source limit, giving P_comp ≲ L_lim/(1 − f_sink) ≲ 10³–10⁴ L_☉. A cool outer swarm evades it: at r ~ 10²–10³ AU the emission peaks at 20–60 μm and beyond, past the coverage of JWST/MIRI's nine imaging filters (F560W through F2550W, the 25.5 μm cutoff); WISE W3/W4 and Spitzer/MIPS, whose 6–18 arcsec beams are confusion- and crowding-limited in the ω Cen core, are the instruments that would bound such a cool swarm observationally, though Appendix C does not draw on them directly. The defensible statement is radius- and temperature-dependent, and the continuous per-filter Appendix C likelihood integrates the two-temperature swarm SED against the real MIRI sensitivity curve rather than marginalizing over instrument-specific placeholder limits, so a sufficiently cool, extended swarm falls below every MIRI filter and registers as undetected rather than excluded. An engineered system operating at the radius-appropriate ceiling is consistent with all current data; deeper MIR photometry tightens the warm wedge linearly, while the cool wedge waits on far-infrared sensitivity. Figure 2 assembles the warm-swarm plane and the temperature-resolved plane, with the transport floor and the fuel ceiling of Section 2.7, into the constraint space that defines the surviving engineered-occupation parameter volume.

3.2 The MAD-regulation signature

The second channel is new to this paper. General-relativistic magnetohydrodynamic simulations establish that natural accretion in the magnetically arrested state is intrinsically episodic: magnetic flux accumulates at the horizon, chokes the inflow, and erupts in quasi-regular flux-expulsion events, producing characteristic variability in jet power and radiative output with a MAD > intermediate > SANE hierarchy in both luminosity and its variance. The eruption mechanism is plasmoid-mediated reconnection of the accumulated horizon flux, resolved ab initio by Ripperda et al. (2022), whose flare cycle of ~10²–10³ r_g/c between flux accumulation and expulsion is the recurrence we adopt; the state-resolved variability tables of the 2026 GRMHD compilation place the MAD eruption power in the same interval at matched mean luminosity. For M = 2×10⁴ M_☉ that cycle is ~10–10² s, comfortably within the cadence of X-ray monitoring.

A system engineered for steady power extraction has an incentive absent in nature: eruptions are interruptions. Regulating the delivered mass and flux (the control variable identified by the simulations is horizon magnetic flux itself) suppresses the eruption cycle and its variability signature. We therefore forward-model the residue as a variability deficit: an accreting IMBH whose X-ray (or optical) power spectrum lacks the flux-eruption band that GRMHD baselines predict for its luminosity and inferred state. Quantitatively, we define the regulation statistic R as the ratio of integrated power in the eruption band (10⁻²–10⁻¹ Hz for the fiducial mass) to the GRMHD-calibrated expectation at matched mean luminosity; natural MADs populate R ~ 1 with factor-of-few scatter, and R ≪ 10⁻¹ sustained over many cycle times has no identified natural mechanism (Figure 3). The band edges follow directly from scaling the Ripperda et al. (2022) flare cycle of 10²–10³ r_g/c to M = 2×10⁴ M_☉ (r_g/c = 0.1 s, hence 10–100 s), with the 2026 GRMHD state tables placing the eruption power in the same interval. The main confusion risk in applying the statistic is contamination from the intermediate magnetic state, whose eruption power is partially suppressed relative to MAD, so the baseline must be conditioned on the inferred state, never on luminosity alone.

Two instrument realities shape where R can be measured. First, at the fiducial mass R is an X-ray/optical statistic; radio is excluded twice over. Synchrotron self-absorption in any compact GHz-emitting flow puts the τ = 1 photosphere at ~10²–10⁴ r_g, low-pass filtering intrinsic variability at Δt ≳ 10–10³ s, directly on top of the eruption band, so a radio non-detection of the band is expected under the quiescent and engineered hypotheses alike and carries no discriminating power. Interstellar scintillation compounds the problem: a microarcsecond-scale source at 5.49 kpc scintillates at GHz frequencies on minutes-to-hours timescales, adding modulation uncorrelated with accretion in and near the band, so even a genuinely regulated (low-R) source would show a measured R biased upward unless the scintillation is modeled out. Second, the statistic has a detectability threshold. Distinguishing R ~ 1 from R < 0.1 requires several counts per ~10 s cycle sustained over many cycles, i.e., F_X ≳ a few ×10⁻¹² erg cm⁻² s⁻¹ for m²-class effective area, corresponding to L_X ≳ 10³⁴–10³⁵ erg s⁻¹ ≈ 4×10⁻⁹–4×10⁻⁸ L_Edd at the fiducial mass. The channel therefore activates in flare and TDE states rather than in deep quiescence, which is consistent with the TDE-as-gift framing of Section 2.6: the events that confirm the hole are the events bright enough to measure R.

The signature has three properties valuable for adjudication. It is conditional: it activates only if accretion is ever detected, so it presently constrains nothing (ω Cen's hole is electromagnetically silent). It is differential: it compares a source against a physics baseline at matched parameters, canceling many systematics. And it is disprovable in place: detection of normal flux-eruption variability from a future ω Cen accretion flare would count against engineered occupation in the framework below, making the channel one of the few that can move evidence in both directions.

3.3 Relic residue: the abandoned swarm

Persistence is conditional on maintenance, and prior work on megastructure collisional cascades has shown what maintenance failure means: once guidance fails, swarm elements collide, and a collisional cascade grinds the population to dust on a timescale of roughly the orbital period divided by the covering fraction. The deep envelope makes this dramatic. At r ~ 1 AU around the fiducial hole the orbital period is ~3 days; for covering fractions 10⁻⁴–10⁻² the first-collision timescale ~ P/f_cov is years to decades, and the full cascade completes on a timescale of order 10³ yr, orders of magnitude faster than for stellar-orbit megaswarms and consistent with the grinding-onset clock of Section 2.4. The end state differs too, and the difference is observationally decisive: cascade debris around a star settles into a long-lived warm dust population, a passive relic technosignature, whereas debris around a black hole is progressively drained, its periapsis distribution fed toward the loss cone by continuing collisions and flyby perturbations and its finest grindings coupled to whatever gas flow exists; the drainage rate remains an estimate, qualified in Section 6.1. The hole removes its own debris. An abandoned engineered-IMBH system therefore passes through a brief bright phase (cascade grinding plus enhanced accretion luminosity as debris drains) and then reverts to a naked quiescent hole, indistinguishable from the gas-starved null except possibly through spin.

Three consequences follow. First, a fourth hypothesis joins the menu at zero structural cost: a formerly engineered, now-abandoned system, whose observables today equal the gas-starved quiescent null plus high spin, and whose prior couples to the goal-stability open problem flagged in Paper B. Second, the searchable relic window is short, so population-level searches for dead installations have low yield around IMBHs specifically; non-detection of relic dust in ω Cen carries almost no evidence either way, and the framework scores it accordingly. Third, spin becomes the only durable observable, and an objection must be met before relying on it: Blandford–Znajek extraction spins the hole down, and MAD jets remove angular momentum faster than accretion supplies it, so one might expect long use to erase the very fossil we propose reading. The rates rescue the argument: in the MAD spin-down calculus, the spin-down per unit accreted mass is of order |Δa_★| ~ ΔM/M, so erasing a high spin requires cycling a mass comparable to the hole's own through the flow, and at the ambient fueling of Section 2.7 that takes M/Ṁ_B ~ 4×10¹¹ yr. Only a civilization importing mass at far above ambient rates for ≫10⁹ yr would measurably despin the hole, and that regime is separately excluded by the waste-heat ceiling. This is also the point of contact with the compact-object signature named in the review literature: Vidal et al. (2026) attach to Penrose-process extraction from a Kerr hole the signature of “a black hole whose rotational rate is slowing down anomalously”. The calculus above prices that signature: the anomalous case requires the extraction or mass-import regime, far above ambient fueling for ≫10⁹ yr, that the waste-heat ceiling of Section 3.1 excludes. Anomalous spin-down is diagnostic of that regime wherever it occurs, and at ω Cen the ceiling predicts its absence for any installation still concealed.

The spin channel needs a careful statement, including an isotropy assumption. The natural expectation is a channel mixture, set by growth history rather than by any single attractor: repeated comparable-mass mergers drive χ toward the ≃0.7 attractor (Fishbach, Holz & Farr 2017; Gerosa & Berti 2017); gas-poor growth by minor mergers and inspirals of compact objects random-walks the spin to low values under isotropic capture; and prolonged coherent disk accretion spins the hole up toward the radiation-limited maximum a_★ = 0.998 (Bardeen 1970; Thorne 1974). Which of the three operated at ω Cen is a question its growth models answer, conditional on the isotropy assumption stated next.

For the minor-merger regime, isotropic capture (⟨cos ι⟩ = 0 for the captured objects' orbital planes) gives a_★,rms = (L̃/√3)√(μ/M_f) (Appendix B); the assumption is load-bearing for everything that follows and we state it here rather than leave it implicit. A net alignment fraction ε_rot = ⟨cos ι⟩ above isotropic adds a coherent term a_★^coh ≈ (1.7–2.0)·ε_rot that carries no √N suppression, so it matches the random walk at ε_rot ≃ 0.026–0.031 (Appendix B) and dominates above it, erasing the low-spin prediction entirely once ε_rot ≳ 0.4–0.5. The prediction below holds only for ε_rot ≲ 0.03. ω Cen is a rotating cluster and the inspiraling population is the mass-segregated heavy-remnant tail, so isotropy is a premise to check rather than a safe default; ⟨cos ι⟩ = 0 is an assumption here, and the González Prieto et al. (2025) realizations cannot supply it, since their Monte Carlo code is spherically symmetric, does not track the orbital plane, and draws transverse directions isotropically. Section 3.3 carries the tolerance as an explicit condition on the green branch.

The González Prieto et al. (2025) realizations (cited here for growth history only; they do not track spin) grow a 500–5000 M_☉ seed to ~5×10⁴ M_☉ primarily by mergers with 30–40 M_☉ black holes, roughly 10³ events at seed-epoch mass ratio q ≃ 0.006–0.08; by the end of the growth history the ratio the final spin depends on is the smaller μ/M_f ≃ 7×10⁻⁴. That is the isotropic minor-merger regime, not the comparable-mass one: the walk reaches the χ ≃ 0.7 attractor only at q ≈ 0.12 (a continuous scaling rather than a literature-sourced cutoff), and ω Cen sits a factor ~175 below that boundary on the final-mass ratio. So the χ ≃ 0.7 attractor does not apply to this target; under isotropic capture the expected outcome, derived rather than imported from a comparable-mass literature value, is a_★ ~ 0.05–0.10: 0.049 ± 0.021 at the González Prieto et al. (2025) endpoint mass, 0.084 at this paper's fiducial mass. For ω Cen the contrast is the sharper low-versus-high one, and the attractor caution belongs to the class-level statement rather than to this system. The information forecast of Section 5 restores the spin channel accordingly, with the caveat that the class-level application to the LISA IMBH population must carry the mixture, since comparable-mass growth histories elsewhere populate the attractor.

