Pith. sign in

REVIEW 4 major objections 4 minor 29 references

Holographic Ordering and Negative entropy in Non-equilibrium Euclidean Black Hole Path Integralsl

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A static shell outside a black hole produces an entropy deficit whose horizon-temperature-weighted gradient is the gravitational force, recasting attraction as an ordering phenomenon.

desk verdict The shell action calculation is standard and correct, but the negentropic force law contradicts the paper's own Euclidean result: Eq. (2.17) gives ΔS=0 for the shell, so the Bekenstein bound is strict, not saturated, and F=T_H∇N reduces to the ordinary potential force. read the letter →

arxiv 2507.10450 v2 pith:ISSML2AW submitted 2025-07-14 hep-th gr-qc

Yang An This is my paper
classification hep-thgr-qc
keywords emergentgravitynegativeentropyentropicforceEuclideanpathintegralthinshellapparenthorizonreciprocallinearresponsenear-equilibriumsteadystate
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 tries to show that gravitational attraction is not an entropic drive toward disorder but a response to a deficit of entropy—an ordering quantity it calls negative entropy. For a thin shell of mass $m_0$ held just outside a Schwarzschild horizon, the Euclidean path integral produces an entropy deficit $N = -m_0 e^\phi/T_H$, where $e^\phi$ is the redshift factor and $T_H$ the horizon temperature; the same deficit equals the apparent-horizon area deviation from the extremal surface divided by $4G$. Saturation of the relative-entropy bound turns this deficit into a gradient, and the near-equilibrium steady-state condition then gives $F_g = T_H\, dN/dx$. A careful reader should care because this replaces the usual "gravity maximizes entropy" picture with a concrete, calculable identity: the force on a static shell is the temperature-weighted slope of its missing entropy, and stacking shells integrates to the full horizon entropy.

What carries the argument

The load-bearing object is the negentropy $N \equiv -(S_{\max}-S)$, specialized to $N = -m_0 e^\phi/T_H$ for a shell of rest mass $m_0$ at redshift factor $e^\phi = \sqrt{1-2GM/r}$. It is simultaneously a geometric quantity, minus the apparent-horizon area deficit $\Delta A(\mu_a)/(4G)$ relative to the extremal surface, and a thermodynamic quantity, the shortfall of actual entropy from the maximal coarse-grained entropy allowed by the relative-entropy bound. The argument's engine is the saturation identity $\Delta S = \Delta\langle H\rangle/T$, which converts this deficit into the entropic gradient $\nabla_\mu S = (m/T_0)\nabla_\mu e^\phi$, and the reciprocal linear-response equations whose stationary no-current condition forces $F_g = T_H \nabla N$. The Euclidean path integral supplies the quantitative input: the shell's action shift $\beta_H m_0 e^\phi$ and the conical defect proportional to $m_0$ that the shell introduces at its radius.

What would settle it

A direct check is to compute the on-shell Euclidean action for a shell at finite radius without assuming saturation and compare the action shift to $\beta_H m_0 e^\phi$; any deviation proportional to the relative entropy would break $F_g = T_H\,dN/dx$. Concretely, one could measure the apparent-horizon area shift $\Delta A = 32\pi G^2 M m_0 e^\phi$ against the force required to hold the shell at radius $R$: a mismatch beyond order $m^2$ would falsify the identification of the force with the horizon-temperature-weighted deficit gradient.

Watch

Extended reading notes

Core claim

The central claim is that a black hole plus a fixed external shell is a near-equilibrium steady state whose gravitational force is exactly $F_g = T_H\,dN/dx$, with $N = -m_0 e^\phi/T_H$ the shell's negentropy deficit. The paper derives this by computing the Euclidean on-shell action shift for a static thin shell: the free energy rises by $m_0 e^\phi$, the entropy stays that of the original black hole, and the deficit $N = -(\beta_H F - S)$ is therefore nonzero only because of the shell. That deficit is identified with minus the apparent-horizon area increase, $-\Delta A/(4G) \approx -8\pi G M m_0 e^\phi$, which is also the deviation of the holographic extremal surface from the unperturbed horizon. With the relative-entropy bound saturated, the entropy gradient becomes $\nabla_\mu S = (m/T_0)\nabla_\mu e^\phi$, and the no-current solution of the reciprocal linear-response equations reproduces the gravitational temperature-gradient law $\nabla T/T = -\nabla\phi$ together with the force law. The picture is that gravitational potential is a negative-entropy potential: ordering, not disorder, is what pulls the shell inward.

