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A subgrid model can follow massive black hole triplets all the way to coalescence inside live galaxy simulations, and encounter geometry alone can decide which pair merges and when.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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RAMCOAL now evolves subgrid massive black hole triplets to coalescence inside live hydrodynamical simulations by mapping chaotic encounters onto Bonetti three-body outcomes.

T0 review reviewed 2026-07-11 challenge →

load-bearing objection Solid methods extension that really does run a live-hydro triplet to a Bonetti-drawn coalescence; the stress-test is right that the resonant phase itself is a library lookup, not integrated dynamics. the 3 major comments →

arxiv 2607.04121 v2 pith:DC5BZSEY submitted 2026-07-05 astro-ph.GA

Set them free: extending RAMCOAL to model massive black hole triplets in hydrodynamical simulations of galaxies

Kunyang Li , Ricarda S. Beckmann , Yohan Dubois , Marta Volonteri This is my paper
classification astro-ph.GA
keywords massive black holesblack hole binariesblack hole tripletshydrodynamical simulationssubgrid dynamicsgravitational wavesgalaxy mergersdynamical friction
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Massive black holes that form binaries and higher-order multiples after galaxy mergers spend part of their lives at separations that cosmological simulations cannot resolve, yet those unresolved stages set merger delays, spins, recoils, and the host-galaxy context of the final gravitational-wave event. This paper extends the staged RAMCOAL framework so that three black holes can be followed as a subgrid system inside a hydrodynamical galaxy simulation: they begin as resolved sink particles, pass through dynamical friction, form bound binaries that harden by stars, gas, and gravitational waves, and, when a hierarchical triplet becomes chaotic, are mapped onto a library of three-body outcomes that updates mergers, exchanges, and ejections while continuing to track accretion and spin. Isolated-galaxy tests with contrasting orbital geometries show that the encounter geometry alone can change which pair finally merges and after how long. A compact third test demonstrates the first full dynamical evolution of a massive black hole triplet from three resolved objects through chaotic interaction to coalescence inside a live hydrodynamical simulation. The result is an end-to-end path from galactic environment to gravitational-wave source parameters for catalogues that can link coalescing black holes to the galaxies that host them.

Core claim

The authors establish that RAMCOAL can now follow subgrid massive black hole triplets self-consistently inside live hydrodynamical galaxy simulations, and they demonstrate for the first time a complete dynamical evolution of such a triplet from three resolved black holes through chaotic three-body interaction all the way to coalescence, with the surviving configuration, accretion, and spin updated on the fly.

What carries the argument

The RAMCOAL triplet extension: a three-stage subgrid treatment (resolved sinks, dynamical-friction pairs, bound binaries) that, when a hierarchical triplet becomes unstable, maps the encounter onto a weighted library of three-body outcomes and updates the surviving system, accretion, spins, and recoils while remaining coupled to the live host galaxy.

Load-bearing premise

When a hierarchical triplet becomes chaotic, the full resonant three-body dance can be replaced by one weighted draw from a fixed scattering library that depends only on primary mass and two mass ratios, with a short fixed interaction timer and an escape-speed ejection velocity.

What would settle it

Run matched live-galaxy simulations in which the chaotic phase is integrated with a direct few-body or regularized N-body method instead of the library draw, and check whether the identity of the coalescing pair, the merger delay, and the residual eccentricity systematically disagree with the library-based outcomes.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper extends the RAMCOAL subgrid framework in RAMSES to treat massive black hole triplets. MBHs evolve as sinks through resolved dynamical friction (stage 0), subgrid DF (stage 1), and bound hardening by stellar scattering, circumbinary torques, and GWs (stage 2). Hierarchical triplets that meet a Mylläri-type instability criterion are mapped onto the Bonetti et al. (2018) scattering library via a weighted draw on (M_pri, q_in, q_out), with a fixed 10^3 yr chaotic timer; remnant spin and recoil follow Rezzolla and Lousto–Zlochower fits. Three isolated-galaxy tests show geometry-dependent partner selection (A vs B) and an exchange merger in a compact active triplet (C), with post-processed GW tracks. The authors claim the first end-to-end triplet evolution to coalescence inside a live hydrodynamical simulation and a route to environmentally linked GW merger catalogues.