3.4 Which accretion architecture spun the hole up

The fuel and power budget of Section 2.7 quotes η_jet ≃ 1.4 at a_★ ≃ 0.99, which is a magnetically arrested flow at near-extremal spin. The spin-up arithmetic of Section 5 assumes Thorne's radiative equilibrium a_★ = 0.998, a thin-disc boundary condition. These are two accretion architectures, distinguished by how the hole was spun up. Extraction at the ambient fueling rate preserves a thin-disc spin: by the despin calculus of Section 3.3, erasing a high spin requires cycling a mass comparable to the hole's own through the flow, which takes M/Ṁ_B ~ 4×10¹¹ yr at ambient fueling. A hole spun up on a thin disc therefore keeps its spin for far longer than a Hubble time while extracting at η ~ 1. The fork lies upstream, in the spin-up itself. On a thin disc the hole climbs to a_★ = 0.998. In a magnetically arrested flow the jet removes angular momentum faster than accretion supplies it above the arrested equilibrium, so spin-up from a low spin stalls at the arrested equilibrium; Narayan et al. (2022) (§3.4) find the spin-up and spin-down equilibrium at a_★^eq ≈ 0.035, note that the shorter simulations of Tchekhovskoy et al. (2012) gave the larger ≈ 0.07, and speculate that a_★^eq → 0 for sufficiently long-lived MAD systems. We carry a_★^eq = 0.035–0.07 for the MAD-mode spin-up (earlier versions quoted 0.1–0.3 for this equilibrium; Appendix G). Every number downstream of the 2.20× growth factor inherits the thin-disc assumption.

We therefore state the fork and price both branches rather than choosing one silently. On the thin-disc branch, the episode is a spin-up to a_★ = 0.998 with no sustained extraction during it, giving M_f/M_i = 2.2024, ΔM = 2.4×10⁴ M_☉ from the fiducial hole, an episode lasting 3.6×10⁷ yr, and endpoints of 4.4×10⁴ M_☉ from the fiducial mass or 1.1×10⁵ M_☉ from the merger endpoint. On the MAD branch, the terminal spin is the equilibrium value and the same Bardeen relation, a_★ = √(2/3) (M_i/M) [4 − √(18(M_i/M)² − 2)] from a_★ = 0, gives M_f/M_i = 1.0097 at a_★ = 0.035 and 1.0198 at a_★ = 0.07: ΔM = 1.9×10² to 4.0×10² M_☉, episodes of 0.43 to 0.88 Myr, and endpoints of 5.05 to 5.10×10⁴ M_☉ from the merger endpoint. These MAD-branch values follow from the closed relation; the script (figs/plc_emix/spin_branch_aeq.py), run at a_★ = 0.035 and 0.07, returns the same values to the digits quoted.

Three of this paper's observables depend on which branch holds. The mass discriminant of Section 5 exists only on the thin-disc branch; on the MAD branch the predicted endpoint is within 2 per cent of the merger-only mass and the channel carries no information at all. Core depletion likewise: ΔM ~ 10² M_☉ is undetectable against any cluster mass model. And the durable residue a_★ ≳ 0.9, which Section 3.3 makes the one lasting observable of an engineered history, becomes a_★ ≈ 0.035–0.07. That range lies within 1σ of the natural-capture null of Appendix B (0.049 ± 0.021 at the endpoint mass; 0.084 at the fiducial mass) and below the T5 trigger a_★ < 0.1. The spin contrast therefore vanishes or inverts: 0.035 lies below the central value of the null and 0.07 above it, both within 1σ, so a measurement would struggle to separate an engineered hole from a natural one, and at 0.035 the engineered hole reads lower. It follows that the spin test T5 of Paper A, which kills the exploitation model at a_★ < 0.1, would fire on an engineered hole that was spun up in MAD mode, so T5 is conditional on the thin-disc branch.

Nor is the fuel ceiling independent of the choice. Section 2.7's 7.6×10⁵ L_☉ is the ceiling at a_★ ≃ 0.99 and carries the η(a_★) ∝ a_★² dependence stated there, so it scales by (a_★/0.99)². At a_★ = 0.035 that factor is 1.25×10⁻³, giving a ceiling of ≈ 9.5×10² L_☉; at a_★ = 0.07 it is 5.0×10⁻³, giving ≈ 3.8×10³ L_☉. They bracket the ADIOS-suppressed natural supply (~2.6×10³ L_☉), each within a factor of 3 of it; Section 2.7 places that supply at or below the warm-swarm waste-heat ceiling of Section 3.1. The engineered case's margin of two orders of magnitude over that ceiling is a thin-disc-branch statement and does not carry to the MAD branch.

We do not adjudicate the fork here, because the observation that would is a spin measurement, and it does not exist. What changes is the bookkeeping: every claim in Section 3 and Section 5 that descends from 2.20, from 1.1×10⁵ M_☉, from 3.5×10⁷ yr, or from a_★ ≳ 0.9 is a thin-disc-branch claim, and is labelled as one from here on. A referee who prefers the MAD architecture should read those numbers as the upper branch of a fork whose lower branch removes the mass channel and the depletion signature, inverts the spin contrast, and lowers the ambient fuel ceiling by a factor of ≈ 200 to 800.

3.5 EMRI dephasing: the mass channel

Every residue channel above constrains power, variability, or spin. One channel constrains engineered mass, and it comes for free with the gravitational-wave observation the series already relies on. Matter in the vicinity of an inspiral imprints secular phase shifts on the waveform; the environmental-effects literature for extreme- and intermediate-mass-ratio inspirals, founded by Barausse, Cardoso & Pani (2014), quantifies how accretion disks, dark-matter spikes, and other mass distributions dephase an inspiral over an observation. An engineered swarm is a mass distribution like any other. If a stellar-mass compact object is ever caught inspiraling into the ω Cen hole, the waveform is a scale: it weighs whatever hardware shares the inspiral's neighborhood, independent of that hardware's luminosity, temperature, or duty cycle.

The channel's reach must be stated carefully, because it is short. The secular dephasing from an exterior axisymmetric mass falls as (r_orb/r_torus)³ per orbit, so a debris torus or swarm at 10²–10³ AU accumulates negligible phase against an inspiral at ~10⁻³–10⁻² AU; we make no claim of debris-torus reach, and the channel does not repair the relic-searchability gap of Section 3.3. What it constrains is mass at or inside the inspiral track, r ≲ 10⁻² AU, and that limitation is well matched to the hypothesis: Paper A's thermodynamic gradient concentrates hardware at exactly those depths, and the envelope of Section 2 permits it there. A LISA-band inspiral through an occupied deep envelope would traverse the swarm itself. An order-of-magnitude scope for the fiducial system: an inspiral accumulates ~10⁵–10⁶ radians of orbital phase over a multi-year observation, and matched filtering resolves phase drifts of order unity, so fractional perturbations at the 10⁻⁶–10⁻⁵ level in the enclosed mass along the track are in principle measurable, a sensitivity to swarm masses far below any other channel's floor. Turning that scope into a forecast requires real waveform modeling, including the degeneracies with the astrophysical environmental effects already cataloged in the literature; we defer that calculation, and any figure it would support, rather than publish an unmodeled curve.

Two properties carry into the adjudication. The channel is two-sided, like R: a measured vacuum inspiral, dephasing consistent with zero environmental mass, counts against engineered occupation at the depths the hypothesis most values, and the framework of Section 4 scores it accordingly. And it inherits the spin channel's schedulability problem: the generic per-IMBH IMRI rate is ~10⁻⁹–10⁻⁶ yr⁻¹ (Mandel et al. 2008; Amaro-Seoane et al. 2018; Fragione et al. 2018), with Arca Sedda, Amaro-Seoane & Chen (2021) the current superseding compilation for dense-cluster rates (LISA detection 0.02–60 yr⁻¹, in-cluster IMRI formation probability 5–50 per cent rising with IMBH mass), and ω Cen's own rate is narrower still, (4–8)×10⁻⁸ yr⁻¹ (Section 5), so the channel is population-level, a constraint LISA accumulates across the Galactic IMBH census rather than a scheduled ω Cen observation. It enters the information forecast of Section 5 as the fourth channel on those terms.

3.6 Predicted-signature table

Table 1 summarizes what each channel looks like under the engineered hypothesis and the two astrophysical nulls defined in Section 4.1.

The table exposes the central inferential difficulty, stated plainly in Paper A and now quantified: silence is predicted by everything. The engineered and quiescent-IMBH hypotheses are near-degenerate in every electromagnetic channel while the system is dormant. The channels that separate them are spin (engineered selection favors a_★ ≳ 0.9; the growth history modeled for ω Cen favors a_★ ~ 0.05–0.10 under isotropic capture, Section 3.3), the R statistic (active only during accretion episodes above the Section 3.2 threshold), EMRI dephasing (active only if an inspiral occurs; Section 3.5), and the waste-heat floor (which separates them only near the sensitivity ceiling). This degeneracy structure, rather than any single measurement, is what the adjudication framework must manage. Two channels proposed elsewhere in the technosignature literature, gamma rays from harvested Hawking radiation (Crane & Westmoreland (2009) ask whether starships or power sources can be built on the Hawking radiation of an artificial black hole; their hole is of million-tonne scale and radiates in gamma rays. The channel is real for manufactured small holes and empty for this paper's target: a 2×10⁴ M_☉ hole has a Hawking temperature of order 10⁻¹² K, about twelve orders of magnitude below the microwave background, so no gamma-ray residue from Hawking output is available to observe.) and gravitational waves from a superradiant Kerr transmitter (Jackson & Benford (2020) propose gravitational-wave beacons, one architecture being a superradiant Kerr power station of the black-hole-bomb type driving a transmitter built from small orbiting black holes, with line frequencies from 100 Hz to the GHz band. Superradiant extraction needs only a spinning Kerr hole, so an IMBH is in principle the right object; the published transmitter architecture is not, being assembled from manufactured small holes, and the channel is directed transmission rather than residue, so it enters the framework of Section 4.1 through the extensible menu rather than as a forward-modeled row of Table 1.), fall outside the table for reasons of scale and of architecture, respectively.

4. The adjudication framework

4.1 Hypothesis space

We adjudicate among three hypotheses about the ω Cen center:

The menu is explicit and extensible; adding hypotheses (an exotic compact object, instrumental systematics on the fast stars, the abandoned-system variant of Section 3.3) changes bookkeeping, never structure. The abandoned-system variant is degenerate with the quiescent-IMBH null on every current channel (Section 3.3) and is therefore not separately scored until a spin measurement exists. We follow the convention that H_eng is reported only as odds against the best-performing null, never against a strawman.

4.2 Per-messenger Bayes factors

For each observational channel i with data d_i, the Bayes factor between hypotheses H_a, H_b is

with channel likelihoods built from the forward models of Section 3 and, following radio-SETI practice, an explicit interference/systematics component mixed into every likelihood so that "instrumental artifact" is priced inside each channel rather than adjudicated informally afterward. Priors on H_eng parameters are pre-registered (Appendix, §B) and varied over stated ranges in sensitivity analysis; the framework's outputs are always reported as (Bayes factor, prior-sensitivity band) pairs, never as bare numbers.

One evaluation rule governs the interface with Paper C's campaign, and it matters more than it looks. Paper C promotes a candidate to adjudication when a pre-registered frequentist trigger fires, and that promotion is data-dependent. The likelihoods above are therefore always evaluated on the complete campaign record, every epoch and every channel including the ones that recorded nothing, never on the promoted subset alone. The trigger layer allocates follow-up effort and carries no inferential weight. Scoring the selected data as though it were the whole dataset would require the conditional form K | trigger = K × P(trigger | H_b)/P(trigger | H_a), and at Paper C's global threshold of p < 5×10⁻⁷ that correction is ~−14 nats against H_eng: large enough to move anything from the candidate band of Section 4.4 to null-favored. The worked example of Section 5 follows the full-record rule, scoring six channels of which five are nulls.

