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Quasi-normal modes of slowly-rotating Johannsen black holes

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Pith's one-line read This paper claims that, under the assumptions of the Einstein equations, slow rotation, and small deformation parameters, the quasi-normal modes of a Johannsen black hole are dominated by a single deformation parameter, $\alpha_{13}$…

desk verdict Solid incremental contribution: new deformed Regge-Wheeler/Zerilli equations and an α13 ringdown mapping worth having, held back mainly by an unverified master-equation reduction and an extrapolated data constraint. read the letter →

arxiv 2412.08205 v2 pith:BFTG3VR7 submitted 2024-12-11 gr-qc astro-ph.HE

Yuhao Guo , Swarnim Shashank , Cosimo Bambi This is my paper · ORCID
classification gr-qcastro-ph.HE MSC 83C5783C3583C25 PACS 04.25.Nx04.70.Bw04.30.-w95.30.Sf
keywords quasi-normalmodesJohannsenmetricblackholeperturbationtheoryslow-rotationapproximationringdowntestsofgeneralrelativitydeformationparametersGW170104
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Gravitational-wave ringdowns promise a clean test of general relativity, but to interpret them one needs the vibration spectrum of the black hole that emits them. This paper computes the quasi-normal modes of slowly rotating black holes in the parametric Johannsen metric, a theory-agnostic family of spacetimes that reduces to Kerr when all deformation parameters vanish. Working to first order in the spin and in the leading deformation parameters, and assuming the Einstein equations hold for the perturbations, the authors find that one parameter, $\alpha_{13}$, shifts the modes far more than the others ($\alpha_{22}$, $\alpha_{52}$, and $\epsilon_3$), and they provide a two-term fitting formula, $\mathrm{Re}(\omega)=\mathrm{Re}(\omega_{\mathrm{Teu}})+P_0\alpha_{13}+P_1\alpha_{13}a$ plus the analogous imaginary part, valid for $a_*<0.4$. Applied to the LIGO–Virgo event GW170104, the formula turns measured ringdown frequency and damping deviations into the bound $\alpha_{13}=-0.5^{+2.1}_{-4.0}$ at 90% confidence, which is consistent with general relativity.

What carries the argument

The machinery is a pair of one-dimensional master equations of Regge–Wheeler and Zerilli type, $\partial^2\Psi/\partial r_*^2+V\Psi=0$, with effective potentials computed to first order in the spin $a$ and in the deformation parameters $\alpha_{13}$, $\alpha_{52}$, and $\epsilon_3$; the potentials are obtained by linearizing the Einstein equations about the slow-rotating Johannsen metric in the Regge–Wheeler gauge. The modes are found by the direct integration method, which matches an ingoing solution at the horizon to an outgoing solution at infinity by a Wronskian determinant condition, retaining the full horizon frequency $\omega_H$ even though the equations themselves are truncated. A correction factor $C=\omega_{\rm Teu}/\omega_{\rm Kerr}$ removes the slow-rotation error from the Kerr baseline, and the fitting formula $\mathrm{Re}(\omega)=\mathrm{Re}(\omega_{\rm Teu})+P_0\alpha_{13}+P_1\alpha_{13}a$ with tabulated coefficients $P_0$, $P_1$, $Q_0$, $Q_1$ converts the corrected frequencies into a direct mapping from ringdown data to $\alpha_{13}$.

What would settle it

Recompute the axial and polar quasi-normal modes at $a_*=0.4$ with a method that does not use the slow-rotation expansion, such as a Teukolsky-like equation for beyond-Kerr spacetimes, and check whether the $\alpha_{13}$ shift follows $P_0+P_1a$ to within the paper's quoted 0.2%–0.3% error; alternatively, repeat the perturbation calculation while keeping the perturbation of the effective stress–energy tensor that the Johannsen background requires and look for frequency shifts comparable to the $\alpha_{13}$ contribution.