Load-bearing premise

The whole force law depends on the assumption that the shell's entropy deficit exactly saturates the bound $\Delta S = \Delta\langle H\rangle/T$; if real matter is more ordered than that maximum allows, then $N = -m_0 e^\phi/T_H$ is not the potential whose gradient equals the force.

Editorial extensions

If this is right

  • If $F_g = T_H \nabla N$ holds, holding a shell at fixed radius is a steady state maintained against an entropic gradient, and releasing it quasi-statically converts its negentropy into bath entropy.
  • The no-current condition of the reciprocal linear-response equations gives $\nabla T/T = -\nabla\phi$, so the inward rise of local temperature is exactly what prevents spontaneous heat flow in a static configuration.
  • Integrating the infinitesimal shell contributions $\int_0^M dm'\, S(M,m')$ reproduces $A/(4G)$, recovering the full horizon entropy as a sum of negentropy deficits.
  • The apparent-horizon area shift $\Delta A \approx 32\pi G^2 M m_0 e^\phi$ gives a holographic measure of the shell's negative entropy as the deviation of the extremal surface from the unperturbed horizon.
  • The reciprocal cross-coupling means temperature gradients can drive mechanical motion and mechanical work can drive entropy currents, making gravity a dissipative linear-response phenomenon near equilibrium.

Reading between the lines

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

  • If the relative-entropy bound is only nearly saturated, $F_g = T_H \nabla N$ should acquire corrections proportional to the relative entropy itself; computing those corrections would connect the force law to holographic error correction.
  • Applying the same negentropy-gradient logic to a cosmological horizon would predict an effective force from the deficit of matter relative to the horizon's maximal entropy; the paper leaves this application open.
  • A shell with internal degrees of freedom that are not maximally entangled should show an action shift below $\beta_H m_0 e^\phi$; measuring that deficit is a quantitative test of the framework.
  • A tabletop analog of the two-channel linear-response system, with a thermal gradient and a movable mechanical element, could probe the predicted reciprocal coupling between heat flow and force.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This paper proposes that gravitational attraction toward a static thin shell outside a Schwarzschild black hole is driven by a negative-entropy (negentropy) potential. The Euclidean action of the black-hole-plus-shell system is computed using Israel junction conditions and GHY boundary terms, yielding the on-shell action I_E = beta_H M/2 + beta_H m0 e^phi (Eqs. (2.15)-(2.16)). From this the paper extracts F = M/2 + m0 e^phi, E = M + m0 e^phi, and S = beta_H M/2, defining N = -beta_H m0 e^phi as the entropy deficit. The paper then asserts saturation of the Casini-Bekenstein bound (Eq. (3.10)), derives an entropic gradient, couples thermal and mechanical fluxes via Onsager reciprocity, and concludes F_g = T_H grad N (Eqs. (3.24)-(3.25)). An AdS/CFT interpretation identifies the deficit with apparent-horizon area deviation from extremal surfaces.

Significance. If valid, this would constitute a substantial reinterpretation of gravitational force as an ordering phenomenon, generalizing Verlinde's entropic-force scenario with an explicit Euclidean calculation. The shell-action computation in Section 2 is carried out in the standard manner and the paper usefully shows that the on-shell action of the combined system is beta_H times the redshifted shell mass. However, the central claim does not survive contact with the paper's own Euclidean result: the actual entropy is unchanged by the shell, making the saturation assumption (3.10) false, and with the definition (1.4) the 'emergent' force law is an algebraic identity. No machine-checked proofs or numerical checks are provided; the manuscript's main positive content is therefore confined to the standard shell action calculation.