One-sided channels have a capacity. Where a channel's data are a non-detection and the null predicts that non-detection with probability one, as the mid-infrared channel does today, the Bayes factor is bounded below by the dormancy prior: ln K_i = ln[f_d + (1−f_d)P(quiet | active)] ≥ ln f_d, with equality in the limit of infinite sensitivity. At the fiducial f_d = 0.5 the whole channel is worth 0.69 nats however deep the photometry goes, and it can never produce evidence in favour of H_eng. We report each such channel's capacity and spent fraction alongside its value, because the quantity a program should be ranked on is the headroom, not the potential.

4.3 Cross-messenger combination

Channels are combined multiplicatively where independent, K_tot = ∏_i K_i, with three corrections. Two are imported from gravitational-wave counterpart methodology. First, a coincidence term: hypotheses that predict correlated anomalies across channels (as H_eng does for MIR excess and low R during an accretion episode) earn a likelihood contribution from the observed coincidence structure itself, computed from the joint forward model rather than the product of marginals. Second, a data-side independence audit: channels sharing calibrators, atmospheric paths, or reduction pipelines are grouped and their shared systematics marginalized jointly before multiplication.

The third is hypothesis-side and is easy to miss, because it is a property of the hypothesis rather than of the instruments. A dormant installation is quiet in every electromagnetic channel at once, so the dormancy fraction f_d is a single nuisance parameter shared across channels, and the product rule marginalizes it once per channel instead of once in total. Any parameter shared this way must be marginalized jointly:

The difference is not cosmetic. With n non-detection channels each returning P(quiet | active) = 0.5 and f_d = 0.5, the joint form gives ln K = −0.47, −0.63, −0.68 at n = 2, 4, 6, saturating at ln f_d = −0.69 as the capacity argument of Section 4.2 requires, while the product of marginals gives −0.58, −1.15, −1.73 and diverges. The product rule would let a menu grow evidence against H_eng without bound by adding degenerate silent channels, which is the mirror image of the mis-scoring the Discussion warns about for omitted hypotheses. The correction is inactive in the worked example below, where the second live channel carries no dormancy parameter, and becomes necessary on the day a second one activates.

All three corrections are conservative in effect: the coincidence term rewards only pre-registered correlation patterns, the data-side audit only ever weakens evidence, and the joint marginalization only ever moves |ln K| down.

4.4 Decision thresholds

We adopt a logarithmic odds scale with pre-registered action bands, stated here for ln K of H_eng over the best null:

These bands are on the Bayes factor, not on posterior odds, and the distinction carries a consequence we state rather than bury. Posterior odds are prior odds times K, and this paper sets no prior on H_eng: the epistemic-status box of Section 1 makes everything conditional on the optimization premise, and that premise is the quantity a prior would have to price. The bands are therefore resource-allocation thresholds, specifying what the campaign does next, and they carry no belief content on their own. A reader who wants a belief statement must supply prior odds; at a plausible 10⁻⁶ per target, ln K = 5 still leaves posterior odds near 1.5×10⁻⁴. The same gap governs class-level use: applied across the candidate hosts of Paper A's Table 1, prior odds must scale as 1/N or the "candidate" band will be crossed by chance somewhere in the sample. Bayes factors supply that multiplicity protection automatically once the prior odds are stated, which is a further reason to state them.

Two design points. The "candidate" band requires multi-messenger support by construction, honoring Paper C's two-messenger rule; no single channel, at any significance, can reach it, because single-channel anomalies are where every historical false alarm has lived. And the kill conditions of Paper A map onto the same scale in the other direction: the pre-registered falsifiers (e.g., LISA measurement of low spin; resolution of the mass tension in favor of H_sub) enter as ordinary likelihood terms and drive ln K negative, so confirmation and falsification run through one pipeline rather than two standards. The framework is deliberately instrument-agnostic: nothing in Sections 4.2–4.4 references ω Cen, and the machinery applies unchanged to any technosignature target with a defined null menu.

5. Worked example: Omega Centauri today

We now run the current ω Cen data through the framework. Inputs: the fast-star kinematics, the N-body modeling, the MSP timing bound (with the TRAPUM 2026 mass limit), the deep radio silence, and the JWST infrared limits. The computation implements the Appendix-C model (deliberately minimal: radius-marginalized hard-threshold MIR likelihood, log-flat priors, 2×10⁶ prior draws); code in paper/figs/fig3_lnk.py, results in Figure 4.

• Quiescent IMBH vs. remnant subcluster: the interesting contest, and the framework's finding is that it is currently a stand-off dominated by the kinematics-versus-timing tension: the fast stars pull toward a point mass, the pulsars toward an extended one, and the combined |ln K| is ≲1 with sign depending on how the two datasets' systematics are weighted. As an independent check, the imbh-constraints aggregation engine of the series' public toolchain, run on its nine-constraint curated compilation (Table 2, Appendix D), returns an empty jointly-allowed mass window (formal tension verdict: the kinematic floor exceeds the tightest model-dependent ceiling), confirming that no point-mass value satisfies all published constraints at face value. The framework adds discipline, and no verdict, to a tension the community already knows; per series policy the tension is recorded, never collapsed.
• Engineered vs. best null: ln K sits in the null-favored band, as it should. The only active channel today is waste heat, and, with the continuous per-filter likelihood of Appendix C marginalized over swarm radius and temperature split against the real JWST/MIRI sensitivity curve, it yields ln K = −0.47 at fiducial priors (prior-sensitivity sweep −1.138, −0.474 and −0.079 at f_d = 0.1, 0.5 and 0.9): the electromagnetic silence sometimes informally cited as consistent with engineering is, in the accounting, mildly against engineered occupation, because the forecast ETC depths exclude 76 per cent of the active-installation prior volume, while the radio, R-statistic, spin, and kinematic channels are inactive or degenerate and contribute nothing. The mass channel is the one exception and is priced below. The radius prior matters: narrowing its upper bound from the fiducial 4×10³ to 10³ AU, excluding the coolest outer configurations, raises detectability to ln K = −0.52, confirming that cool, extended swarms are what preserve the engineered hypothesis's viability. The leak prior's lower bound is an engineering estimate (Section 3.1, Appendix A.3): repeating the computation with the prior extended down to 1 − f_sink = 10⁻⁶ gives ln K = −0.37 instead of −0.47, so a softer transport bound softens the conclusion without reversing it. A second sensitivity is the fuel ceiling: the fiducial prior extends P_comp to the unsuppressed 7.6×10⁵ L_☉, and truncating it at the ADIOS-suppressed natural ceiling of Section 2.7 (~2.6×10³ L_☉) removes much of the detectable prior volume, moving the mid-infrared ln K to −0.37 (Appendix C). Rather than pick a corner, we also run the ceiling as a mixture, drawing it per sample from the spin fork (a_★ = 0.998 on the thin-disc branch, a_★ ~ U(0.035, 0.07) on the MAD branch, the equilibrium band of Section 3.4, with η ∝ a_★²) and from the suppression state (unsuppressed, or suppressed by a factor log-uniform over 10²–500), at equal weights on each. That gives ξ = 0.608 and ln K = −0.363. The corner weights recover the endpoints exactly: all thin-disc and unsuppressed returns the fiducial −0.474, all MAD and suppressed returns −0.244. Conditioning the ceiling on spin rather than fixing it at the engineered corner is therefore worth about +0.11 nats to the engineered hypothesis, and the fiducial number is the least favorable of the three to it. A third, and now the largest of the three, is the lower edge of the same prior: P_comp is drawn from 1 L_☉ upward, a boundary that coincides numerically with the warm instrument limit and has no physical warrant, and extending it to 10⁻³ L_☉ moves ln K to −0.32. Since ln K here is a ratio of prior volumes, every boundary is a lever and all are now tabulated in Appendix C rather than only the two varied at first (Appendix G). Two things dominate the total, and both are structural rather than observational: the present evidence budget is set by the dormancy prior (the data enter only through the excluded active-prior fraction ξ = 1 − P(quiet | active) = 0.755, which is the part the data determine), and the channel is capped at ln f_d = −0.69 in total, of which 68 per cent is spent. The framework's present value is the forecast, not the number. The surviving configurations are dormant or low-power, deep-envelope systems (Figure 2), and they are also post-spin-up. Paper A's Phase 4 drives the hole toward high spin by sustained near-Eddington feeding, which at the fiducial mass is 6.6×10⁸ L_☉: four to five orders of magnitude above the concealment ceiling of Section 3.1, and not concealable by any architecture in the envelope. Everything in this paragraph and the next is a thin-disc-branch statement in the sense of Section 3.4, and on the MAD branch the growth factor is 1.010–1.020 instead of 2.20 and the consequences below largely vanish. The episode's duration is set by the mass it must deliver, not a round number: reaching Thorne's radiation-limited a_★ = 0.998 from a_★ ≈ 0 requires M_f/M_i ≈ 2.20 (Bardeen 1970; Thorne 1974). (The 2.20 figure, 2.2024 to five significant figures, is our own numerical integration of the standard prograde-ISCO accretion spin-up relation d(ln M)/da_★ = 1/[L_ISCO(a_★)/E_ISCO(a_★) − 2a_★] from a_★ = 0 to Thorne's a_★ = 0.998 equilibrium, using the standard Bardeen, Press & Teukolsky (1972) ISCO energy and angular-momentum formulas; it asymptotes to √6 ≈ 2.449 as a_★ → 1. Bardeen 1970 and Thorne 1974 are cited jointly as the standard attribution for, respectively, the accretion spin-up mechanism and the 0.998 radiative-equilibrium limit, not as the source of a closed-form 2.20.) At the Salpeter e-folding time t_S ≈ 4.5×10⁷ yr for η = 0.1 accretion, that mass gain takes ln(2.20) t_S ≈ 3.5×10⁷ yr. The engineered hypothesis at ω Cen therefore requires that the spin-up episode has already ended, which is a statable constraint rather than a loose reading of "dormant", and it carries a second observable, secular core depletion, which the Discussion quantifies and finds weaker than earlier drafts implied. It also carries a mass prediction the spin-up requirement makes on its own: the ≈2.20× growth factor applies on top of whatever mass the merger history alone delivers, so the engineered hypothesis at ω Cen predicts M ≈ 2.20 × M_merger ≈ 1.1×10⁵ M_☉ against the merger-only M(gas-starved null) ≈ 5×10⁴ M_☉. The channel is priced here against the TRAPUM 2026 ceiling of 10⁵ M_☉ (Colom i Bernadich et al. 2026) (figs/fE_mass_channel.py), and the result retires the earlier claim that it was already observable (Appendix G). The two hypotheses inherit the same growth history, so the same merger-mass prior has to describe it under each; using this section's own log-uniform [10⁴, 5×10⁴] M_☉ merger-mass prior, the late-spin-up branch runs over [2.2×10⁴, 1.1×10⁵] M_☉ and only its top 6.0 per cent clears the TRAPUM ceiling. Reading TRAPUM's 90 per cent limit as a likelihood ratio across the boundary and weighting the late branch at w = 0.5, the channel returns ln K_mass = −0.03 (band −0.005 to −0.05 over w = 0.1–0.9). The mass channel is therefore a live discriminant in principle and a negligible one on 2026 data; reading the merger endpoint as a point value rather than a range instead gives ln K_mass = −0.59, and the whole weight of the channel sits in the difference between those two treatments of one prior. Per series policy we do not collapse the mass tension in either direction, and the kinematic mass measurement is deliberately not scored here, since doing so would arbitrate the two astrophysical nulls as a side effect; only the timing upper limit enters.
• Information forecast: channels are ranked by expected |Δln K| per unit cost against their remaining headroom, not their nominal potential, since the mid-infrared channel has 0.22 nats left in total (Section 4.2) and further depth cannot buy more. The stated metric is an expectation, conditional value times probability of arrival, and the spin channel's two levels of that expectation diverge sharply enough that they need separate rankings rather than one merged list. Target-level, ω Cen itself: the ω Cen-specific IMBH–compact-object capture rate from the González Prieto et al. (2025) models is (4–8)×10⁻⁸ yr⁻¹, giving ~3×10⁻⁷ probability of ω Cen yielding its own inspiral over a four-year mission. Even against the large conditional value of a decisive spin measurement (a null at a_★ = 0.049 ± 0.021, or 0.084 at the fiducial mass, makes an engineered a_★ ≳ 0.9 a many-σ excursion), the expected |Δln K| this channel delivers for ω Cen itself is ~10⁻⁶ nats, negligible against the other target-level channels. The target-level leaders are therefore (1) resolution of the mass tension by continued MSP timing plus Gaia DR4 astrometry, which arbitrates the two astrophysical nulls and thereby moves the engineered hypothesis's denominator; (2) any detection of accretion at any level, which activates the R statistic, the only cheap channel with large discriminating power in both directions; (3) EMRI/IMRI environmental dephasing (Section 3.5), the only channel constraining engineered mass, which moves evidence in both directions since a vacuum inspiral counts against engineered occupation at the depths the hypothesis most values; spin itself ranks last at the target level, its 3×10⁻⁷ arrival probability attached rather than left implicit. Class-level, the LISA IMBH population: here spin leads, because the constraint accumulates across every ω Cen-like system LISA catches rather than waiting on this one target, with arrival probability ≈1 for the population even though it is ~10⁻⁷ for any single target; the class-level version must carry the spin mixture, since comparable-mass growth histories elsewhere populate the χ ≃ 0.7 attractor, and it inherits the isotropy caveat of Section 3.3 (Appendix B). The relic analysis of Section 3.3 raises the stakes on this channel: spin is the one fossil that survives abandonment, so the LISA-era population measurement adjudicates the engineered-history hypotheses for the whole Galactic IMBH population at once, and the per-target frameworks of this paper compose naturally into that population-level test. Deeper radio silence, by contrast, now moves ln K weakly for every pairing: for this hypothesis menu the existing limits are already deep enough that further depth buys little discrimination.