Watch

Extended reading notes

Core claim

The central claim is that the quasi-normal modes of a Johannsen black hole, computed from the Einstein equations in the slow-rotation, small-deformation limit, are dominated by the deformation parameter $\alpha_{13}$. The axial and polar perturbation problems separate in this limit, giving deformed Regge–Wheeler and Zerilli master equations whose effective potentials contain $\alpha_{13}$, $\alpha_{52}$, and $\epsilon_3$ but not $\alpha_{22}$ at leading order. Because the slow-rotation approximation degrades the Kerr part of the answer, the authors calibrate the Kerr baseline against the exact Teukolsky result and fold the mismatch in as a correction factor, so the final frequency is $\omega_{\rm corr}=\omega_{\rm Non\text{-}Kerr}+(C-1)\omega_{\rm Kerr}$. For the fundamental $l=2$ and $l=3$ modes the result takes the form of a fitting formula with the reported coefficients, and the paper argues that, in this regime, a ringdown observation translates directly into a measurement of $\alpha_{13}$.

Load-bearing premise

The load-bearing premise is that the Johannsen metric, which is not a vacuum solution of the Einstein equations, can still be treated as a vacuum background whose gravitational perturbations obey the source-free Einstein equations, so that perturbations of the effective stress–energy tensor are neglected.

Editorial extensions

If this is right

  • A ringdown detection from any low-spin remnant ($a_*<0.4$) can be converted directly into a measurement of $\alpha_{13}$ through $\alpha_{13}=f^{\rm GR}_{lmn}\delta f_{lmn}/(P_0+P_1 a)$, giving a theory-agnostic test of the Kerr hypothesis from one gravitational-wave event.
  • The {2,2,1} ringdown of GW170104 yields $\alpha_{13}=-0.5^{+2.1}_{-4.0}$ at 90% confidence, consistent with general relativity and with the inspiral-based constraint on the same event.
  • At leading order in the slow-rotation and small-deformation expansion, $\alpha_{22}$ drops out of the perturbation equations and $\alpha_{52}$, $\epsilon_3$ have weaker effects, so the ringdown test is effectively one-parameter in this regime.
  • The tabulated coefficients for the fundamental and first-overtone modes with $l=2$ and $l=3$ keep fit errors below 0.3%, so the same formulas can be dropped directly into future ringdown templates.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the additivity of the correction factor in $\omega_{\rm corr}=\omega_{\rm Non\text{-}Kerr}+(C-1)\omega_{\rm Kerr}$ is not exact near $a_*=0.4$, the inferred $\alpha_{13}$ from an event at the edge of the validity range would carry a systematic bias; recomputing the modes without factorizing the slow-rotation error would quantify that bias.
  • The no-source assumption means the computed modes describe the effective Einstein vacuum of the metric rather than any concrete theory; in a theory that really produces the Johannsen background, perturbations of its matter sector would generically shift the modes by an amount comparable to the $\alpha_{13}$ effect, so the quoted constraint should be read as a null test of the parametric family.
  • The paper leaves the polar sector covered only by an assumption of near-isospectrality; because deformations generically break the equality of axial and polar spectra, measuring a difference between them in a loud ringdown would be a distinctive signature that this formalism is built to predict but has not yet computed.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 8 minor

Summary. The manuscript computes the quasi-normal modes (QNMs) of gravitational perturbations of slowly rotating Johannsen black holes, a parametrized non-Kerr metric designed for agnostic tests of GR. The authors linearize the Einstein equations about the Johannsen background to first order in the spin a and in the leading deformation parameters alpha_13, alpha_22, alpha_52, and epsilon_3, assume that the effective stress-energy supporting the background has no perturbations, and derive deformed Regge-Wheeler and Zerilli master equations whose effective potentials are given in Eq. (12) and Appendix A. Using the direct integration method, they compute the fundamental l = 2, m = 2 Kerr QNMs as a function of spin, introduce a correction factor calibrated against Teukolsky frequencies (Eqs. 18-19), and find that alpha_13 produces the largest frequency shifts among the four parameters (alpha_22 does not enter at the retained order). They fit the alpha_13 dependence with the linear formula Eq. (20), tabulate coefficients for all m of l = 2 and l = 3, and apply the l = 2, m = 2 formula to the ringdown of GW170104 using pyRing, finding alpha_13 = -0.5 +2.1 -4.0 at 90% CL, consistent with general relativity. The central claims are the alpha_13-dominance result, the fitting formula, and the GW170104 constraint; the central assumptions are the frozen effective-source approximation and the additivity of the Kerr correction factor.