Significance. If the framework holds under cosmological application, it would fill a genuine gap between pure post-processing delay models and expensive regularized N-body treatments (e.g. KETJU), enabling PTA/LISA/TianQin catalogues that retain live gas, accretion, spin, and recoil history. Strengths include an explicit seven-state classification, on-the-fly coupling of accretion/spin/feedback to the subgrid orbit outside the chaotic interval, validation of the Bonetti selector for one cell (N_seed=100), and a transparent demonstration that encounter geometry alone can change which pair merges. The work is timely given PTA backgrounds and LISA adoption. The main caveat is that the resonant three-body phase itself is not dynamically integrated in the live run, so the strongest wording of the end-to-end claim needs careful qualification.

major comments (3)
  1. [Abstract; §5.2; §6.4; Conclusions] Abstract, §6.4, and Conclusions claim the first triplet evolution “all the way to coalescence inside a live hydrodynamical simulation.” In case C the chaotic phase is not integrated: once Eqs. (16)–(19) are satisfied, a single Bonetti draw is taken and coalescence is assumed after a fixed 10^3 yr timer (§5.2); the GW track is reconstructed only in post-processing (Fig. 22). Live gas/stars/feedback therefore never act during the resonant interaction that decides partner and timing. The capability is real for pre- and post-chaotic stages, but the wording overstates what is demonstrated. Soften the claim and state explicitly that the resonant phase is a library lookup plus timer.
  2. [§5.2] §5.2 sets the chaotic interaction timer to a fixed 10^3 yr (called tunable) and sets ejection velocity to the local escape speed from the resolution sphere rather than drawing from the Bonetti N-body distributions. Both choices are load-bearing for merger delays and the wandering/offset population that the paper advertises for dual-AGN and multimessenger science. Either justify the numerical values against the Bonetti completion-time and velocity distributions, or present a short sensitivity test (e.g. timer ×10 and ÷10) so that the reported coalescence times in cases A–C can be interpreted.
  3. [§5.2; §7.3] The Bonetti grid assumes M_out < M_pri + M_sec (q_out ≤ 1); the opposite case is deferred (§5.2). In hierarchical assembly the intruder can be more massive. The manuscript should quantify how often this configuration is expected in the intended cosmological application and state how such systems will be handled (reject, force merge, or flag) so that the catalogue claim is not silently incomplete.
minor comments (5)
  1. [§5.4, Eq. (34)] Eq. (34) lists the Rezzolla coefficients as empty placeholders (“s4 =, s5 =, t0 =…”). Insert the numerical values from Rezzolla et al. (2008).
  2. [Table 3; §6.4] Table 3 and the text give slightly inconsistent initial separations/velocities for case C; reconcile the numbers and state whether the resolution sphere is 2Δx or 4Δx for that run.
  3. [Fig. 6] Fig. 6 caption and text note that the plotted quantity switches from instantaneous separation to semi-major axis at the stage 1–2 boundary; make this change of variable explicit in the figure legend itself to avoid an apparent jump.
  4. [§6.3 and passim] Several typos: “RMACOAL” (§6.3), “set them free” title is fine but “set them free” vs body consistency, and occasional missing spaces around units (e.g. 0.39 kpc).
  5. [§5.3] The CBD preferential-accretion coefficients p0, p1, p2 (Eqs. 22–23) and η_s2 = 0.01 are free parameters; a one-sentence statement of their provenance (or that they are held fixed from Duffell et al.) would help reproducibility.

Circularity Check

1 steps flagged

No significant circularity: external libraries and staged prescriptions are applied, not fitted to the paper's own outputs; the end-to-end claim is overstated but not circular.

specific steps
  1. self citation load bearing [§3 (opening); abstract; conclusions]
    "Before introducing the triplet extension, we summarize the RAMCOAL model for MBHBs as presented in paper I (Li et al. 2025). ... We demonstrate the first triplet MBH dynamical evolution all the way to coalescence inside a live hydrodynamical simulation."