6. Discussion

6.1 Limitations

The feasibility envelope depends on cluster-center parameters that carry real uncertainty (mass and number fraction of the remnant tail; the bound-cusp population inside the influence radius) and on a material-strength scale taken from present technology; both are varied in the Appendix. The largest lever is the remnant tail, and within it the perturber mass rather than the number fraction: for the unbound comparison bracket, the 10→31 M_☉ correction is worth a factor 8.0 in ⟨m_★²⟩, against a factor ~10 across the whole 0.1–3 per cent fraction bracket; for the bound channel that now sets the operative floor, jointly anchoring f_BH and m_BH on the extended dark mass narrows the equivalent bracket to a factor 3.1 in kick variance (Appendix A). The cusp existence question is closed by the González Prieto et al. (2025) realizations, and the modeling gap it left behind, that the Monte Carlo modeled only the decoupled unbound flyby channel while the channel that survives, perturbation by bound cusp members, was treated analytically, is now closed: the bound-member diffusion channel and its secular cadence are both Monte Carlo outputs (Section 2.4, Appendix A). What remains a modeling limitation rather than a parameter is the quasi-static cusp assumption, perturber positions are drawn once per history although the cusp itself relaxes on the same 0.03–0.6 Gyr segregation timescale that builds it, which biases toward longer lifetimes than a resampled population would give; the remaining parameters move the boundaries by factors of a few. The MAD-regulation channel inherits the systematic uncertainties of the GRMHD baseline literature, which is simulation-calibrated rather than observationally calibrated at IMBH masses. The adjudication framework shares the standard vulnerability of Bayesian model selection to unmodeled hypotheses: it ranks the menu it is given, and a surprise outside the menu (an astrophysical mechanism not yet imagined) would be mis-scored until added. The mitigation is procedural rather than mathematical: the menu is public, extensible, and versioned. That procedural claim is no longer unvalidated: Appendix F scores the machinery itself, not the ω Cen models, against three historical cases with an independently known resolution, including a positive control (the 1967 pulsar discovery) that the machinery registers at ln K = +7.9 before correctly collapsing to −0.5 once the rotating-neutron-star hypothesis enters the menu. Two mundane cases (Project Hephaistos, Boyajian's star) are checked for false positives and pass; one data-driven collapse test falls short of its pass threshold as constructed, reported rather than tuned away, and detailed in Appendix F.

Several inputs to this paper come from companion papers, which are preprints and not yet refereed: the in-cluster fuel census and Phase-4 spin-up episode from Paper A, the 3× crowding factor and the p < 5×10⁻⁷ trigger from Paper C, and the central density and 4×10⁴ M_☉ hole used in the giant-count sensitivity from Paper F. The numbers that depend on them are provisional, and a reader who declines the companions should treat those numbers as conditional.

Two numbers used above deserve explicit qualification at the claim level. The transport floor. The geometric content of the bound 1 − f_sink ≳ 10⁻⁴ is now a calculation (capture cone, beaming gain, étendue, and the plasma-cutoff carrier condition; Appendix A.3); what remains estimated is the re-interception term, the covering-fraction floor on power re-thermalized within the swarm, which a full radiative-transfer treatment of the swarm interior would replace. Its influence on the adjudication is bounded and signed: extending the leak prior down to 10⁻⁶ moves ln K from −0.474 to −0.371, well inside the dormancy-prior band of [−1.138, −0.079], so no conclusion of Section 5 turns on the exact value.

Core depletion is a weaker signature than the series has been claiming. Paper A and Paper C both list secular core depletion as a Phase-4 observable, and neither had quantified it. Doing so changes its standing, on the thin-disc branch of Section 3.4; on the MAD branch the delivered mass is ~2×10²–4×10² M_☉ and there is no signature to discuss. Spinning the fiducial hole up to Thorne's radiation-limited a_★ = 0.998 (Section 5; M_f/M_i ≈ 2.20) requires a mass gain of ΔM = 1.20 × 2×10⁴ = 2.4×10⁴ M_☉, and the delivered stellar mass is larger still, (2.7–3.4)×10⁴ M_☉ once 10–30 per cent radiated and jetted loss is counted. The enclosed-mass model has to be stated, because no single one covers the range: inside the core radius we use the adopted flat core (ρ₀ = 3×10³ M_☉ pc⁻³ inside r_c = 3.6 pc), and beyond it a Plummer sphere of the same r_c carrying the total cluster mass 3.55×10⁶ M_☉, since the flat core extrapolated to 10 pc is not a model of anything. On the flat core the delivered mass is 191 per cent of the mass now inside 1 pc, 24 per cent inside 2 pc, and 4.1 per cent inside the core radius; on the Plummer profile it is 0.91 per cent inside 10 pc. Earlier drafts quoted 1.0 per cent at 10 pc (Appendix G) without saying that the number came from a different model than the three beside it; the flat core extrapolated there would give 0.19 per cent. The fractional deficit is therefore set entirely by the delivery radius, and Paper A's own fuel census puts it at ~10 pc, where a sub-per-cent deficit sits below what a cluster surface-density profile can isolate from mass-function and distance systematics. Only fuel drawn from within ~2 pc would leave a deficit large enough to see, and that reservoir does not hold enough stars to supply the phase.

One further point closes the channel while a second reopens it as a discriminant rather than a dead end. The removed stellar mass is not lost from the cluster: it becomes central point mass, so the total enclosed-mass profile outside the feeding region is preserved and only the luminous-to-dark split changes, which is the quantity the mass-tension programme of Section 5 already measures. The endpoint is not degenerate with ordinary growth: 2×10⁴ × 2.2 ≈ 4.4×10⁴ M_☉ is below the González Prieto et al. (2025) merger-only mass of ~5×10⁴ M_☉, so a spin-up episode starting from this fiducial hole does not reproduce the mass the models reach without one, and the final mass carries information about which history produced it. Applied instead to the merger-only endpoint itself as the pre-spin-up mass, the same ≈2.20× factor gives the M ≈ 1.1×10⁵ M_☉ prediction of Section 5; the two calculations bracket the endpoint mass depending on when in the growth history the spin-up episode runs. The late-spin-up branch reaches M ≈ 1.1×10⁵ M_☉ only if the merger endpoint is read as a point value; carried over the same [10⁴, 5×10⁴] M_☉ prior both hypotheses share, the branch spans [2.2×10⁴, 1.1×10⁵] M_☉ and only its top 6 per cent clears the TRAPUM 2026 ceiling. The fiducial branch, 4.4×10⁴ M_☉, sits below the ceiling throughout. Section 5 scores the resulting channel at ln K_mass = −0.027: a live discriminant, and on 2026 data a very weak one. What remains of the depletion signature is a lookback limit worth stating on its own: a region depleted at the ~10 pc delivery radius refills on the half-mass relaxation time, ~12 Gyr (log t_rh = 10.09; Harris 1996), and the episode itself lasts only a small fraction of that (3.5×10⁷ yr, under one per cent; Section 5), so the signature persists for a time of order the cluster age after the episode ends. The shorter ~4 Gyr core clock (log t_rc = 9.60) applies only to a deficit inside ~2 pc. Combined with the post-spin-up requirement of Section 5, the observable and the requirement are in tension only for fuel drawn from that inner region, where a system old enough to be quiet is old enough to have erased its own depletion.

The drainage timescale. The ~10³ yr figure of Section 3.3 is an estimate. The asymmetry it rests on is robust, since relic dust persists around a star while debris around a hole is fed toward the loss cone and removed, but the rate is not. Closing that gap requires a loss-cone refill calculation for the debris population, tracking collisional periapsis diffusion and the coupling of the finest grindings to the ambient flow; we state the gap rather than supply a number that calculation has not yet produced.

6.2 Relation to the series and beyond

Within the series, this paper closes the constructive quadrant: Paper A argued the destination is attractive, and Section 2 finds the destination is habitable by infrastructure, with the thermodynamically preferred region also the dynamically safest. It also arms Paper C's campaign with the scoring machinery its two-messenger policy presupposed, and returns to Paper D a refined survival input (p_s now decomposable into transit and residence terms, the latter bounded here). One dependency flag on that export: the residence-survival figure Paper D imports (≈0.8 over 10⁸ yr) was derived from the impulsive diffusion floor. It is now anchored on the bound-remnant channel of Appendix A, the operative one because unbound flybys never bind: 75 per cent of histories survive to 10⁸ yr at the envelope edge and 97 per cent at 400 AU, so the ≈0.8 figure is bracketed by 0.75 at the edge and 0.97 at 400 AU. Paper D should cite it against that channel. The adiabatically corrected unbound-flyby curve, whose envelope minimum is 7.6×10⁸ yr, describes the non-operative channel, and the impulsive floor remains the conservative alternative. Beyond the series, the two exportable products are the envelope method, applicable to any proposed megastructure environment with a stated perturber population, and the adjudication framework, applicable to any technosignature program willing to pre-register its nulls.

6.3 Conclusion

The engineered-IMBH hypothesis survives its first engineering audit and its first quantitative adjudication, in both cases by narrowing: infrastructure persists only within ~4×10³ AU of the hole, and the surviving parameter space after the forecast ETC depths is dormant, low-power, and deep. Every forthcoming measurement listed in Section 5 shrinks it further or breaks it open. Either outcome is progress that a hypothesis without an envelope, a residue model, and a scoring rule could not deliver.