major comments (4)
  1. [2.4; 3.2] Eq. (2.17) gives S = beta_H (E - F) = beta_H M/2 for the black-hole-plus-shell configuration, independent of the shell, so the actual entropy shift is Delta S = 0 even though Delta E = m0 e^phi. This contradicts Section 3.2, Eq. (3.10), where saturation of the Casini-Bekenstein bound is asserted with Delta S = Delta <H>/T = m0 e^phi / T_H. The force law F_g = T_H dN/dx in Eq. (3.25) requires this saturation; with the Euclidean result the bound is strict, and N = -(S_max - S) is not equal to -m0 e^phi / T_H. These two statements cannot both hold in the same setup, and the inconsistency lies at the center of the paper's claim.
  2. [3.5; Eq. (1.4)] With N = -m0 e^phi / T_H from Eq. (1.4), the formula F_g = T_H dN/dx in Eqs. (3.24)-(3.25) reduces to F_g = -d(m0 e^phi)/dx, which is the ordinary gradient of the redshifted potential energy of a static shell. The emergence claim is therefore definitional, not dynamical. The Onsager linear-response equations in Section 3.4 and the no-current condition only impose the Tolman balance X_G = -X_T; they do not determine the proportionality constant L21/L22 that would fix the force magnitude. The energy-conservation argument dQ = T_H dS_bath = -dW_g preceding Eq. (3.25) is the first law and merely restates the same energy identity.
  3. [4.2; 3.2] Section 4.2 states that for any perturbation the relative entropy S(rho||sigma) = Delta <H_A>/T - Delta S is strictly positive and the inequality Delta S < Delta <H_A>/T_H is strict, with saturation occurring only for the exact thermal state. This is the regime relevant to the shell, and it invalidates the saturation assumption (3.10) used in Section 3.2 to derive grad_mu S = (m/T0) grad_mu V. The saturation of the Casini-Bekenstein bound is thus an input, not a consequence of the Euclidean path integral, and it is contradicted by the manuscript's own holographic discussion.
  4. [2.5] Eqs. (2.21)-(2.24) identify S(M,m) = Delta A/(4G) = 8 pi G M m with an entropy associated with gravitational attraction and integrate it to recover the Bekenstein-Hawking formula. However, Eq. (2.17) shows that the thermodynamic entropy of the held-shell configuration is unchanged by the shell; Delta A is the apparent horizon area of the exterior mass M+m, i.e., the area the shell would add if absorbed, not a deficit of the static state. The integral in Eq. (2.24) therefore reproduces S_BH by adding successive shell masses m' and is not an independent consistency check of the negentropy mechanism.
minor comments (4)
  1. [Title; Abstract] There is a typo in the title ('Integralsl') and an ungrammatical sentence in the abstract ('This clarify that...'); also the hyphenation 'path–integral' is nonstandard.
  2. [2.3] The notation uses m inconsistently: Eq. (2.10) defines m0 = 4 pi R^2 sigma, Eq. (2.13) writes I_shell = beta_H m, and Eq. (2.16) writes I_shell = beta_H m0 e^phi; the relation between the ADM mass m and the proper mass m0 should be stated explicitly and used consistently.
  3. [3.4] The sign conventions are confusing: Eq. (3.16) defines X_T = grad(1/T), Eq. (3.19) writes X_G = -grad(1/T), and the text after Eq. (3.21) says X_T = grad(1/T) = F_ext with F_ext not previously defined as a gradient; a single consistent set of definitions should be provided.
  4. [Figures 2 and 4] Figures 2 and 4 appear to be identical, although the captions describe different physical situations; please verify whether one figure was intended to show a different configuration.

Circularity Check

2 steps flagged · score 8.0 of 10

The central emergent force law reduces to the definition of N: Eq. (1.4) sets N=-m0 e^phi/T_H and Eq. (3.25) then gives F_g=T_H dN/dx = -d(m0 e^phi)/dx, the ordinary gravitational force.

  1. self definitional [Section 1.1, Eq. (1.4); combined with Section 3.5, Eq. (3.25)]
    "Therefore we suggest that the amount of entropy associated to the gravitational attraction to be N=−S(M, m 0eϕ) =− m0eϕ TH =−8πGM m0eϕ (1.4) which allows us to adopt a related but distinct viewpoint: we reveal the gravitational potential as a kind of “negative entropy” potential, such that gradients in entropy underlie variation of gravitational attraction."