    The binary stages (DF, stellar hardening, CBD torques, GW) are taken wholesale from the authors' own prior paper rather than re-derived. This is ordinary sequential self-citation and is not load-bearing for the new triplet claim (the Bonetti mapping and test-case C demonstration stand independently), so it contributes only a minor score increment.

full rationale

The paper's derivation chain is a staged subgrid model that imports independent external results (Bonetti et al. 2018 scattering probabilities, Lousto & Zlochower 2013 recoils, Rezzolla et al. 2008 remnant spins, Duffell et al. 2020 preferential accretion) and applies them on the fly inside RAMSES. It does not fit those libraries to its own simulation outputs, nor does it redefine a target observable in terms of a fitted parameter and then call the result a prediction. Self-citations to paper I (Li et al. 2025) and prior RAMSES spin/accretion work supply the binary baseline; the new triplet machinery (Mylläri-type instability criterion, 49-combination reduction, Bonetti weighted draw) is an independent extension demonstrated in controlled isolated-galaxy tests. The strongest claim—that a triplet is followed "all the way to coalescence inside a live hydrodynamical simulation"—is weaker than advertised because the chaotic resonant phase is replaced by a library lookup plus a fixed 10^3 yr timer rather than integrated live, but that is an overstatement of capability, not a circular reduction of a prediction to its inputs. Score 1 for ordinary self-citation of the binary baseline that is not load-bearing for the new triplet results.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 2 invented entities

The central claim rests on the validity of replacing chaotic three-body dynamics by a discrete library draw, on several tunable numerical thresholds, and on the parent RAMSES galaxy-formation model. No new physical entities are postulated; free parameters are algorithmic switches and fixed timers rather than fits to the paper’s own data. Domain assumptions inherited from classical MBHB theory and from Bonetti et al. (2018) dominate the ledger.

free parameters (4)
  • chaotic interaction timer = 10^3 yr
    Fixed at 10^3 yr for all triplets; authors note it is tunable and will be made parameter-dependent in future work (§5.2).
  • η_s2 mini-disc suppression threshold = 0.01
    Eddington-ratio threshold below which both mini-discs may accrete; set to 0.01 for this work (§5.3).
  • preferential-accretion polynomial coefficients p0,p1,p2 = 0.8054, 0.9840, 0.3818
    Adopted from Duffell et al. (2020) fits and used to partition CBD inflow (§5.3); not re-fit here but control mass-ratio evolution.
  • resolution-sphere multiplier (4Δx or 2Δx) = 4Δx (fiducial); 2Δx (case C)
    Defines when pairs/triplets enter subgrid; reduced to 2Δx for compact test C, affecting when the chaotic channel activates.
axioms (5)
  • domain assumption A hierarchical triplet becomes chaotic once the Mylläri et al. (2018) instability proxy Q_st < Q_st,0, after which secular evolution is no longer valid.
    Invoked in §5.1.1 to promote state 4 → state 5; the proxy is taken from stellar dynamics literature without re-derivation for gas-rich galactic nuclei.
  • domain assumption Chaotic three-body outcomes are fully captured by the seven-channel Bonetti et al. (2018) probability tables binned only on (M_pri, q_in, q_out) with q_out ≤ 1.
    Core of §5.2; the model does not integrate resonant dynamics and leaves M_out > M_pri+M_sec to future work.
  • ad hoc to paper Ejection velocity may be set to the local escape speed from the resolution sphere rather than drawn from N-body scattering data.
    Stated explicitly in §5.2; simplifies bookkeeping but severs the link to the calibrated kick distribution.
  • domain assumption A bound MBHB is always embedded in a circumbinary disc whose torques follow the adopted viscous-drag prescription, independent of local gas geometry.
    Used throughout stage-2 hardening (§3.3, §6.3); authors flag it as possibly optimistic for inclined systems.
  • standard math Standard Peters GW, Quinlan stellar hardening, and Lousto–Zlochower recoil formulae remain valid once the subgrid binary is formed.
    Classical inputs reused without modification (§3.3, §5.4).
invented entities (2)
  • Seven-state subgrid classification (single, stage-1 pair, bound binary, three stage-1, hierarchical triplet, active triplet, pending ejection) no independent evidence
    purpose: Provides a finite-state machine that reduces arbitrary sink encounters to at most three active MBHs and routes them into the Bonetti channel.
    Bookkeeping construct introduced in §4 and Table 1; no independent physical evidence required beyond algorithmic closure.
  • Two-channel CBD reservoir (mini-disc supply vs gap/clump reservoir) with feedback-regulated mini-disc suppression no independent evidence
    purpose: Subgrid representation of cavity-edge lump and preferential accretion when the disc is unresolved.
    Motivated by Farris/Duffell hydro results but implemented as a new RAMCOAL partition (§5.3); falsifiable only via future resolved CBD comparisons.

