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REVIEW 3 major objections 3 minor 34 references

Fractional Schwarzschild-Tangherlini black hole with a fractal event horizon

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read By replacing the second derivative in the Wheeler-DeWitt equation with a Riesz fractional derivative of order $\alpha$, the paper derives a Schwarzschild-Tangherlini black hole with event-horizon dimension $1+\alpha/2$ between 1 and 2 and…

desk verdict The paper relabels D-dimensional Schwarzschild-Tangherlini as a 'fractal' black hole by defining D and G_tilde by fiat, and the one new calculation has an internal inconsistency in the classical limit. read the letter →

arxiv 2506.06031 v1 pith:QJTE7RVY submitted 2025-06-06 gr-qc hep-th

S. Jalalzadeh , H. Moradpour , G. R. Jafari , P. V. Moniz This is my paper
classification gr-qchep-th MSC 83C5783C4526A33 PACS 04.70.Dy04.60.-m
keywords fractionalblackholefractaleventhorizonWheeler-DeWittequationRieszderivativeSchwarzschild-Tangherlinimetricblack-holetemperatureLevyparameterHausdorffmeasure
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

The paper sets out to show that a fractional, non-local version of the Wheeler-DeWitt equation, obtained by replacing the kinetic term with a Riesz fractional derivative of order $\alpha$, turns the ordinary Schwarzschild black hole into a Schwarzschild-Tangherlini black hole whose event horizon is a fractal surface of dimension $D-2=\alpha/2+1$, ranging from 1 to 2. The same construction yields a temperature $T=(D-3)/(4\pi R_S)$ that becomes arbitrarily small as $\alpha\to 0$, so these black holes would radiate far more slowly than ordinary ones and could be nearly eternal. The paper also derives the associated metric, gravitational potential, ground-state remnant mass, and gravitational-wave emission frequencies, all of which depend on the fractional parameter. A sympathetic reader would care because the calculation connects quantum-gravity nonlocality directly to observable thermodynamic and gravitational signatures.

What carries the argument

The carrier of the argument is the fractional Wheeler-DeWitt equation, in which the ordinary second derivative is replaced by the Riesz fractional derivative $(-d^2/dx^2)^{\alpha/2}$, a nonlocal operator defined through the Fourier transform. In momentum space this becomes a fractional harmonic oscillator, whose semiclassical Bohr-Sommerfeld spectrum gives the $\alpha$-dependent mass levels. The link from quantum spectrum to geometry is the adiabatic invariant $I=\int dM/\omega$, which reproduces the entropy-area relation and forces the horizon area to be the Hausdorff measure of a $(D-2)$-dimensional unit sphere, with $D=\alpha/2+3$; the metric then uses the fractal horizon line element. The effective $D$-dimensional gravitational constant $\tilde G$ absorbs the $\beta$-function factors so that the entropy and temperature have the same form as in the integer-dimensional Tangherlini case.

What would settle it

Measure the gravitational-wave ringdown or quasinormal-mode spectrum of a stellar-mass black hole: a $10\,M_\odot$ hole should show the fractional family's prediction (roughly 128 Hz at $D=4$, dropping to 10 mHz or far lower for smaller $D$), so finding a standard 128 Hz peak with no lower-frequency branch would rule out the claimed $\alpha$-dependence. Alternatively, compute the entropy of the fractal metric by standard field-theoretic methods; if it does not equal $\Omega_{(D-2)}R_S^{D-2}/(4\tilde G)$, the central identification fails.