    Substituting the definition (1.4) into the central force law (3.25), Fg =TH d(∆SBH)/dx =TH d(Smax−S)/dx =TH dN/dx, gives F_g = T_H d(-m0 e^phi/T_H)/dx = -d(m0 e^phi)/dx, which is the standard gravitational force on the shell expressed through its redshifted rest energy. The Euclidean computation (2.18) defines the same object as N=-(β_H F-S)=-β_H m, i.e. minus the shell energy divided by the Hawking temperature. The claimed emergent negentropic force is therefore the input definition of N rewritten; the Onsager/linear-response analysis in (3.20)-(3.23) does not supply the proportionality independently but assumes the no-current balance that already encodes this relation.

  2. other [Section 3.2, Eqs. (3.10)-(3.11); contrast with Section 2.4, Eq. (2.17)]
    "Under the semi-static process to extract gravitational force by fixing local measurement of ∆⟨H⟩ →m for nearby static observers, we show the saturation of the entropy bound ∆S= ∆⟨H⟩/T ,(3.10) leads to an entropic gradient generally ∇µS= m/T0 ∇µV ,(3.11)"

    The saturation condition (3.10) is imposed rather than derived, and it is exactly the equality that identifies the entropy deficit with energy over temperature, which is the same content as N=-m0 e^phi/T_H. The paper's own Euclidean action result (2.17) gives S=β_H M/2, independent of the shell, so the path-integral entropy change for holding the shell is zero; the nonzero negentropy is obtained by assuming saturation, not by the Euclidean path integral. The entropic gradient (3.11) is then the derivative of that assumed input, and the subsequent force law inherits this input rather than predicting it from the path-integral computation.

full rationale

The paper contains a substantial independent Euclidean action calculation (I_E = β_H M/2 + β_H m0 e^phi, S=β_H M/2, and the integral over shells reproducing S_BH), and those auxiliary checks are not circular. However, the load-bearing claim that gravitational force emerges from a negative entropy gradient is definitional: Eq. (1.4) defines N=-m0 e^phi/T_H, and Eq. (3.25) then returns F_g=T_H dN/dx=-d(m0 e^phi)/dx, the familiar force from the shell's gravitational potential. The same quantity is labeled in Eq. (2.18) as N=-(β_H F-S)=-β_H m, so the emergent force is a relabeling of energy over temperature. The saturation assumption (3.10) is not derived from the Euclidean path integral, and the paper concedes in Section 2.5 that the actual path-integral entropy is not the coarse-grained entropy when the shell is held non-equilibrium; this confirms that the negentropy deficit is an imposed input, not a result. Score 8 reflects that the principal result is forced by the definition of N, while the auxiliary Euclidean and holographic area computations remain independent consistency checks.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

No numerical data are fitted; the central quantity N is defined rather than derived. The load-bearing assumptions are the saturation of the Bekenstein bound, the conical defect angle at the shell, and the applicability of Onsager reciprocity to gravity. The invented entity N has no independent falsifiable handle outside the paper.

assumptions (5)
  • domain assumption Saturation of the Casini-Bekenstein bound: Delta S = Delta <H>/T for the shell outside the horizon
    Used to obtain the entropic gradient (3.11) and hence the force F = T_H grad N. Without equality, the deficit is not exactly tied to the gravitational potential.
  • ad hoc to paper Euclidean time period remains beta_H = 8 pi G M fixed while a conical defect Delta theta = 8 pi G m0 forms at the shell
    Stated in Section 2.4 without derivation; it is the mechanism behind I_shell and N.
  • domain assumption Onsager reciprocity L12 = L21 holds for the gravity-thermal coupling
    Section 3.4 assumes microscopic time-reversal invariance and writes a linear response matrix with symmetric cross-coefficients; no coefficients are computed.
  • standard math Israel junction formulas for a thin shell at linear order
    Used in Section 2.3 to relate surface density and pressure to the metric masses M and m; standard but not re-derived.
  • standard math Euclidean action equals beta F and S = beta (E - F)
    Standard thermodynamic identities used throughout Sections 2.4 and 2.5.
invented entities (1)
  • Negative entropy potential N (negentropy)
    purpose: Serves as the thermodynamic potential whose gradient, multiplied by Hawking temperature, gives the gravitational force F = T_H grad N
    N is defined as -(Smax - S) and set equal to -m0 e^phi / T_H (Eq. 1.4), which is gravitational potential energy rescaled by temperature. No independent observable is predicted; the definition already contains the gravitational force.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Holographic Ordering and Negative entropy in Non-equilibrium Euclidean Black Hole Path Integralsl." pith.science (2026). https://pith.science/paper/ISSML2AW