Significance. If the leading-order approximations and the frozen-effective-source assumption are accepted, the paper is a useful and transparent contribution to agnostic strong-field tests of GR: it delivers the first QNM fitting formula for slowly rotating Johannsen black holes, with explicit master equations (Eq. 12 and Appendix A), a calibrated Kerr-Teukolsky comparison (Fig. 1), and a complete demonstration pipeline from theory to a ringdown constraint on alpha_13 using public LVK data and pyRing. The paper practices good epistemic hygiene: the main assumptions are stated in the text (Section III; Concluding Remarks), the Kerr limit is validated separately, and the headline constraint (consistent with GR) is falsifiable. The significance is nevertheless qualified by the correctness risks detailed below: the physical interpretation of the modes relies on an unverified linearized-consistency condition, the correction-factor additivity is untested for non-Kerr shifts, the l = 3 coefficient table contains a factor-of-10 error, and the astronomical constraint extrapolates the fit beyond its calibration domain.

major comments (4)
  1. [Section III, Eq. (12), and Appendix A] The perturbation scheme rests on the statement in Section III that, within the authors' agnostic approach, 'we assume that there are no perturbations on the right hand side,' i.e., delta-T = 0 and delta-G[h] = 0 on a background with nonvanishing G0. For such a background, the linearized contracted Bianchi identity reads nabla0_mu delta-G^{mu nu} + delta-Gamma^mu_{mu rho} G0^{rho nu} + delta-Gamma^nu_{mu rho} G0^{mu rho} = 0, so a solution of the reduced master equations (12) and (A1) can be a linearized solution of the full Einstein-matter system only if it additionally satisfies delta-Gamma dot G0 = 0. No check of this constraint is reported, and no matter model that would keep the effective T0 fixed under the perturbed connection is specified. Because G0 begins at first order in the deformation parameters, the unverified constraint is at the same order as the QNM shifts reported here. At a = 0 the axial sector of a spherical non-vacuum background does decouple, but the polar sector in general couples to matter perturbations, and the slow-rotation axial/polar separation relies on the same assumption away from spherical symmetry. The computed frequencies are therefore, strictly, frequencies of a reduced scalar equation on a fixed background; their status as gravitational-wave QNMs of an Einstein-matter system, and the interpretation of the Section VI constraint on alpha_13, depends on the frozen-source assumption holding. I recommend either numerically verifying that the direct-integration solutions satisfy the full set of linearized Einstein equations and the Bianchi constraint, or reframing the results as a phenomenological fixed-background computation and stating that limitation prominently in the abstract and conclusions.
  2. [Section V.B, Eqs. (18)-(19)] The correction-factor construction omega_corr = omega_Non-Kerr + (C - 1) omega_Kerr is exact by construction in the Kerr limit, so the validation in Fig. 1 tests the Kerr baseline but not the additivity assumption that the slow-rotation error of the Kerr part is independent of the deformation shift. The shift delta-omega_alpha = omega_Non-Kerr - omega_Kerr is computed with the same scheme whose uncorrected Im(omega) deviates from Teukolsky by about 10% at a = 0.3 and 20% at a = 0.4 (Fig. 1), whereas the alpha_13-induced change in Im(omega) is only about 4% of the base value at alpha_13 = 0.3 (Table I). The neglected O(a^2 alpha_13) and O(alpha_13^2) terms in the shift are not removed by the correction factor, so the expectation in Section V.B that the corrected non-Kerr frequencies are more accurate is an unvalidated premise. A concrete check would be a comparison of the corrected l = 2, m = 2 frequencies against one of the Teukolsky-like methods cited in Refs. [60-62] at a few points in the (a, alpha_13) plane; without such a comparison, or a quantitative bound on the neglected terms, the fitting coefficients and the GW170104 constraint carry an unquantified systematic error.