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Extended reading notes

Core claim

The paper's central claim is that applying the fractional Wheeler-DeWitt equation $$\frac12\left(-\frac{$d^{2}$}{$dx^{2}$}\right)^{\$\alpha$/2}\psi+\frac12 $x^{2}$\psi=\frac{$2M^{2}$}{$m_P^{2}$}\psi,\qquad 0<\$\alpha$\le 2,$$ to a Schwarzschild geometry generates, in the semiclassical limit, a mass spectrum whose level spacing depends on $\alpha$. Using the adiabatic-invariant form of the entropy-area relation, the paper then identifies the entropy as $S=\Omega_{(D-2)}R_S^{D-2}/(4\tilde G)$ with $D=\alpha/2+3$, which is exactly the entropy of a Schwarzschild-Tangherlini black hole whose horizon is a $(D-2)$-dimensional fractal sphere with dimension between 1 and 2. The temperature follows from the first law as $T=(D-3)/(4\pi R_S)$, so for fixed mass it decreases as $D$ approaches 3, and the metric and Newtonian potential acquire the corresponding non-integer radial power. The paper concludes that fractionality is not confined to the quantum regime: because the Riesz derivative is nonlocal, the classical geometry itself carries the fractal signature.

Load-bearing premise

The argument rests on treating the non-integer horizon dimension as a genuine fractal surface with the specified Hausdorff measure and line element, and on keeping the entropy-area relation $S=\text{Area}/(4\tilde G)$ with the analytically continued sphere area; if that measure is not the physical one, the fractal horizon and its extremely low temperature do not follow.

Editorial extensions

If this is right

  • For a black hole of fixed mass, the temperature $T=(D-3)/(4\pi R_S)$ falls steeply as $D$ approaches 3, so fractional black holes would evaporate far more slowly and could appear almost eternal.
  • The ground-state remnant mass remains near $0.5\,m_P$ but varies slightly with $D$, with a maximum at $D\approx 3.1238$ and a minimum at $D\approx 3.9258$, so the end state of evaporation is still a Planck-scale remnant.
  • Gravitational-wave emission frequencies from a $10\,M_\odot$ black hole shift from about 128 Hz at $D=4$ to 10 mHz at $D=3.9258$ and to $10^{-277}$ Hz at $D=3.1238$, making the fractional parameter potentially visible in ringdown or inspiral data.
  • The Newtonian potential becomes $V(r)\propto r^{-(D-3)}$, so gravity would deviate from the inverse-square law at scales where the fractional regime dominates, with interstellar data suggesting $2.9<D\le 4$.
  • The heat capacity remains negative for the allowed range $3<D\le 4$, so the fractal black hole is thermally unstable in the same way as the standard Schwarzschild-Tangherlini black hole.

Reading between the lines

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

  • Beyond the paper, if the entropy really is carried by a fractal Hausdorff measure rather than an integer-area sphere, the holographic counting of horizon microstates would need to be revised: information per unit fractal area would not be constant in the usual sense.
  • The same fractional-Wheeler-DeWitt construction could be applied to rotating or charged black holes; the paper announces the Kerr case as future work, and the natural inference is that horizon fractality would modify quasinormal-mode spectra and gravitational-wave echoes beyond the static case.
  • If the fractional parameter acts as an effective spacetime dimension through $D=\alpha/2+3$, then independent probes such as galaxy rotation curves or short-scale gravitational tests could constrain $\alpha$ without ever resolving the horizon.
  • One testable extension would be to compute the stress-energy tensor or the quasinormal spectrum directly from the fractal metric given in the paper; if those calculations disagree with the temperature formula, the assumed Hausdorff measure would be the place to reconsider.
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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

3 major / 3 minor

Summary. The paper studies a fractional generalization of the Wheeler-DeWitt equation for a Schwarzschild black hole, in which the ordinary second derivative is replaced by a Riesz fractional derivative of order α/2. Using Bohr-Sommerfeld quantization in the semiclassical limit, it obtains a mass spectrum Mn, defines an effective spacetime dimension D = α/2 + 3, and then rewrites the entropy, horizon radius, and temperature in a form resembling a D-dimensional Schwarzschild-Tangherlini black hole. The central claim is that the event horizon is a fractal of dimension D-2 between 1 and 2, leading to substantially lower temperatures and longer lifetimes for astrophysical black holes.