reviewed 2026-07-11 · how reviews work

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Cite this review

Pith. "Pith review of Set them free: extending RAMCOAL to model massive black hole triplets in hydrodynamical simulations of galaxies." pith.science (2026). https://pith.science/paper/DC5BZSEY

@misc{pith2026260704121,
  author       = {Pith},
  title        = {Pith review of: Set them free: extending RAMCOAL to model massive black hole triplets in hydrodynamical simulations of galaxies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DC5BZSEY}},
  note         = {Machine review of arXiv:2607.04121}
}
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read the original abstract

Massive black hole binaries (MBHBs), and the higher-order multiples produced by repeated galaxy mergers, spend part of their lives in dynamical regimes that cosmological simulations cannot resolve, even though these regimes set their merger delays, spins, recoils, and host-galaxy context. We extend the RAMCOAL framework to follow such subgrid massive black hole triplets directly within hydrodynamical galaxy simulations. As in the original staged binary model, the black holes start as sink particles, pass through a dynamical-friction phase, and settle into bound binaries that harden through stellar scattering, gas torques, circumbinary-disc coupling, and gravitational-wave emission. When a hierarchical triplet becomes chaotic, RAMCOAL maps the encounter onto a library of three-body outcomes from direct N-body experiments and updates the surviving system, following the resulting mergers, exchanges, and ejections together with the accretion and spin evolution of each black hole. Using isolated-galaxy tests with contrasting geometries, we show that the encounter geometry alone can change which pair finally merges, and after how long. We demonstrate the first triplet MBH dynamical evolution all the way to coalescence inside a live hydrodynamical simulation. This establishes an end-to-end capability to predict triplet-driven MBH coalescences self-consistently coupled to the evolving host galaxy. Because each MBHB coalescence carries its environmental history through the subgrid phase, RAMCOAL offers a route toward merger catalogues that link the gravitational-wave signatures of coalescing black holes to the galaxies in which they form.

Figures

Figures reproduced from arXiv: 2607.04121 by Kunyang Li, Marta Volonteri, Ricarda S. Beckmann, Yohan Dubois.