7. Appendices: Monte Carlo, spin channel, priors, the nine-constraint compilation, reproducibility, E6 validation, revision record

A. Flyby Monte Carlo: method and parameters

Method (implemented in paper/figs/fig1_envelope.py; fixed seed 20260717, offset per configuration so that no curve depends on the order in which the configurations are run). For each of 40 logarithmic radius bins a ∈ [6 r_g, 4×10³] AU (the grid begins at the Schwarzschild ISCO): (1) 2×10⁵ encounters are drawn with impact parameter from the gravitationally focused cumulative distribution Q(b) ∝ v_rel²b² + 2GMb truncated at b_max = 30a (contributions beyond carry <10⁻³ of the kick variance), relative speed drawn as v = 0.3σ + x with x a Rayleigh variate of scale σ = 21 km s⁻¹, i.e., a Rayleigh distribution shifted upward by 0.3σ (mean 1.25σ + 0.3σ = 1.55σ; the JSON output of Appendix E records the exact parameterization), below the single-population Maxwellian mean of a star's speed relative to an effectively stationary hole, √(8/π)σ ≈ 1.60σ; since slower encounters deliver larger kicks, the choice is conservative for swarm survival, and mass from the perturber mass function. The baseline function has two stellar components (0.35 M_☉ with weight 0.7; 0.6 M_☉ white-dwarf tail with weight 0.3); the fiducial function adds segregated remnants, 1.4 M_☉ neutron stars at number fraction 0.02 and 31 M_☉ black holes (Section 2.4) at number fraction f_BH ∈ {0.001, 0.01, 0.03} with the stellar weights renormalized. The total number density is set per branch by the observed core mass density, n_★ = ρ₀/⟨m(f_BH)⟩, rather than held at a fixed 10⁴ pc⁻³: adding heavy remnants at fixed number would raise the implied mass density well above the adopted 3×10³ M_☉ pc⁻³, since the 10⁴ pc⁻³ figure itself assumed ⟨m⟩ = 0.3 M_☉, not the mean of the function this Monte Carlo draws from. Renormalizing lowers Γ ∝ n_★ by 1.42, 1.58, 2.50, and 4.54 for the baseline and the three f_BH fractions, lengthening every lifetime by the same factors; the fixed-number convention is retained as a diagnostic switch, and a 10 M_☉ configuration is retained in the output so that the change of perturber mass is auditable. (2) Each encounter contributes an impulsive eccentricity kick δ = 2(m_★/M) min[1,(a/b)²](v_orb/v), i.e., δ = δv_tid/(2v_orb) with the O(1) geometric coefficient set to unity; that choice sits on the optimistic side by a factor of ~2 in the δ² timescale, offset by the conservative velocity sampling above and by the neglect of adiabatic suppression, so the net sits within the stated factor-few accounting. The baseline curves apply no adiabatic suppression; a parallel run applies the Gnedin & Ostriker (1999) kernel A(x) = (1+x²)^−5/2 to δ² per encounter, with x = (b/a)(v_orb/v), and is the physical estimate (Section 2.4). We use the power-law kernel rather than the exp(−x²) form of Spitzer (1987) because the latter over-suppresses relative to N-body results; the power law is the conservative choice. (3) Per history (2×10⁴ per bin), survival time is the random-walk first-passage to e = 0.5: encounters to threshold n_★ = e²/⟨δ²⟩ with CLT scatter, divided by the total encounter rate Γ(<b_max), capped at 12 Gyr. The bookkeeping is thus: the 2×10⁵ encounter draws per bin characterize the per-encounter kick-variance statistics; each of the 2×10⁴ histories then draws its encounter count analytically from that first-passage distribution rather than re-simulating individual encounters, so a history at the median spans ~n_★ ~ 10⁴–10⁶ encounters depending on bin. Single-passage disruption is impossible over the part of the grid where the impulsive kernel is meaningful: at a ≳ 1 AU the saturated kick is δ ≲ 5×10⁻³ for stellar perturbers and ≲ 0.4 for the rare 31 M_☉ black-hole perturbers (0.13 for the retired 10 M_☉ run). Inside ~0.2 AU the saturated form returns δ > 1 for a heavy penetrating passage (δ = 4.0 at 10⁻² AU; 1.3 for 10 M_☉), and the first-passage step count n_★ = e²/⟨δ²⟩ falls below 10²; neither the small-kick premise nor the random-walk premise holds there, which is a further reason the impulsive curves are a floor rather than an estimate in the deep envelope, and is why the adiabatic run is carried alongside them. Direct star–element scattering requires approach within ~2Gm_★/(v_relv_orb) and contributes negligibly at any realistic covering fraction. This unbound-field calculation uses a core-average perturber density (10⁴ pc⁻³) with gravitational focusing capturing only the unbound flux; it is the curve retained in Figure 1(a) as the measure of that channel's adiabatic decoupling, not as an operative floor.

Bound-cusp channel (panel b). Implemented in the same script; species density profiles and joint anchoring in common.py. The relaxed Bahcall–Wolf cusp that the González Prieto et al. (2025) realizations form raises the local density of bound perturbers, four species (main-sequence and white-dwarf stars at γ_★ = 1.3; neutron stars, 1.4 M_☉, interpolated γ_NS = 1.5; black holes, γ_BH = 2.0), each following N_s(<r) = [4πn_s,0r_infl³/(3−γ_s)](r/r_infl)^3−γ_s normalized against the core-average density n_s,0 = f_sN_★,core at r_infl. The black-hole number fraction is anchored jointly rather than assembled from separate sources: f_BH,global = (M_dark/m_BH)/N_★,cluster with N_★,cluster = 7×10⁶, and f_BH,central = 10×f_BH,global with the ×10 segregation enhancement adopted as a stipulated normalization, not derived here. Varying the enhancement over 3–30 moves the joint-anchored central fraction over 0.35–3.5 per cent and the envelope-edge presence probability over 20–90 per cent, with the conditional diffusion floor spanning 1×10¹⁰ to 2×10⁷ yr; the stipulated factor sets where in that range the headlines sit, and the qualitative conclusion (bound remnants set the operative clock; flybys never bind) holds at every setting. The González Prieto et al. (2025) models publish cusp slopes for the black-hole population (γ_BH ≃ 2.0) but no central number fraction, so the factor remains a stipulation anchored on the r_infl matching convention rather than a derived quantity; the slopes imply the contrast grows inward as (r/r_infl)^−0.7, a factor 5 at the stripping radius. At the fiducial M_dark = 2.5×10⁵ M_☉, m_BH = 31 M_☉: f_BH,central = 1.15 per cent. Because kick variance scales as n_BH⟨m_BH²⟩ ∝ M_darkm_BH, a lighter perturber is linearly, not quadratically, less damaging at fixed dark mass; the bracket {M_dark = 2.0×10⁵ M_☉, m_BH = 15 M_☉} to {3.0×10⁵ M_☉, 31 M_☉} spans a factor 3.1 in kick variance, against the factor 30 the unbound comparison bracket spans. Since γ_BH = 2 gives a shell count linear in radius, the number of bound black holes inside the stripping radius has expectation ≲1 everywhere in the envelope; each history draws its perturber count per species as an independent Poisson variate from the expected shell count λ_s(a) = N_s(<√2a) − N_s(<a/√2) rather than using a smooth rate, which is what makes the resulting lifetime distribution bimodal rather than unimodal. Relative velocities are drawn as Maxwellian with the local Keplerian dispersion s_rel,s = √(2GM/[(1+γ_s)a]) (species-dependent, since x ~ 1 for bound members and σ does not apply); impact parameters are uniform in b² out to b_max = 3a, set by where the adiabatic kernel's contribution to the kick variance drops below 1 per cent; gravitational focusing is dropped (it uses the perturber's own mass for an already-orbiting body and is a ~10⁻³ correction at b = a). The same Gnedin & Ostriker kernel applies, since x ~ 1 rather than ≪ 1 for these perturbers; an un-kerneled run is retained in the JSON (bound_fid_nokernel) as a diagnostic only. Survival is the same random-walk first-passage criterion as the unbound channel, with the total rate summed over the four species' Poisson-drawn counts for that history.

Secular cadence. t_sec(a) = (M/m_pert)(r_p/a)³P_orb(a) for the outermost bound black hole, fixed at r_p = 4×10³ AU (where N_BH(<r) = 1); quenched by relativistic apsidal precession, t_GR(a) = [a/(3r_g)]P_orb(a), and by mass precession from the light cusp population enclosed within a (drawn per history from the same N_s(<a) profile, excluding the perturbing black hole), t_mass(a) = [M/M_enc(<a)]P_orb(a); active only where t_sec < min(t_GR, t_mass) and within the isotropic-cusp resonance window, probability 0.775 for mutual inclination in (39.2°, 140.8°).

Results. For the stars-plus-WDs baseline, median impulsive lifetimes are 4.8×10⁹, 4.7×10⁹, and 3.6×10⁹ yr at a = 10, 10², 10³ AU: flat, by the scale cancellation ⟨δ²⟩ ∝ a⁻¹, Γ ∝ a discussed in Section 2.4. Adding the remnant components lowers the floor in proportion to the added ⟨m_★²⟩: medians at the same radii are (1.6, 1.5, 0.92)×10⁹ yr at f_BH = 0.001, (1.7, 1.6, 1.3)×10⁸ yr at the fiducial f_BH = 0.01, and (9.8, 9.0, 8.8)×10⁷ yr at f_BH = 0.03, with envelope minima of 1.6×10⁸, 6.9×10⁷, and 5.2×10⁷ yr respectively. The perturber mass dominates the fraction bracket: the same fiducial run at the earlier 10 M_☉ assumption gives (9.8, 9.2, 6.2)×10⁸ yr with an envelope minimum of 3.6×10⁸ yr, so the 10→31 M_☉ substitution is worth a factor 8.0 in ⟨m_★²⟩ and displaces the floor further than the whole 0.1–3 per cent bracket does at fixed mass. Varying the mean stellar mass by ×2, the density by ×3, and e_cross over 0.3–0.7 moves the floor by additional factors of a few. The quoted minima are min-of-noisy-bins statistics; the script reports the spread over five seeds directly, giving ±20–40 per cent (for example the f_BH = 0.01 minimum spans 6.9–9.3×10⁷ yr), so no hard floor is claimed at better than a factor of two.

The adiabatically corrected run, which is the physical estimate for the unbound channel, sits far above all of these: at the fiducial mass function its median reaches the 12-Gyr cap for a ≲ 3 AU and returns 1.1×10¹⁰, 9.9×10⁹, and 4.3×10⁹ yr at a = 10, 10², 10³ AU, with an envelope minimum of 7.6×10⁸ yr at the outer edge where the suppression is weakest. The adiabatic parameter at b = a runs from 4.1×10³ at the ISCO to 2.2 at the stripping radius. The operational reading is that the unbound-flyby channel is removed from the problem across the envelope, and that any process that does bind must involve perturbers bound to the hole, which panel (b) now supplies directly rather than as an analytic estimate.