Significance. If the central claim were established, the paper would offer a concrete phenomenological window onto fractional quantum gravity: a one-parameter family of black holes with tunable horizon dimension and extremely low Hawking temperatures. The paper has a clear strength: the Bohr-Sommerfeld derivation of the mass spectrum in Eq. (11) is explicit, internally consistent, and reduces to the ordinary Schwarzschild spectrum at α = 2. However, the advertised physical conclusion—a fractal event horizon emerging from the fractional WDW equation—is not actually derived; it is obtained by defining D through Eq. (13) and by choosing the effective gravitational constant in Eq. (16) so that the entropy takes the standard area-law form. The low-temperature prediction is therefore a restatement of the free parameter α rather than an independent consequence of the fractional dynamics. In addition, the ground-state mass formula in Eq. (14) does not reduce to the ordinary D = 4 result, contradicting the text and figure. These issues concern the central claim and materially reduce the significance of the paper in its current form.

major comments (3)
  1. [Section 4, Eqs. (15)–(18)] The entropy and temperature are not derived from the fractional Wheeler-DeWitt equation. The effective gravitational constant G̃ in Eq. (16) is chosen so that Eq. (15) takes the standard D-dimensional area law S = Ω_{D-2} R_S^{D-2}/(4G), and Eq. (17) is then the ordinary Tangherlini radius for that D. Since D itself is defined by fiat in Eq. (13) as α/2 + 3, the temperature T = (D-3)/(4π R_S) in Eq. (18) is a direct reparameterization of the free parameter α, not a prediction of the fractional dynamics. The claim in the abstract and Section 5 that the fractional WDW equation uniquely characterizes a fractal horizon is therefore unsupported.
  2. [Section 4, Eqs. (19)–(20)] The Hausdorff measure in Eq. (19) and the line element in Eq. (20) are imported from Refs. [21] and [22] with no demonstrated connection to the wave function of the fractional WDW equation. The statement that the horizon is a fractal with dimension D-2 = α/2 + 1 is an interpretive label attached to an analytically continued sphere area, not a geometric property shown to follow from the fractional Laplacian. Many different measures on the two-sphere integrate to the same total Ω_{D-2}, so the specific measure (19) and line element (20) do not establish that the physical event horizon has Hausdorff dimension D-2.
  3. [Section 3, Eq. (14) and Fig. 1] Equation (14) does not reproduce the ordinary D = 4 ground-state mass. Evaluating Eq. (14) at D = 4 gives M0 = sqrt(π)/sqrt(Γ(1/3)) mP ≈ 0.813 mP, not the value M0 = mP/2 from Eq. (2) that the text invokes. The stated range 0.4998 ≤ M0 ≤ 0.5234 in the text and Fig. 1 also does not match the formula as written. This quantitative inconsistency affects the claimed connection between the fractional spectrum and the standard Schwarzschild limit and must be corrected or explained.
minor comments (3)
  1. [Eq. (12)] The symbol B in Eq. (12) is not defined in the text; it presumably denotes a Beta function related to Eq. (11), but the expression as printed is ambiguous.
  2. [Section 3, Eq. (13)] The phrase 'spacetime dimension' for D = α/2 + 3 is misleading: this is a parameterization of the fractional order, and the paper never constructs a spacetime manifold with this dimension. The notation should be qualified accordingly.
  3. [Section 4, text after Eq. (18)] The sentence defining heat capacity would benefit from stating explicitly whether C is the usual specific heat dM/dT or the rescaled quantity (1/M)dM/dT; the latter appears to be used, but this is not made clear.

Circularity Check

2 steps flagged · score 7.0 of 10

The fractal event horizon and its low temperature reduce to a reparametrization of the free Lévy parameter α: D is set by fiat in Eq. (13), and G̃ is defined in Eq. (16) so that the entropy and temperature become those of a D-dimensional Tangherlini black hole by construction.

  1. self definitional [Introduction and Section 3, Eq. (13)]
    "these quantities depend on the fractional parameter 0 < α ≤ 2 that determines that the spacetime dimension D is expressed as D ≡ α/2 + 3. ... where we defined D ≡ α/2 + 3, 3 < D ≤ 4."