Figure 1
Figure 1. Figure 1: Three-stage RAMSES–RAMCOAL treatment of MBH-pair evolution, progressing from left to right. To read the schematic, follow the stage numbers in increasing order (Stage 0 → Stage 1 → Stage 2), tracking a single pair from left to right as its separation shrinks toward coalescence. In stage 0 (left), above the resolution sphere, the MBHs are standard RAMSES sink particles whose positions are not pinned to pote… view at source ↗
Figure 2
Figure 2. Figure 2: Flowchart of the RAMCOAL chaotic three-body classification and outcome pipeline, read as five numbered steps (left) with two reference legends (right): the seven physical states and the seven interaction outcomes. (1) Candidate pairs are identified from the active MBH systems at that time step: inactive or already-consumed MBH (MBH that already belonging to another subgrid system) are dropped, pairs are te… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic representation of the stage-2 circumbinary-disc (CBD) accretion partition. The primary (𝑀1) and secondary (𝑀2) are each sur￾rounded by a mini-disc and are embedded in a circumbinary gas disc with a central hollow (cavity), while an overdense clump orbits near the cavity edge. In the subgrid model, the subgrid inflow 𝑀¤ inner is divided into a cavity-edge / CBD component, 𝑀¤ gap, and a mini-disc-f… view at source ↗
Figure 4
Figure 4. Figure 4: compares the outcome frequencies recovered from these draws (coloured bars, with Poisson error bars) against the tabu￾lated probabilities of Bonetti et al. (2019) for the corresponding (𝑀pri, 𝑞in, 𝑞out) cell (grey bars). For this configuration the selec￾tor is dominated by the two single-hole ejection channels: ejection of MBH2, which leaves the MBH1–MBH3 binary (outcome 4, 44/100), and ejection of MBH1, w… view at source ↗
Figure 5
Figure 5. Figure 5: Gas-density slices through the mid-plane of the galaxy disc with MBH trajectories for triplet test case A (in-plane initial orbital configuration). The four panels each capture a qualitatively distinct phase of the evolution rather than a continuous time sequence, at 𝑡 = 0.1, 0.221, 0.858, and 1.604 Gyr (the last zoomed to the central 1 kpc). The coloured curves are the projected sink-particle trajectories… view at source ↗
Figure 6
Figure 6. Figure 6: Pairwise-separation evolution for triplet test case A with the in-plane initial orbital configuration. The figure presents the resolved stage 0 pairwise separations, the subsequent RAMCOAL stage 1 and stage 2 evolution of the selected inner pair, and the labelled coalescence time. The curve labelled “xsink tracer 1 – MBH3” is the separation between MBH3 and the centre of mass of the inner MBH1–MBH2 pair: o… view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of the hardening terms for triplet test case A. The diagnostic displays the loss-cone (LC), viscous-disc (VD), and gravitational-wave (GW) contributions to the semi-major-axis and eccentricity evolution [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Subgrid orbital elements for the inner binary in triplet test case A. The figure presents the stage 2 semi-major axis and eccentricity evolution and marks the coalescence-scale values reported by the diagnostic. binary disk always increase the eccentricity in this case, since the eccentricity is lower than the attractor eccentricity as shown in a various CBD simulations (Zrake et al. 2021; D’Orazio & Duffe… view at source ↗
Figure 9
Figure 9. Figure 9: Accretion and mass-ratio evolution for triplet test case A. The upper panel shows the component accretion rates of MBH1 (blue) and MBH2 (orange), with accretion rate at each time step shown in light shading tracks. The solid smoothed tracks overplotted show the centred running mean over a 20 Myr window with each point averages the rate within ±0.01 Gyr of that time, i.e. a local running mean rather than an… view at source ↗
Figure 10
Figure 10. Figure 10: Spin-magnitude evolution for triplet test case A. The vertical dashed line marks the transition from the resolved stage 0 to the subgrid stage 1/2 evolution. Spin of the secondary MBH becomes larger than that of the primary due to preferential accretion which leads to more repeat growth of spin. which 𝑑𝑎∗/d𝑡 = 0 in this regime is small but positive, and the MBHs sit below this threshold, so the net torque… view at source ↗
Figure 11
Figure 11. Figure 11: Gas-density slice snapshots with MBH tracks for triplet test case B, using the inclined initial orbital configuration. Top row present the 𝑧-direction slices for the three selected snapshots, while the bottom row display the corresponding 𝑥-direction slices for the same snapshots.The black cross in the last panel at 𝑡 = 199 Myr marks the starting position of the stage 1 RAMCOAL subgrid evolution, and the … view at source ↗