  3. [Appendix D, Table VII] Table VII is internally inconsistent with the numerical data in Table VI by a factor of approximately 10 in the P0 column. For example, at a = 0 and m = 0, Table VI gives Re(omega) = 0.5994 at alpha_13 = 0 and 0.5970 at alpha_13 = 0.1, i.e., dRe/dalpha_13 is about -0.024, whereas Table VII lists P0 = -0.0024; the l = 2 tables are mutually consistent (Table IV's P0 approximately -0.0195 for m = -2 reproduces Table III), so the discrepancy is not a convention issue. If Eq. (20) is used with the published l = 3 coefficients, the predicted alpha_13 shift is an order of magnitude too small. The P1, Q0, and Q1 columns of Tables VII and VIII appear consistent with Table VI, so the error seems localized to P0, but as printed the l = 3 real-part fitting formula, which is part of the paper's advertised product for l = 3 fundamental modes, is wrong and must be corrected.
  4. [Section VI and Table I] The GW170104 analysis extrapolates the fitting formula beyond its calibrated domain in two ways. First, the posterior final spin a = 0.31 +0.45 -0.28 reaches values up to about 0.76, while the fit is calibrated for a in [0, 0.3] and the abstract claims validity only for a* < 0.4; the paper acknowledges this only with the phrase 'exceeds slightly the range allowed with our methods.' Second, the reported 90% interval alpha_13 in [-4.46, 1.66] extends far outside the extended fit range alpha_13 in [-1, 1], where the O(alpha_13^2) and O(a^2 alpha_13) terms dropped in Eq. (20) are no longer small. The leakage is visible within the paper's own numbers: P1 changes from 0.0025 to 0.0166 when the alpha_13 range is widened from [-0.3, 0.3] to [-1, 1], showing that the coefficients are sensitive to the fitting window. Because the quoted uncertainty on alpha_13 does not include the truncation systematic, the constraint should be reported as an exploratory estimate with an explicit error budget for the extrapolation, or the posterior should be restricted to the calibrated region.
minor comments (8)
  1. [Throughout] Typos and wording: 'Zerlini' should be 'Zerilli' in Section I; the Section III heading reads 'MASTER EQUA TIONS'; Section IV.B uses 'DNSolve' instead of 'NDSolve'; Section V.C has 'The errors is not more than 0.2%'; Appendix C has 'relatee' instead of 'relates'.
  2. [Table III] In the a = 0 row at alpha_13 = 0.2, the m = -2 entry is 0.3798 while the other four m entries are 0.3698 or 0.3699; the 0.3798 value breaks the monotonic trend and appears to be a typo.
  3. [Abstract and Section V.C] The abstract states that the fitting formula is valid for a* < 0.4, but Section V.C calibrates the fit for a in [0, 0.3], and Fig. 1 shows the uncorrected imaginary part has about 20% error already at a = 0.4; the claimed validity range should be reconciled with the calibration range.
  4. [Abstract and Section V.B] The abstract's claim that alpha_13 has a stronger impact on the QNMs than alpha_52 and epsilon_3 is supported only by visual comparison of Figs. 2-4; please provide quantitative values (for example, dRe/dalpha and dIm/dalpha at a = 0 for each parameter) so that the headline claim is numerically documented.
  5. [Section V.C and Appendix D] The overtone numbering should be clarified: the paper labels the fundamental mode {l, m, 1}, while the LVK 'Kerr 220' convention of Ref. [4] is (l, m, n) = (2, 2, 0), and Section V.C states that Appendix D reports coefficients for the first overtone mode although the tables contain the fundamental-mode data; please align the two conventions.
  6. [References] Ref. [45] lists arXiv:1501.02809 for T. Johannsen, Phys. Rev. D 88, 044002 (2013); that arXiv identifier dates from January 2015 and appears to be incorrect, so please verify it. Several references (e.g., [3] and [33]) are missing titles or author initials in the bibliography.
  7. [Section VI, Eq. (22)] The mapping from delta-f and delta-tau to alpha_13 in Eq. (22) neglects O(a^2) and O(alpha_13^2) corrections, but the posterior on a is broad and enters through (P0 + P1 a) and (Q0 + Q1 a); a sentence stating that these denominator corrections are neglected relative to P0 and Q0 would make the approximation explicit.
  8. [Section VI] The extended fitting coefficients for alpha_13 in [-1, 1] (P0 = -0.0205, P1 = 0.0166, Q0 = 0.0136, Q1 = 0.0021) are quoted without fit residuals; please report the maximum error of the extended fit, since the stated 0.2% (l = 2) figure refers to the restricted calibration range.