    The central claimed emergence — a non-integer spacetime dimension and hence a fractal horizon with dimension D−2 = α/2+1 — is not derived from the fractional Wheeler-DeWitt equation. Eq. (13) simply defines D as a function of the free parameter α. All later statements about a horizon of dimension 'between 1 and 2' are a restatement of this definition, so the headline result is equivalent to the input parameter.

  2. fitted input called prediction [Section 4, Eqs. (15)–(18)]
    "Employing the results (11)-(13) in the adiabatic invariant (7), it allows us to present the entropy of the fractional BH as S(Frac) H−B = Ω(D−2)R_S^{D−2}/(4G̃), where ... G̃ = ... is the effective D-dimensional gravitational constant, and R_S = ... is the effective horizon (Schwarzschild–Tangherlin) radius."

    G̃ is defined in Eq. (16), not determined by an independent measurement or by the wave equation, precisely so that the entropy derived from the mass spectrum takes the standard (D−2)-sphere area form of a Tangherlini black hole. R_S and then T_H^(Frac) = (D−3)/(4πR_S) follow from this imposed identification. The advertised arbitrarily low temperature for D near 3 is therefore a direct restatement of choosing α, not an emergent prediction of fractal geometry.

full rationale

The paper's derivation chain has real content: Eq. (11) is a Bohr-Sommerfeld spectrum for the fractional oscillator (8), and the adiabatic-invariant route (7) does produce an entropy function of M and α. The circularity enters when this entropy is relabeled. In Eq. (13) the spacetime dimension D is defined as α/2+3; no argument from the fractional WDW equation fixes D, so the 'non-integer dimension 1<D−2≤2' is a parametrization of the free Lévy parameter, not an emergent prediction. In Section 4 the effective constant G̃ is defined by Eq. (16) so that the entropy takes the standard Tangherlini area form, and R_S and T=(D−3)/(4πR_S) are then exactly the known D-dimensional Tangherlini formulas with the redefined D. The claimed low-temperature behavior for D→3 is therefore a direct restatement of choosing α→0, and the imported Hausdorff measure (19) and line element (20) are asserted from Refs. [21,22] without a link to the wave function. The central advertised result reduces to definitions and a fitted constant, warranting a score of 7; the independent mass-spectrum calculation keeps the paper from being entirely vacuous.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The central result rests on the choice of alpha (free), the fractional Wheeler-DeWitt postulate, and an ad hoc identification of the horizon as a (D-2)-dimensional fractal sphere. The effective gravitational constant and horizon radius are defined to match the standard area law, so the low-temperature conclusion is a reparameterization rather than a prediction.

free parameters (1)
  • alpha (Levy fractional parameter)
    All derived quantities depend on alpha, which is not constrained by any independent measurement. The paper illustrates alpha values corresponding to D = 3.1238 and 3.9258 in Section 3.
assumptions (5)
  • domain assumption The fractional Wheeler-DeWitt equation (8) with the Riesz derivative is a valid quantum gravity description for a Schwarzschild black hole.
    Invoked in Section 3, Eq. (8), without derivation from a more fundamental theory.
  • standard math Bohr-Sommerfeld quantization applies to the fractional oscillator in the semiclassical limit M >> m_P.
    Used to obtain the mass spectrum (11); standard old quantum theory, valid in the large-n limit.
  • domain assumption The entropy of the black hole equals the adiabatic invariant integral dM/omega (Eq. 7).
    Assumed at the end of Section 2; identifies Bekenstein-Hawking entropy with an adiabatic invariant without further justification.
  • ad hoc to paper The event horizon is a (D-2)-dimensional sphere with the Hausdorff measure (19) and line element (20).
    Introduced in Section 4 to justify a 'fractal' area; no derivation from the fractional Wheeler-DeWitt equation.
  • ad hoc to paper The spacetime dimension is defined as D = alpha/2 + 3 (Eq. 13).
    Chosen so that the mass spectrum scaling matches the Tangherlini entropy-area scaling; not derived from first principles.
invented entities (1)
  • Fractal event horizon of non-integer dimension D-2
    purpose: Explains deviations from standard black hole thermodynamics and much lower temperatures.
    The fractal nature is asserted via the Hausdorff measure (19), but no independent observable fixes alpha or the fractal dimension, so the entity has no falsifiable handle outside the model.