Figure 12
Figure 12. Figure 12: Gas-density slice snapshots with subgrid MBH tracks for test case B. The cyan track show the orbit of MBH3 around the center of mass of the binary evolving in stage 1 of RAMCOAL, and the magenta track is that for MBH1. These panels illustrate the subgrid evolution of the MBH pair after they enter the RAMCOAL regime following the last panel of [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Pairwise-separation evolution for triplet test case B with the inclined initial orbital configuration. The figure presents the resolved stage 0 pairwise separations, the stage 1 and stage 2 evolution of the MBH1–MBH3 inner binary, the wider-orbit tracer of the surviving MBH2, and the labelled coalescence time; it should be read together with the slice-with-track sequence in [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 14
Figure 14. Figure 14: Evolution of the hardening terms for triplet test case B. The diagnostic displays the loss-cone (LC), viscous-disc (VD), and gravitational-wave (GW) contributions to the semi-major-axis and eccentricity evolution for the inclined case [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Subgrid orbital elements for the final binary in triplet test case B. The figure traces the stage 2 semi-major axis and eccentricity evolution and labels the MBH1–MBH3 coalescence. Loeb 2007), and the Bonetti post-Newtonian suite that RAMCOAL follows statistically finds that ∼ 20-30% of otherwise stalled binaries coalesce within a Hubble time (Bonetti et al. 2018). Because RAM￾COAL does not resolve the ch… view at source ↗
Figure 17
Figure 17. Figure 17: Spin-magnitude evolution for triplet test case B. The diagnostic separates the resolved stage 0 spin history from the later RAMCOAL stage 1/2 evolution and marks the onset of stage 2 [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Evolution of the subgrid spin growth for MBH1 and MBH3 after the binary enters stage 1. The plotted quantity is log10 (𝑎∗ − 𝑎∗,0 ), where 𝑎∗,0 is the spin magnitude of each MBH at the onset of stage 1. The vertical dashed line marks the transition from stage 1 to stage 2 [PITH_FULL_IMAGE:figures/full_fig_p020_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Spin–gas-disc alignment for triplet test case B. The panel follows cos 𝜃spin−disk, the cosine of the inclination between each MBH spin and the local gas-disc angular-momentum direction, for MBH1 (red) and MBH3 (blue) as a function of time, with the stage 1 and stage 2 transitions marked. The inset zooms in on the final approach to coalescence [PITH_FULL_IMAGE:figures/full_fig_p020_19.png] view at source ↗
Figure 21
Figure 21. Figure 21: Separation evolution of the three MBHs in triplet test case C. The system passes from resolved RAMSES evolution (grey shaded) to three subgrid stage 1 MBHs (blue shaded, RAMCOAL state 3, ∼ 99.5 Myr), then to a hierarchical triplet: a bound inner pair (MBH2 and MBH3) plus an outer MBH1 (orange shaded, RAMCOAL state 4, ∼ 104 Myr), and finally to a chaotic triplet interaction at ≃ 108.8 Myr (star) by the end… view at source ↗
Figure 23
Figure 23. Figure 23: Component accretion-rate evolution for triplet test case C (MBH1 green, MBH2 blue, MBH3 magenta). The left axis gives the absolute rate 𝑀¤ and the right axis the Eddington ratio 𝑀¤ /𝑀¤ Edd; raw histories are shown in light shading and smoothed running means with a running mean window of 0.18 Myr as solid lines. Accretion remains strongly sub-Eddington through￾out. in a minority a new and typically more ec… view at source ↗
Figure 24
Figure 24. Figure 24: Mass-ratio evolution for triplet test case C, shown as frac￾tional deviations from the initial values: the inner-pair ratio 𝑞in = min(𝑀2, 𝑀3 )/max(𝑀2, 𝑀3 ) (initial 𝑞0 = 1) and the outer ratio 𝑞out = 𝑀1/(𝑀2 + 𝑀3 ) (initial 𝑞0 = 0.5) [PITH_FULL_IMAGE:figures/full_fig_p022_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Spin-magnitude evolution for triplet test case C (MBH1 green, MBH2 blue, MBH3 magenta). The spins grow slowly, in steps tied to the accretion episodes, and remain small (𝑎∗ ∼ 10−4 ). few gigayears after the galaxy merger while the remaining binary persisted at parsec separations. RAMCOAL differs from KETJU in that it does not resolve the few-body dynamics directly but instead carries calibrated subgrid pr… view at source ↗

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This paper was first reviewed by grok-4.5 on July 11, 2026.