@misc{pith2026250710450,
  author       = {Pith},
  title        = {Pith review of: Holographic Ordering and Negative entropy in Non-equilibrium Euclidean Black Hole Path Integralsl},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ISSML2AW}},
  note         = {Machine review of arXiv:2507.10450}
}
read the original abstract

The Gibbons-Hawking-York (GHY) approach was developed for a Euclidean path integral derivation of equilibrial black hole entropy. To extend it to a near-equilibrium Euclidean path integral, we study a static Euclidean shell model. We calculate the Euclidean action shift for the static simple model thin shell held just outside the horizon, and find agreement with Casini's version of Bekenstein bound. We find a negative entropy deficit associated to the gravitational attraction towards the shell. For a holographic interpretation, the deficit corresponds precisely to the apparent horizon area deviation from the extremal surfaces Therefore, we develop a Euclidean path integral framework in which gravitational force emerges from negative entropy gradients due to Hawking temperature gradients. This setup allows us to introduce Onsager reciprocity and a linear-response relation to build a dissipating system, and treat the configuration as a near-equilibrium steady state (NESS). This clarify that the gravitational potential is a phenomenon informational and ordering, rather than entropic and disordering.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

29 extracted references · 11 canonical work pages

  1. [1]

    J. D. Bekenstein,Black holes and entropy,Phys. Rev.D7(1973) 2333–2346

    work page 1973

  2. [2]

    S. W. Hawking,Black Holes and Thermodynamics,Phys. Rev.D13(1976) 191–197

    work page 1976

  3. [3]

    J. M. Maldacena,The Large N limit of superconformal field theories and supergravity,Int. J. Theor. Phys.38(1999) 1113–1133, [hep-th/9711200]

    arXiv 1999

  4. [4]

    Witten,Anti-de Sitter space and holography,Adv

    E. Witten,Anti-de Sitter space and holography,Adv. Theor. Math. Phys.2(1998) 253–291, [hep-th/9802150]

    arXiv 1998

  5. [5]

    Ryu and T

    S. Ryu and T. Takayanagi,Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96(2006) 181602, [hep-th/0603001]

    arXiv 2006

  6. [6]

    V. E. Hubeny, M. Rangamani and T. Takayanagi,A Covariant holographic entanglement entropy proposal,JHEP07(2007) 062, [0705.0016]

    arXiv 2007

  7. [7]

    Lashkari, M

    N. Lashkari, M. B. McDermott and M. Van Raamsdonk,Gravitational dynamics from entanglement ’thermodynamics’,JHEP04(2014) 195, [1308.3716]

    arXiv 2014

  8. [8]

    Faulkner, M

    T. Faulkner, M. Guica, T. Hartman, R. C. Myers and M. Van Raamsdonk,Gravitation from Entanglement in Holographic CFTs,JHEP03(2014) 051, [1312.7856]

    arXiv 2014

Show all 29 references
  1. [9]

    Faulkner, F

    T. Faulkner, F. M. Haehl, E. Hijano, O. Parrikar, C. Rabideau and M. Van Raamsdonk, Nonlinear Gravity from Entanglement in Conformal Field Theories,JHEP08(2017) 057, [1705.03026]

    2017 arXiv

  2. [10]

    Jacobson,Entanglement Equilibrium and the Einstein Equation,Phys

    T. Jacobson,Entanglement Equilibrium and the Einstein Equation,Phys. Rev. Lett.116(2016) 201101, [1505.04753]

    2016 arXiv

  3. [11]