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Pith. "Pith review of Fractional Schwarzschild-Tangherlini black hole with a fractal event horizon." pith.science (2026). https://pith.science/paper/QJTE7RVY

@misc{pith2026250606031,
  author       = {Pith},
  title        = {Pith review of: Fractional Schwarzschild-Tangherlini black hole with a fractal event horizon},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QJTE7RVY}},
  note         = {Machine review of arXiv:2506.06031}
}
read the original abstract

We demonstrate that the implementation of the fractional and non-local Wheeler--DeWitt (WDW) equation within the context of Schwarzschild geometry leads to the emergence of a Schwarzschild--Tangherlini black hole (BH), which is uniquely characterized by an event horizon that exhibits fractal properties and is defined by a non-integer dimension that lies in the continuum between the values of 1 and 2. Our calculations further reveal that this intriguing fractional BH may potentially possess a temperature that is substantially lower than that of a conventional BH, thereby suggesting a significant deviation from the expected thermodynamic properties of standard BHs. These remarkable characteristics, which are intrinsically linked to the non-integer dimensionality of the event horizon, likely arise from applying the Riesz fractional derivative as a sophisticated non-local operator, thus introducing fascinating dynamics into the theoretical framework of BH physics.

Figures

Figures reproduced from arXiv: 2506.06031 by the authors.

Figure 1
Figure 1. The ground state mass of a fractional black hole as a function of D in the Planck units, i.e., mP = 1. As [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. The figure schematically represents the standard interpretation of the entropy-area relationship for a black hole [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. A cartoon schematically representing the entropy-area of a fractional black hole. over the space [21] defined by dµH = π D−3 2 Γ( D−3 2 ) | cos θ| D−4 sin θdθdφ, (19) in the spherical coordinates (θ, φ). Also, the event horizon’s line element is given by [22] dl2 H =  cos2 θ + [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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Works this paper leans on

34 extracted references · 21 canonical work pages

  1. [21]

    Tarasov V E 2011 Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media (Springer Science & Business Media)

    work page 2011

  2. [22]

    Varieschi G U 2021 Universe 7 387 (Preprint 2109.02855)

    arXiv 2021

  3. [1]

    Trivedi O, Bidlan A and Moniz P 2024 Phys. Lett. B 858 139074 (Preprint 2407.16685)

    arXiv 2024

  4. [2]

    Generalized Ghost Pilgrim Dark Energy Fractal Cosmology with Observational Constraint

    Bhoyar S R, Ingole Y B and Kale A P 2024 Phys. Scripta 100 015026 (Preprint 2409.14171)

    work page Pith review arXiv 2024

  5. [3]

    Socorro J, Rosales J J and Toledo-Sesma L 2024 Universe 10 192 (Preprint 2402.08169)

    work page Pith review arXiv 2024

  6. [4]

    Socorro J, Juan Rosales J and Toledo-Sesma L 2024 arXiv e-prints arXiv:2405.17350 ( Preprint 2405.17350)

    work page Pith review arXiv 2024

  7. [5]

    El-Nabulsi R A and Anukool W 2023 Chaos Solitons and Fractals 167 113097

    work page 2023

  8. [6]

    7 854 (Preprint 2310.09464)

    de Oliveira Costa E W, Jalalzadeh R, da Silva Junior P F, Rasouli S M M and Jalalzadeh S 2023 Fractal Fract. 7 854 (Preprint 2310.09464)

    arXiv 2023

Show all 34 references
  1. [7]

    Jalalzadeh S 2022 Phys. Lett. B 829 137058 (Preprint 2203.09968)

    2022 arXiv

  2. [8]

    44 63–66

    Mukhanov V F 1986 JETP Lett. 44 63–66

    1986

  3. [9]

    Xiang L 2004 Int. J. Mod. Phys. D 13 885–898

    2004

  4. [10]