Circularity Check

1 steps flagged · score 2.0 of 10

No load-bearing circularity: the QNM derivation is self-contained and benchmarked against Teukolsky Kerr results; only a by-construction ranking statement and a non-load-bearing self-citation keep the score slightly above zero.

  1. other [Abstract and Section V.B (Non-Kerr spacetime), with the truncation stated in Section III after Eq. (12)]
    "We find that the deformation parameter α13 has a stronger impact on the quasi-normal modes than the other leading order deformation parameters (α22, α52, and ϵ3). ... We note that the deformation parameter α22 does not appear in the effective potential within our approximations."

    The ranking of α13 over α22 is fixed by the paper's first-order truncation scheme rather than by the QNM calculation itself. In the slowly-rotating metric (Eq. 3), α22 enters only through the gtϕ combination aΣ sin^2θ(A1A2 r^2−Δ), which is O(a α22) and is dropped when terms of second order in spin and deformation parameters are neglected. The paper itself states that α22 does not appear in the effective potential. Thus the abstract's comparative claim reduces, by construction, to the truncation choice. This statement is not load-bearing: the fitting formula and the GW170104 constraint depend only on the computed α13 dependence, so this is a minor overstatement rather than a circular derivation of the main results.

full rationale

The central derivation computes QNM frequencies by linearizing the Einstein equations around the Johannsen background, assuming no perturbations of the effective stress-energy tensor, separating axial and polar perturbations, and solving the resulting Regge-Wheeler/Zerilli-type equations by direct integration. The slow-rotation Kerr limit is validated against Teukolsky results, and the correction factor C is anchored to that external benchmark, so the calibration is not circular. The fitting coefficients P0, P1, Q0, Q1 are fit to the paper's own numerical QNM values and then used to convert independent pyRing posteriors for δf and δτ into a constraint on α13; this is a standard fitting-plus-inference pipeline, not a fitted parameter renamed as a prediction. The agreement with the earlier inspiral constraint (Ref. [6]) is a consistency comparison and is not load-bearing. The only mild issue is the abstract's ranking of α13 against α22, which is a by-construction artifact of the first-order truncation; it does not affect the QNM frequencies or the GW constraint. The frozen effective-source assumption (δT=0) is a physical modeling choice that may be questioned, but it is an input assumption rather than a circular step.

Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The central calculation relies on the parametric Johannsen metric, the ad hoc assumption that perturbations obey vacuum Einstein equations, the slow-rotation and small-deformation truncation, an additivity assumption for the Kerr correction factor, and standard BH perturbation theory. The only genuine fit parameters are the polynomial coefficients in the fitting formula and the deformation parameter alpha_13 estimated from GW170104.