Bound-cusp results. At the envelope edge (a = 4×10³ AU), a bound black hole is drawn in 24.8 per cent of 2×10⁴ histories (p_BH,present; the analytic expectation from 1 − e^−λ_BH at the joint-anchored fiducial). The outcome is bimodal by construction, not merely broad, so the conditional and unconditional statistics are reported separately: conditional on presence the diffusion floor has median 4.3×10⁷ yr, while the unconditional median is the 12-Gyr cap and 72.3 per cent of histories reach it because no black hole (or a distant one) was drawn. Interior to the edge, presence becomes rare and the conditional floor becomes severe: at a = 4×10² AU, p_BH,present = 2.8 per cent with a conditional floor of 1.4×10⁶ yr, and 97 per cent of histories survive to 10⁸ yr against 75 per cent at the edge; this is the trade the analytic single-perturber estimate could only gesture at, rare but severe inward, common but mild at the edge. The perturber-mass bracket of the joint anchoring moves the conditional edge floor from 1.8×10⁸ yr at {2.0×10⁵ M_☉, 15 M_☉} to 4.3×10⁷ yr at fiducial and 4.3×10⁷ yr at {3.0×10⁵ M_☉, 31 M_☉}, the factor-3.1 variance bracket stated above; the un-kerneled diagnostic run drops the conditional edge floor to 5.9×10⁶ yr, confirming the adiabatic suppression matters by a factor of a few for the bound channel too, as expected since x ~ 1 rather than ≪ 1 there. The secular clock reaches t_sec = t_GR ≃ 3.7×10⁷ yr at a ≃ 4.0×10² AU, its GR-quenched inner edge, and t_sec ≃ 1.2×10⁷ yr at a ≃ 9×10² AU against t_GR ≃ 2.6×10⁸ yr there, unquenched; it is drawn in Figure 1(b) only over 4×10²–4×10³ AU and is not a survival curve at any point on that range.

A.3: transport bound. The floor 1 − f_sink ≳ 10⁻⁴ is a calculation in three steps: delivery geometry, carrier physics, and re-interception, with only the last carrying an estimated coefficient.

Delivery. The photon-capture cross-section of the hole is σ = 27π r_g² (critical impact parameter 3√3 r_g), so from a platform at r = 10² r_g the disposal channel subtends Ω_c/4π = 6.75×10⁻⁴: isotropic emission delivers less than a thousandth of its power to the horizon, and f_sink → 1 requires a beaming gain ≳1.5×10³. That gain is optically trivial. The capture cone has half-angle θ_c ≃ 0.052 rad, so diffraction demands only an aperture D ≳ 23λ, and the étendue (concentration) limit 1/sin²θ_c ≈ 3.7×10² puts the optic area at a few hundred radiator areas. Delivery is not the binding term.

Carriers. The minimum-energy vacuum carrier, wavelength ~r_g, energy hc/r_g ≃ 7×10⁻³³ J, is a ~10 Hz wave and cannot propagate: it lies below the plasma frequency of any realistic environment (Paper A, this revision). Inside a fed envelope with n_e ~ 10⁴–10⁸ cm⁻³, ν_p ≈ 1–90 MHz, so the minimum propagating carrier energy runs 6×10⁻²⁸ to 6×10⁻²⁶ J, five to seven decades above the vacuum figure. The delivered per-bit disposal cost is therefore set by the plasma cutoff together with multi-bit-per-photon coding at the channel capacity derived in the appendix of Paper A v2.12, which assumes carrier modes kept colder than the 2.7 K sky, and matter carriers, cold mass dropped down the capture cone, are a fallback that evades the cutoff entirely at a higher cost per bit. None of this changes the geometric floor below; it fixes the cost per bit, not the intercepted fraction.

Re-interception. Beamed flux traverses the swarm on its way to the cone and is intercepted with probability of order the swarm covering fraction seen from a typical element, ≳10⁻⁴ for the covering fractions that make the swarm computationally worthwhile; the intercepted power is re-thermalized and radiated at swarm temperature. This is the binding term, and its coefficient is the one number a full radiative-transfer treatment of the swarm interior would refine (Section 6.1); the sensitivity analysis of Appendix C shows the adjudication does not turn on it.

B. Spin channel: random-walk derivation and net-alignment sensitivity

Derivation. For a capture of mass μ ≪ M onto a hole of spin parameter a_★ = Jc/(GM²), the change in a_★ per event is δa_★ = (μ/M)[L̃ cos ι − 2a_★Ẽ], with L̃ the specific angular momentum deposited (units GM/c) and ι the inclination of the captured orbit to the existing spin axis. Under isotropic arrival, ⟨cos ι⟩ = 0, the mean drift comes only from the mass term and a_★ is set by the vector random walk of J against a denominator M² that grows with each capture. At a_★ ≪ 1 the walk is free of the Hughes–Blandford damping term, which does not act until a_★ approaches its ~0.3 equilibrium, not reached here. Each capture deposits |δJ| = μL̃(GM_i/c) in a random direction, with M_i = M₀ + iμ; summing in quadrature over N = (M_f − M₀)/μ events,

so that

which for a quasi-circular capture from the Schwarzschild ISCO (L̃ = 2√3) reduces to a_★,rms = 2√(μ/M_f). The M_i² weighting means the walk is controlled by the final mass: the last captures are both the largest angular-momentum depositors and the ones normalized against the largest M².

Internal consistency against Mandel (2008)'s scaling. Mandel et al. (2008) give the general result that IMRI spin-up scales as χ ~ m/M and rarely exceeds χ = 0.3 for the neutron-star-capture channel; we did not locate a specific stated 123 M_☉/a_★ = 0.2 ± 0.08 data point in the accessible text to validate against directly, so we check our own formula's behaviour against the scaling instead. The isotropic random walk for a hole gaining half its mass from neutron-star captures, M_f = 2M₀, gives Σ M_i² = 0.2917 M_f³/μ and a_★,rms = 1.871√(μ/M_f). Setting a_★ = 0.2 with μ = 1.4 M_☉ gives M_f = 123 M_☉, an M_f/μ ratio of ~88, consistent with Mandel's χ ~ m/M scaling at order of magnitude and within his stated sub-0.3 ceiling. The spread checks too: a 3-D random walk gives a Maxwellian |a_★| distribution with σ/mean = √(3 − 8/π)/(2√(2/π)) = 0.42, comparable to the ~0.4 relative spread generic to this class of random-walk spin-up. The formula's internal behaviour is consistent with the general scaling the literature supplies; we adopt it on that basis rather than importing a specific external number, since Mandel's own worked examples use M_f/M₀ and mass ratios that differ from ω Cen's by more than an order of magnitude each.

Applied to ω Cen. With μ = 30–40 M_☉ (González Prieto et al. 2025), M_f = 2×10⁴ (this paper's fiducial mass) to 5×10⁴ M_☉ (the González Prieto et al. 2025 endpoint), and L̃ = 2√3 (circularized capture) to 4 (direct plunge on a marginally bound orbit):

Converting the central rms value to the Maxwellian mean and dispersion gives a_★ = 0.049 ± 0.021 (90th percentile 0.077); the full derived range across all corners is a_★ ~ 0.05–0.10. The corners mix two masses and the mass has to travel with the number: the 0.049 and 0.053 entries assume the González Prieto et al. (2025) endpoint M_f = 5×10⁴ M_☉, while this paper's own fiducial 2×10⁴ M_☉ gives 0.084 at the same μ and L̃. We quote 0.049 ± 0.021 at the endpoint mass and 0.084 at the fiducial mass throughout, and no longer round either to a single ≈0.06, which belonged to neither. This replaces the imported literature value a_★ ~ 0.1–0.3 (Mandel et al. 2008), which is the outcome of a different growth factor and mass ratio never derived for this target.

The mass-ratio boundary. No numeric cutoff for the random-walk regime exists in Hughes & Blandford (2003) or its successors; the boundary is not sharp because the underlying physics is continuous in mass ratio. Setting a_★,rms = 0.7, the numerical-relativity attractor value, in the scaling above with q ≡ μ/M_f gives 2√q = 0.7, so q ≈ 0.12: the random walk alone reaches attractor-level spin at q ≈ 0.12 and produces less below that, as √q. Two limits of the construction should be stated with it. The scaling equates a random-walk r.m.s. to an attractor value that numerical relativity obtains from a different mechanism, so q ≈ 0.12 marks where the walk alone could reach attractor-level spin, not where the attractor takes over. And at q = 0.12 the walk runs over N = M_f/μ ≈ 8 events, where the √N averaging the derivation rests on is barely established. Fishbach, Holz & Farr (2017)'s tested range is q ≳ 0.7, which does not overlap 0.12, so the boundary is not corroborated by that work and we do not claim consistency with it; their √q ≲ 0.32 anti-aligned bound is quoted here only as the one other quantitative q-scaling in the literature. ω Cen sits at q = μ/M_f ≈ 7×10⁻⁴, a factor ~175 below the boundary.

Net-alignment sensitivity. Relaxing ⟨cos ι⟩ = 0 to a net alignment fraction ε_rot = ⟨cos ι⟩, the coherent deposit is J_coh = ε_rot(L̃μG/c)Σ_i M_i, with Σ_i M_i ≃ M_f²/(2μ), giving

for L̃ = 2√3 to 4. The mass dependence cancels: unlike the random walk, the coherent channel carries no √N suppression, so it dominates as soon as it exists. It equals the random walk's 0.053 at ε_rot ≈ 0.026–0.031, reaches 3× the random walk at ε_rot = 0.08–0.09, crosses the entire imported 0.1–0.3 band between ε_rot ≈ 0.05 and ≈ 0.17 (so at ε_rot = 0.20 it exceeds the band's upper edge by 15–33 per cent), and reproduces the attractor value by this separate route at ε_rot ≈ 0.35–0.40. Above ε_rot ≈ 0.45–0.5 the low-spin prediction is gone. ω Cen is a rotating cluster and the inspiraling population is mass-segregated by dynamical friction, a process with no obvious reason to randomize the angular momentum a captured object arrived with, so isotropy is an assumption to test rather than a default. The González Prieto et al. (2025) realizations cannot test it: their code is spherically symmetric and draws transverse directions isotropically, so ⟨cos ι⟩ = 0 is built into them.