    Coates A, V¨ olkel S H and Kokkotas K D 2019Phys. Rev. Lett. 123 171104 (Preprint 1909.01254)

    1909 arXiv

  5. [11]

    Wald R M 2010 General relativity (University of Chicago press)

    2010

  6. [12]

    Kunstatter G 2003 Phys. Rev. Lett. 90 161301 (Preprint gr-qc/0212014)

    2003 arXiv

  7. [13]

    Junior P F d S, Costa E W d O and Jalalzadeh S 2023 Eur. Phys. J. Plus 138 862 ( Preprint 2309.12478)

    2023 arXiv

  8. [14]

    Jalalzadeh S, Costa E W O and Moniz P V 2022 Phys. Rev. D 105 L121901 (Preprint 2206.07818)

    2022 arXiv

  9. [15]

    Jalalzadeh S and Vargas Moniz P 2022 Challenging Routes in Quantum Cosmology (World Scientific) ISBN 978-981-4415-06-4

    2022

  10. [16]

    Laskin N 2002 Phys. Rev. E 66 056108 (Preprint quant-ph/0206098)

    2002 arXiv

  11. [17]

    Riesz M 1949 Acta Mathematica 81 1 – 222

    1949

  12. [18]

    Kirichenko E V and Stephanovich V A 2018 Phys. Rev. E 98 052127 (Preprint 1810.08422)

    2018 arXiv

  13. [19]

    27 636–651

    Tangherlini F R 1963 Nuovo Cim. 27 636–651

    1963

  14. [20]

    Khosravi N 2011 Phys. Lett. B 695 343–349 (Preprint 1002.4120)

    2011 arXiv

  15. [23]

    Giusti A 2020 Phys. Rev. D 101 124029 (Preprint 2002.07133)

    2020 arXiv

  16. [24]

    Varieschi G U 2021 Mon. Not. Roy. Astron. Soc. 503 1915–1931 (Preprint 2011.04911) Fractional Schwarzschild-Tangherlini black hole with a fractal event horizon 10

    2021 arXiv

  17. [25]

    Space Sci

    Moradpour H, Jalalzadeh S and Javaherian M 2024 Astrophys. Space Sci. 369 98 ( Preprint 2409.12869)

    2024 arXiv

  18. [26]

    Moniz P V and Jalalzadeh S 2020 Mathematics 8 313

    2020

  19. [27]

    Rasouli S M M, Jalalzadeh S and Moniz P V 2021 Mod. Phys. Lett. A 36 2140005

    2021

  20. [28]

    Dark Univ

    Jalalzadeh R, Jalalzadeh S, Jahromi A S and Moradpour H 2024 Phys. Dark Univ. 44 101498 (Preprint 2404.06986)

    2024 arXiv

  21. [29]

    Jalalzadeh S, da Silva F R and Moniz P V 2021 Eur. Phys. J. C 81 632

    2021

  22. [30]

    6 655 (Preprint 2210.00909)

    Rasouli S M M, Costa E W O, Moniz P V and Jalalzadeh S 2022 Fractal Fract. 6 655 (Preprint 2210.00909)

    2022 arXiv

  23. [31]

    Bidlan A, Moniz P and Trivedi O 2025 Eur. Phys. J. C 85 520

    2025

  24. [32]

    Jalalzadeh S and Moniz P V 2022 Challenging Routes in Quantum Cosmology (WORLD SCIENTIFIC) ( Preprint https://www.worldscientific.com/doi/pdf/10.1142/8540) URL https://www.worldscientific.com/doi/abs/10.1142/8540

    2022 doi

  25. [33]

    Canedo D L, Moniz P and Oliveira-Neto G 2025 Fractal and Fractional 9 ISSN 2504-3110 URL https://www.mdpi.com/2504-3110/9/6/349

    2025

  26. [34]

    Calcagni G 2021 Class. Quant. Grav. 38 165005 [Erratum: Class.Quant.Grav. 38, 169601 (2021)] (Preprint 2106.15430)

    2021 arXiv

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Reviewed August 7, 2026 · model on record in the stance chip above.