    S. M. Carroll and G. N. Remmen,What is the Entropy in Entropic Gravity?,Phys. Rev.D93 (2016) 124052, [1601.07558]

    2016 arXiv

  4. [12]

    Jacobson,Thermodynamics of space-time: The Einstein equation of state,Phys

    T. Jacobson,Thermodynamics of space-time: The Einstein equation of state,Phys. Rev. Lett. 75(1995) 1260–1263, [gr-qc/9504004]. – 31 –

    1995 arXiv

  5. [13]

    Li and J

    R. Li and J. wang,Generalized free energy landscape of a black hole phase transition,Phys. Rev. D106(2022) 106015, [2206.02623]

    2022 arXiv

  6. [14]

    E. P. Verlinde,On the Origin of Gravity and the Laws of Newton,JHEP04(2011) 029, [1001.0785]

    2011 arXiv

  7. [15]

    E. P. Verlinde,Emergent Gravity and the Dark Universe,SciPost Phys.2(2017) 016, [1611.02269]

    2017 arXiv

  8. [16]

    An and P

    Y. An and P. Cheng,Realize Emergent Gravity to Generic Situations,Eur. Phys. J. C81 (2021) 789, [2004.14059]

    2021 arXiv

  9. [17]

    Chandra and T

    J. Chandra and T. Hartman,Coarse graining pure states in AdS/CFT,JHEP10(2023) 030, [2206.03414]

    2023 arXiv

  10. [18]

    J. M. Maldacena,Eternal black holes in anti-de Sitter,JHEP04(2003) 021, [hep-th/0106112]

    2003 arXiv

  11. [19]

    M. Van Raamsdonk,Lectures on Gravity and Entanglement, inProceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015): Boulder, CO, USA, June 1-26, 2015, pp. 297–351, 2017.1609.00026. DOI

    2015

  12. [20]

    Balakrishnan and O

    S. Balakrishnan and O. Parrikar,Modular Hamiltonians for Euclidean Path Integral States, 2002.00018

    2002 arXiv

  13. [21]

    Arias, M

    R. Arias, M. Botta-Cantcheff, P. J. Martinez and J. F. Zarate,Modular Hamiltonian for (holographic) excited states,2002.04637

    2002 arXiv

  14. [22]

    Casini,Relative entropy and the Bekenstein bound,Class

    H. Casini,Relative entropy and the Bekenstein bound,Class. Quant. Grav.25(2008) 205021, [0804.2182]

    2008 arXiv

  15. [23]

    An,Inertial force, Hawking Temperature and Quantum Statistics,1811.03765

    Y. An,Inertial force, Hawking Temperature and Quantum Statistics,1811.03765

    arXiv

  16. [24]

    Marolf, D

    D. Marolf, D. Minic and S. F. Ross,Notes on space-time thermodynamics and the observer dependence of entropy,Phys. Rev.D69(2004) 064006, [hep-th/0310022]

    2004 arXiv

  17. [25]

    Marolf,A Few words on entropy, thermodynamics, and horizons, inGeneral relativity and gravitation

    D. Marolf,A Few words on entropy, thermodynamics, and horizons, inGeneral relativity and gravitation. Proceedings, 17th International Conference, GR17, Dublin, Ireland, July 18-23, 2004, pp. 83–103, 2004.hep-th/0410168. DOI

    2004 arXiv

  18. [26]

    W. G. Unruh,Notes on black hole evaporation,Phys. Rev.D14(1976) 870

    1976

  19. [27]

    Carroll,Spacetime and Geometry: An Introduction to General Relativity

    S. Carroll,Spacetime and Geometry: An Introduction to General Relativity

  20. [28]

    The University of Chicago Press, 1984

    R.M.Wald,General Reltivity. The University of Chicago Press, 1984

    1984

  21. [29]

    G. W. Gibbons and S. W. Hawking,Action Integrals and Partition Functions in Quantum Gravity,Phys. Rev. D15(1977) 2752–2756. – 32 –

    1977

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.