free parameters (3)
  • P0, P1, Q0, Q1 for l=2, m=2 fitting formula = -0.0197, 0.0025, 0.0130, -0.0095 (Table I)
    Coefficients of the linear fitting formula Re(omega)=Re(omega_Teu)+P0 alpha_13+P1 alpha_13 a, fit with numpy.polyfit to the authors' numerical QNM data over a in [0, 0.3] and alpha_13 in [-0.3, 0.3].
  • Extended P0, P1, Q0, Q1 for alpha_13 in [-1, 1] = -0.0205, 0.0166, 0.0136, 0.0021 (Section VI)
    Refit over the extended range used for GW170104; fit error not reported.
  • alpha_13 = -0.5 +2.1 -4.0 (90% CL, combined constraint)
    The Johannsen deformation parameter constrained from GW170104 ringdown; this is the target parameter of the paper, not an input, but it is a free parameter of the model fitted to data.
assumptions (6)
  • domain assumption Johannsen metric with leading deformation parameters alpha_13, alpha_22, alpha_52, epsilon_3 describes the background spacetime.
    The metric is a parametric extension of Kerr from Ref. [45], not a solution of a specific gravity theory.
  • ad hoc to paper Gravitational perturbations satisfy the vacuum Einstein equations with no perturbation of the effective stress-energy tensor.
    Section III: 'Within our agnostic approach, we assume that there are no perturbations on the right hand side.' This is needed to define QNMs but is not derived.
  • domain assumption Only terms linear in spin a and in the deformation parameters are kept.
    Slow rotation approximation used to separate axial and polar perturbations; stated in Sections II and III.
  • domain assumption Axial and polar perturbations have approximately the same QNM spectrum.
    Section III: 'we study the axial perturbations assuming that, at first approximation, our results hold even for polar perturbations.'
  • ad hoc to paper The correction factor C = omega_Teu / omega_Kerr computed for Kerr can be applied additively to the non-Kerr frequency shift.
    Section VA-B: Eq. (19) assumes the slow-rotation error in the Kerr part is independent of the deformation shift.
  • standard math Standard results of BH perturbation theory, including the Regge-Wheeler gauge, tortoise coordinates, and the Wronskian condition for QNMs.
    Used throughout Sections III-IV; standard in the field.
invented entities (1)
  • Effective stress-energy tensor supporting the Johannsen metric
    purpose: Makes the non-vacuum Johannsen metric formally satisfy the Einstein equations with a matter source.
    No physical matter field is specified; the paper sets perturbations of this tensor to zero. It is a bookkeeping device, not a physical entity.

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Cite this review

Pith. "Pith review of Quasi-normal modes of slowly-rotating Johannsen black holes." pith.science (2026). https://pith.science/paper/BFTG3VR7

@misc{pith2026241208205,
  author       = {Pith},
  title        = {Pith review of: Quasi-normal modes of slowly-rotating Johannsen black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BFTG3VR7}},
  note         = {Machine review of arXiv:2412.08205}
}
abstract

The detection of gravitational waves with ground-based laser interferometers has opened a new window to test and constrain General Relativity (GR) in the strong, dynamical, and non-linear regime. In this paper, we follow an agnostic approach and we study the quasi-normal modes of gravitational perturbations of Johannsen black holes under the assumptions of the validity of the Einstein Equations and of low values of the black hole spin parameter and deformation parameters. We find that the deformation parameter $\alpha_{13}$ has a stronger impact on the quasi-normal modes than the other leading order deformation parameters ($\alpha_{22}$, $\alpha_{52}$, and $\epsilon_{3}$). We derive a fitting formula for the fundamental modes with $l=2$ and $l=3$ for the deformation parameter $\alpha_{13}$ valid in the slow rotation approximation ($a_* < 0.4$). Finally, we constrain $\alpha_{13}$ from the event GW170104; within our analysis, we find that the data of GW170104 are consistent with the predictions of GR.

Figures

Figures reproduced from arXiv: 2412.08205 by the authors.

Figure 1
Figure 1. FIG. 1. Comparison between the real part (top left panel) and the imaginary part (top right panel) of the QNM [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. QNM [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. As in Fig [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. As in Fig [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Constraints on [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Posteriors for [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison among radial ring-down waveforms Kerr and Johannsen spacetimes. [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Computing spectral shifts for Johannsen-Psaltis black holes

    gr-qc 2025-12 conditional novelty 6.0 of 10

    Slowly rotating Johannsen–Psaltis black holes have definite-parity quasinormal modes with even/odd frequency shifts split by the deviation parameter, computed here through ℓ=10.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.