C. Priors, likelihoods, and computation

Implemented in paper/figs/fig3_lnk.py (2×10⁶ prior draws; fixed seed). H_eng parameters: P_comp log-uniform on [1, P_fuel] L_☉ with P_fuel = 7.6×10⁵ L_☉ (the ambient Bondi ceiling of Section 2.7, quoted as 8×10⁵ in the text). That ceiling is the engineered-unsuppressed, near-extremal-spin value and is not cosmetic: it falls ~5× at a_★ = 0.5 through the η(a_★) dependence of Section 2.7, and ~10²–500× under natural outflow suppression, so a fully stated prior is a mixture over (a_★, suppressed/unsuppressed). We instead quote the sensitivity band, since the arithmetic is analytic in ln K = ln[f_d + (1 − f_d) P(quiet | active)]: at the fiducial unsuppressed ceiling, P(quiet | active) = 0.245 and ln K_MIR = −0.47; truncating the ceiling at the ADIOS-suppressed ~2.6×10³ L_☉ removes much of the detectable prior volume, raising P(quiet | active) to 0.377 and moving ln K_MIR to −0.37; the a_★ = 0.5 ceiling sits between. Other parameters: 1 − f_sink log-uniform on [10⁻⁴, 1]; swarm radius r log-uniform over the envelope [2×10⁻², 4×10³] AU (an uninformative choice within the allowed region; the thermodynamic gradient of Paper A would weight it deeper, which is the conservative direction for the engineered hypothesis since deep swarms are warm and more detectable); dormancy probability f_d = 0.5, varied over [0.1, 0.9] for the sensitivity band. The value 0.5 is a stipulated fiducial, not derived from this paper's inputs, so the reported result is the sweep ln K = −1.138, −0.474, −0.079 at f_d = 0.1, 0.5, 0.9. A conditional bound exists and is not adopted: if "active" means the spin-up episode alone (ln(2.20) t_S ≈ 3.55×10⁷ yr, Section 5) and the occupation lasts at least 10⁸ yr (Section 6), then f_d ≥ 1 − 3.55×10⁷/10⁸ = 0.645 and ln K ≥ −0.31. Post-spin-up operation is also active and its duration is not bounded here, so the condition is an assumption and the sweep stands. MIR channel likelihood (v1.0, continuous, OCS-MIRI-1): the waste luminosity L_waste = (1 − f_sink)P_comp splits into a warm component f_warmL_waste and a cool component (1 − f_warm)L_waste, each re-radiated as a blackbody from the same swarm radius r via T_eff(L, r), giving a two-temperature SED with f_warm log-uniform on [10⁻³, 0.999] (the one new free parameter this build introduces) as the only lever beyond (P_comp, f_sink, r). Each component's flux density at every MIRI imaging filter's pivot wavelength (F560W through F2550W, the 25.5 μm imager cutoff) is compared against the JDox ETC-6.0 point-source sensitivity, penalized by the core crowding factor of 3× that Paper C already applies; a configuration is detected iff the summed flux exceeds the crowded limit in any filter, combined as a logical OR, never a sum, over the nine filters. This replaces the piecewise step in T_eff. Filter bandpass width is not sourced and is not folded in; each filter is treated as a delta function at its pivot wavelength, a stated approximation. The crowding factor is inherited from Paper C's NIRCam-calibrated recovery and has not been verified for the MIRI point-spread function, but the adjudication does not turn on it: removing the penalty altogether (3× → 1×) moves ln K_MIR by 0.014 nats, an order of magnitude below every prior boundary in the sensitivity table below. The far-infrared corner (swarm cold enough that its Wien tail is negligible at every MIRI wavelength) is not clamped by an assumed placeholder limit; it falls out of the physics as simply undetected by any filter in the array, which is the conservative state, since no far-infrared instrument is currently operating (OCS-MIRI-DATA). Data = non-detection in all nine filters; P(quiet | active) = 0.245, giving ln K_MIR = ln[f_d + (1 − f_d) × 0.245] = −0.47 at fiducial f_d, band [−1.14, −0.08]. A logistic softening of the per-filter threshold, width 0.5 dex in log₁₀(F_ν/F_lim), combined across filters as P(nondetect) = product over filters of (1 − p_det,i), moves ln K by −0.012 nats, so the threshold shape remains immaterial. Two statistics are reported alongside that number because it is not, on its own, a measure of what the data said. The first is the data-only quantity ξ = 1 − P(quiet | active) = 0.755, the fraction of the active-installation prior volume the forecast ETC depths exclude; no dormancy prior enters it. The second is the channel capacity: because P(no detection | H_q) = 1, this channel is one-sided and bounded by ln K ≥ ln f_d = −0.69 (Section 4.2), of which 68 per cent is now spent, against 41 per cent under the hard-threshold model this build replaces. Further mid-infrared depth can buy at most 0.22 nats more, and can never buy evidence for H_eng.

Prior-edge sensitivity. ln K here is a ratio of prior volumes, so every prior boundary is a lever and all of them are reported. Each row re-runs the same computation with one boundary moved; each is a command-line switch on fig3_lnk.py.

The largest lever is the lower edge of the prior. The P_comp prior runs upward from 1 L_☉, a boundary that coincides numerically with the warm instrument limit and has no physical warrant; extending it three decades downward is worth +0.153 nats, more than the transport-floor sensitivity (+0.103) and now the fuel-ceiling one as well (+0.101). The temperature-split floor is the weakest lever measured, +0.006 nats, because the total waste luminosity and the swarm radius, not its partition between the two components, set most of the detectability. Stated plainly: the evidence budget is still dominated by the dormancy prior, the data entering only through ξ, but the continuous per-filter likelihood raises ξ from 0.50 to 0.76 against the same forecast ETC depths, because the real JWST/MIRI depths (sub-μJy at the warm filters) exclude far more of the active-installation prior volume than the old order-of-magnitude piecewise limits did; this moves ln K_MIR from −0.29 to −0.47 and the spent capacity fraction from 41 to 68 per cent, without touching the capacity bound ln f_d itself. The same forward model, a two-temperature swarm SED integrated against the real per-filter MIRI sensitivities, supplies the point-source SED discriminant Paper C's mid-infrared program needs (Program 1): the warm/cool flux ratio across F560W through F2550W is the model's falsifiable prediction for any point-source excess the program detects. Radio, R, spin, kinematic, and timing channels are inactive or degenerate between the engineered and quiescent-IMBH hypotheses on current data and contribute ln K = 0 each, as Table 1 requires. The quiescent-vs-subcluster contest is checked against the imbh-constraints v1.0.0 aggregation engine (nine curated constraints; empty joint window, tension verdict), whose mass-window math is anchored by the series' replayed golden and CI parity gate.

D. The nine-constraint compilation

The aggregation-engine cross-check of Section 5 (the imbh-constraints library, nine curated constraints, in the OCS toolchain archive v1.0, Zenodo DOI 10.5281/zenodo.20689279) draws on the compilation of Table 2. The jointly allowed window is the span between the strongest lower bound and the weakest upper bound at fiducial parameters; it is empty (8,200 M_☉ against ≈709 M_☉, tension verdict), the formal statement that no point-mass value satisfies all published constraints at face value. The deep radio limit (Mahida et al. 2026) is not among the nine: it constrains accretion efficiency rather than mass directly.

E. Reproducibility

Figure scripts and outputs are published alongside this page at /papers/figs/ (common.py for shared constants, one script per figure — fig1_envelope.py, fig2_constraint_plane.py, fig3_lnk.py, fig4_r_statistic.py — plus fig1_results.json for machine-readable Monte Carlo output and a README). The full LaTeX source for all five papers (.tex + .bib) is at /papers/source/. All random draws use fixed seeds; all quoted numbers regenerate from the scripts. The measurement compilation and aggregation library are part of the public OCS toolchain archive (v1.0, which includes the imbh-constraints library), Zenodo DOI 10.5281/zenodo.20689279; interactive calculator versions of the Figure 1 and Section 2.7 computations are the site's flyby-survival.html (single-point), flyby-survival-simulator.html (full envelope sweep, this paper's Figure 1 reimplemented in browser JS), and imbh-fuel-budget.html tools (see cross-check note below).

A guided walkthrough chaining all four interactive tools in the paper's own argument order (envelope sweep → single-point drill-down → fuel ceiling → adjudication) is at scenario-paper-e-feasibility.html.

F. E6 validation of the adjudication machinery

Section 4's machinery is tested here against three historical cases where the resolution is independently known, per the protocol committed in advance (OCS-E6-PROTOCOL) and implemented in figs/appendix_d_validation.py. The object under test is the scoring machinery itself, the explicit-menu requirement, the per-channel Bayes factor, the priced catch-all null, and the action bands, never the ω Cen forward models; what varies per case is the channel likelihood the machinery is fed. LGM-1, the discovery of the first pulsar, is the positive control: the only case that can drive ln K strongly positive, and the only one that carries a menu-completion test. Project Hephaistos and Boyajian's star are the mundane cases, scored for false positives against a menu that already contained the resolving alternative before the resolving datum arrived. 'Oumuamua is excluded per the protocol: its lightsail proposal has no settled community verdict to calibrate against. Priors and menus for each case were sealed in a dedicated commit containing no scoring code before any ln K was computed (paper/e6-priors/priors.json, commit f82e11d); the scoring commit (96e0d1e) follows it. Every channel construction is stated in that file and in the script's comments; several are explicit modeling choices rather than re-measurements of the primary data, flagged as such below.

LGM-1. At Epoch 1a (late November 1967, after the fast-recorder capture, before the orbital-Doppler test), the menu is terrestrial interference, artificial signal, and a natural-source catch-all. The catch-all is priced against the fastest coherent celestial periodicity catalogued at that date, of order one hour; CP 1919's coherent 1.337-s period sits 3.4 dex beyond that boundary, and at the fiducial inflation λ = 1 this channel alone gives ln K = +7.90 against the natural catch-all, satisfying P1 (ln K ≥ 3). Its magnitude falls in the candidate range, but the candidate band is a two-part rule, ln K > 5 and support across at least two messengers, and Epoch 1a is a single channel; LGM-1 therefore registers at strong anomaly by the band rule however large the number gets, correcting an earlier draft that called +7.90 "comfortably inside the candidate band" three pages after stating that no single channel can reach it. The framework tolerates catch-all generosity up to λ = 2.63 before P1 fails; the (π_nat, λ) grid confirms ln K is invariant in π_nat, as a likelihood ratio must be. Two collapses follow, reported separately per protocol. Collapse B, menu completion: adding the rotating-neutron-star hypothesis to the Epoch-1a menu, priced as though it had existed in November 1967, drives ln K to −0.50, a 8.40-nat drop that satisfies P2. Collapse A, data with the menu held fixed: adding the null orbital-Doppler test and the discovery of a second independent pulsating source moves ln K to 6.69 and 4.04 respectively, 2.83 with both; the multiplicity channel is the larger mechanism (−3.86 nats against −1.20 for the Doppler null). This is a finding, not a pass: P3 requires ln K below 1 after Collapse A and the constructed channels do not reach it, so F3 is reported rather than repaired by re-tuning a prior after seeing the number. The achievable range at Epoch 1a, read off the λ sweep bounds, is [2.63, 8.00], 5.37 nats (not itself a passed criterion, since its lower end is set by the top of the λ sweep and that top is a choice). P6 crosses one nat at λ_max ≈ 1.5 and cannot fail above it, but no pre-registered rule fixes λ_max, so P6 is reported as informational rather than passed.

Project Hephaistos. The dossier's open question, whether Suazo et al. (2024) already discussed background-galaxy contamination at publication or whether Ren et al. (2024) introduced it three weeks later, is resolved by reading the discovery paper's own full text (§3.1, §4): the paper estimates ~2 contaminants expected among the seven candidates from Hot DOG chance alignment and states plainly that "the possibility of perfect alignments cannot be ruled out." Background contamination is therefore on the Epoch-1 (2024-05-06) menu by construction, not a later addition, which sets every subsequent trajectory. Scored per candidate for the three with a direct-imaging resolution: G reaches ln K = +0.92 (uninformative) at Epoch 1 and collapses to −3.08 once the off-star VLBI radio position resolves it (2024-05-23), never crossing the strong-anomaly band before that datum existed. D and E follow the same Epoch-1 read, rise to +2.42 (anomaly, not strong anomaly) once the 2026-07-03 archival diagnostics find no contamination signature for them specifically, in contrast to B and C, and collapse to −3.58 once JWST/MIRI resolves both as background galaxies (2026-07-10). All three satisfy P4: no case reaches the strong-anomaly band (ln K ≥ 3) before its resolving datum is public, and the anomaly-band excursion for D and E is exactly what that band exists to register.

Boyajian's star. Scope is the short-timescale dipping only, per the dossier's cut; the secular dimming and the contested plate-fading claim are excluded and named as such. The megastructure hypothesis is not on the 2015-09-11 discovery-paper menu; it enters 2015-10-15 (Wright et al.), so the wavelength-dependence channel is inactive at both early epochs and ln K = 0 there, a null result correctly attributed to the channel not yet existing rather than to evidence. The continuous form required by §B is implemented physically rather than as a hard threshold: an anomalous-diffraction extinction efficiency Q_ext(x) is integrated over a log-uniform dust particle-size prior spanning Boyajian et al. (2018)'s own stated constraint region, and compared against the solid-occulter prediction of zero wavelength dependence at any macroscopic scale. At the 2018-01-02 multiband result the achromaticity statistic is read at the 90th percentile of the dust-prior's own predictive, a conservative rather than most-favorable reading of the paper's "at least some dips" language, giving ln K = −17.4 for the artificial hypothesis, decisively null-favored and consistent with the paper's own "inconsistent with... in-line with" language. P4 is satisfied; the channel was never live before the resolving datum. The exact measured reddening slope and its uncertainty were not extracted from the primary photometry in this pass; the scoring instead uses the qualitative constraint the paper itself commits to, a stated limitation rather than a claimed measurement, in the same sense Appendix C's filter-bandpass approximation is stated rather than sourced.

Result. The terminal criteria are P1, P2, P4, and P5, and all four pass; none of F1, F2, or F4 fires, so the framework does not fail validation. P6 is reported as informational rather than passed, for the λ_max-dependence reason given above. Three changes to the criteria themselves are worth stating, because each removed a way the exercise could not have failed. P5 is now a check rather than an assertion: the script reads the committer times of the sealed-priors file and of its own source from git and fails if the seal does not strictly predate the scoring code, where earlier drafts printed a pass unconditionally. And every hard-coded likelihood ratio in the three cases is now written as an explicit pair of predictives rather than as a ratio with the artificial-hypothesis predictive silently fixed at unity by convention; the decomposition reproduces all three of P1, P2, and P3 to the digits quoted here, as it should, since making an assumption visible ought not to move a number. The Boyajian channel keeps one construction that no decomposition repairs: its measurement dispersion is defined so that the megastructure model sits at six standard deviations, so the −17.4 nats it returns restates that choice. Across 2 to 8 standard deviations the channel runs from −1.6 to −31.3 nats for the artificial hypothesis (the script's ln_r printout carries the opposite sign, +1.6 to +31.3, because it is the log ratio in favor of the dust alternative), and the verdict is insensitive to the choice only in sign. P3 fails as constructed: Collapse A's data-driven test leaves LGM-1 at ln K = 2.83 rather than below 1, meaning the discriminating data the discoverers themselves treated as decisive (the Doppler null and the second source) carry real but not overwhelming weight in this channel decomposition, a genuinely reachable outcome the protocol was designed to surface rather than to guarantee against. The framework registered LGM-1's anomaly about six months before the community's own resolution (November 1967 detection against the May 1968 Gold paper), tracked Hephaistos candidates G, D, and E through anomaly and back to null without ever crossing strong-anomaly ahead of their resolving data, and correctly returned zero evidence for Boyajian's star before the wavelength-dependence channel existed. The construction choices flagged throughout, most consequentially the one-hour natural-coherence reference boundary for LGM-1 and the unmeasured Boyajian reddening slope, are stated rather than sourced from a primary catalog or spectrum; a future pass with access to a machine-readable 1967 radio-source catalog and the raw Boyajian et al. (2018) photometry could tighten both without changing the qualitative result.

G. Revision record

This appendix lists what changed between versions, so that the body and abstract can state current results only.

v2.8 (2026-10-24). Fact fixes from the Round R10 referee report, adjudicated by the farm (D301, D305, D306) and checked independently. The shear-to-precession comparison is now made at one threshold: precession reaches 2π in 1.7×10³ orbits, about 12 days, against 33 orbits for shear, a factor 50 (was a factor 8, which set one radian against 2π). Latency across the swarm is seconds, not milliseconds (r/c = 9.8 s at 100 r_g), and the ISCO period is 9 s. The distance in the scintillation sentence is 5.49 kpc (was 5.4), and the X-ray threshold is 4×10⁻⁹–4×10⁻⁸ L_Edd (was 10⁻⁸–10⁻⁷). The fuel-ceiling lever is +0.101 (was +0.121). The mid-infrared exclusion is attributed to the forecast ETC depths, not to current limits (also in the forward-model table caption and row). The abstract's −2.3 is labelled ln f_d, the channel capacity, not a Bayes factor. The Zenodo record cited for the aggregation library is described as the OCS toolchain archive v1.0 that includes the imbh-constraints library, not as a standalone package with an ASCL entry. The envelope edge 0.1 r_infl is declared a stipulated prior, with r_J = 0.81 pc. The tidal-disruption rate bracket is attributed to Fragione et al. (2018) Eqs. 27 and 8 with an assumed occupation fraction 0.01 (it was attributed to Stone and Metzger), and the spin-up duration quoted from Paper A is 4.2–6.6×10⁷ yr (was 3.4–5.3). The dormancy prior is reported as the sweep −1.138, −0.474, −0.079 at f_d = 0.1, 0.5, 0.9, with the bound f_d ≥ 0.645 given only as a conditional assumption. The worked-example sentence on the joint-marginalisation correction now says the second live channel carries no dormancy parameter.

v2.7 (2026-10-23). Fact fixes from the farm audit D267. The dynamic range between horizon scale and cluster scale is eight orders of magnitude in radius (the paper's own endpoints span 8.3 dex; was nine). The tidal-disruption bracket is 5×10⁻⁸–5×10⁻⁷ yr⁻¹, the printed cluster-density bracket 5×10⁻⁶–5×10⁻⁵ times the IMBH-hosting fraction 0.01 (was 10⁻⁸–10⁻⁷); the ω Cen-specific 5×10⁻⁸ sits at its lower end. Posterior odds at ln K = 5 and prior odds 10⁻⁶ are e⁵×10⁻⁶ = 1.5×10⁻⁴ (was 2×10⁻⁴). The composite ln K at f_cov = 10⁻² is −0.61 in the text (−0.6145, printed as −0.615 in Appendix C; was −0.62). The saturated-kick amplitudes of Appendix A are restated at the adopted 31 M_☉ perturbers: δ ≲ 0.4 at a ≳ 1 AU and δ = 4.0 at 10⁻² AU, with δ > 1 inside ~0.2 AU since δ ∝ a^−1/2 (was 0.14, 1.4 and ~1 AU, the retired 10 M_☉ run). The orbital velocities quoted beside the station-keeping budget are ~70 km s⁻¹ at 4×10³ AU and ~4×10³ km s⁻¹ at 1 AU (was 10²–10³, true only between about 18 and 1800 AU).

v2.6 (2026-10-23). Fact fixes from the farm audits D271 and D253. The van der Marel and Anderson (2010) bound of 1.2×10⁴ M_☉ is a 1σ limit (it was labelled 3σ; the primary gives 1.8×10⁴ at 3σ). The cross-references to Paper A are Appendix A of Paper A for the staged architecture and Section 5.2 of Paper C for the wander program. The natural-refilling shortfall is stated against the Paper A requirement of ~1–2×10⁻³ stars yr⁻¹ (6.8×10⁴ stars over 3.4–5.3×10⁷ yr), i.e. four to five orders of magnitude (was 10⁻⁴–10⁻³ and three to four). The MAD-endpoint spin range is described as within 1σ of the natural-capture null (was “at or below”; the null is 0.049 ± 0.021 and the range tops out at 0.07).

v2.5 (2026-10-22). The carrier paragraph of the delivery discussion is aligned with Paper A v2.12: dense coding is bounded by the pure-loss channel capacity, not the Bekenstein bound, and no numerical gain is quoted here.

v2.4 (2026-10-22). The ceiling mixture of Section 5 was rerun on the MAD equilibrium band a_★ ~ U(0.035, 0.07): the mixture moves from ξ = 0.649, ln K = −0.393 to ξ = 0.608, ln K = −0.363, the all-MAD-and-suppressed corner from −0.306 to −0.244, and the total with the mass channel from −0.420 to −0.390; the thin-disc corner is unchanged. The MAD-branch growth factors, episodes and endpoints of Section 3.4 were reproduced by the script rather than by hand. The statements that ⟨cos ι⟩ could be read from the González Prieto et al. (2025) realizations were withdrawn: the isotropy of captured orbital planes is an assumption, because that code does not track the orbital plane.

v2.3 (2026-10-20). The abstract was shortened and now opens with the statement that no electromagnetic null can adjudicate H_eng; the −0.69 cap belongs to the mid-infrared channel and not to the engineered-mass channel. Section 3.4 was rewritten: the fork is the spin-up architecture, extraction at ambient fueling preserves a thin-disc spin (Section 3.3), the MAD equilibrium is a_★ = 0.035–0.07 (Narayan et al. 2022) in place of the earlier 0.1–0.3, and the MAD-branch growth factor, episode, endpoints and fuel ceiling were re-priced by hand (1.0097–1.0198; 9.5×10²–3.8×10³ L_☉). The a_★ ~ U(0.1, 0.3) ceiling mixture of Section 5 was computed on the earlier band and its rerun is pending. The residence-survival export was re-anchored on the bound-remnant channel (0.75 at the edge, 0.97 at 400 AU) and the flyby-curve minimum corrected from 3×10⁸ to 7.6×10⁸ yr. The refill clock for the 10 pc deficit was changed from ~4 to ~12 Gyr. In Appendix F the Boyajian sensitivity range was given its correct sign and −17.9 was reconciled to −17.4. Typographical fixes were made in four places and the sodium-absorption source of n_e was removed.

Earlier versions. (i) Differential precession was first given as the decoherence mechanism for co-radial formations; Keplerian shear decoheres a spread faster (33 orbits against 2.7×10²) and applies across the whole envelope. (ii) Presence of a bound black hole at the edge was 53 per cent under a fixed-number convention and is 24.8 per cent under the drawn convention of Appendix A. (iii) The giant count inside the influence radius was first taken as ~10³, a count of stars there rather than of stars on the giant branch. (iv) The mid-infrared bound began as a single number that assumed the warm case, and the hard-threshold likelihood excluded 50 per cent of the active prior volume against 76 per cent for the continuous per-filter likelihood. Placeholder instrument limits and the "Wien-peak band" and "cross-band leakage" language, which the script never implemented, were retracted. (v) The MAD eruption-power band was first extended a decade lower than the cited cycle supports. (vi) The spin discussion first reduced the discriminating contrast to ≃0.7 versus ≳0.9 by importing the generic hierarchical-merger caution, and left the isotropy assumption implicit. (vii) The spin-up episode was first given a round duration about three times the computed 3.5×10⁷ yr. (viii) The engineered-mass channel was first described as already observable and left unscored, with the kinematic and timing channels at ln K = 0 each; it is priced here at −0.03 nats. (ix) The radius grid first began at 5 r_g, inside the ISCO, where circular orbits do not exist. (x) Only two prior boundaries were first varied in the sensitivity tables, and the P_comp lower edge was not among them. (xi) LGM-1 at +7.90 was first called comfortably inside the candidate band, which contradicted the two-part band rule. (xii) Cusp existence was first left open and carried as a bracket and a second sensitivity lever; the realizations of González Prieto et al. (2025) closed it. (xiii) A 10 M_☉ perturber run printed 2.3 against 0.54 for a velocity-diffusion comparison and was retired at the referee-5 revision. (xiv) The 1.0 per cent deficit at 10 pc came from a different enclosed-mass model than the neighboring figures. (xv) The unconditional median was first reported under the conditional label for the bound-cusp floor, while p_BH,present exceeded one half.

Acknowledgements and disclosure

The author thanks the maintainers of the NASA Astrophysics Data System and arXiv, on which the citation verification for this work relied. AI assistance disclosure: drafting, citation verification, derivation checking, and figure preparation for this manuscript were performed with substantial assistance from a large language model (Claude, Anthropic), under the author's direction; the author reviewed and takes full responsibility for all claims, derivations, and references. Interactive calculators implementing the quantitative material in Sections 2–4 are available as supplementary material at omegacentauri.me.

Data availability

No new observational data were generated. All cited measurements are available in the referenced publications. The figure scripts, shared constants, and machine-readable Monte Carlo output described in Appendix E are mirrored publicly at omegacentauri.me/papers/source/ and omegacentauri.me/papers/figs/, alongside the interactive calculator versions of the same computations.

References