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Mass spectra and wave functions of toponia

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Solving the Salpeter equation for toponium yields fully degenerate spin multiplets.

desk verdict Useful toponium potential-model calculation, but the printed decay-width equations don't match the quoted numbers—fix before publish. read the letter →

arxiv 2411.17955 v2 pith:F2OFZMO5 submitted 2024-11-27 hep-ph hep-ex

Guo-Li Wang , Tai-Fu Feng , Ya-Qian Wang This is my paper · ORCID
classification hep-phhep-ex
keywords toponiumBethe-SalpeterequationSalpeterCornellpotentialCoulombmassspectrumdecaywidthtopquark
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 claims that toponium, a top-antitop bound state, can be described by solving the instantaneous Bethe-Salpeter equation with either the Cornell potential or the pure Coulomb potential at a top-quark mass of 172.7 GeV. Because the top quark is so heavy, the paper finds that all spin-dependent splittings vanish: the pseudoscalar and vector S-wave states share one mass, and the P-, D-, F-, and G-wave triplet and singlet states are each pairwise degenerate. The paper then uses the resulting wave functions to predict the two-photon, two-gluon, and dilepton decay widths of the ground-state toponia, finding values that differ by about a factor of two depending on whether the linear confinement term is included. These predictions matter because recent LHC data near the top-pair threshold show hints that a pseudoscalar toponium state may be contributing to the observed events.

What carries the argument

The machinery is the instantaneous Bethe-Salpeter (Salpeter) equation, solved with a specified relativistic wave-function ansatz for each $J^{PC}$ channel and a running strong coupling $\alpha_s(\vec q)$ with $\Lambda = 0.10$ GeV. The Cornell potential $V_{\text{Cornell}}(r)=\lambda r - \frac{4}{3}\frac{\alpha_s}{r}$ supplies the confining and Coulomb interaction, while the Coulomb-only case drops the linear term. The numerical solution yields mass eigenvalues and radial wave functions $\varphi_S$, $\varphi_P$, $\varphi_D$; because the top quark is so heavy, the paper reduces the relativistic wave functions to nonrelativistic forms and uses them, together with standard triangle and current matrix-element formulas, to compute the decay widths.

What would settle it

A high-statistics scan of the $t\bar t$ invariant mass near 343–345 GeV that resolves a narrow $1S$ resonance and measures the $^1S_0$–$^3S_1$ mass separation; any spin splitting above roughly 1 MeV would contradict the claimed degeneracy. Alternatively, a measured $\Gamma(\eta_t\to\gamma\gamma)$ well outside the predicted 7.56–15.9 keV band, or an observed excited-state mass pattern different from the degenerate tables, would falsify the wave-function and potential choices.

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Extended reading notes

Core claim

The central claim is that, for toponium with $m_t = 172.7$ GeV, the mass spectrum is spin-degenerate: $M(n^1S_0)=M(n^3S_1)$, $M(n^3P_0)=M(n^3P_1)=M(n^3P_2)=M(n^1P_1)$, and analogous degeneracies for D, F, and G waves. Solving the Salpeter equation with the Cornell potential ($\lambda = 0.18$ GeV$^2$) gives a $1S$ mass of 343.62 GeV, while the Coulomb-only potential gives 343.59 GeV; excited states lie between roughly 344.3 and 345.3 GeV, all below the $2m_t$ threshold. The paper further claims that the decay widths of the ground states are $\Gamma(\eta_t\to\gamma\gamma)=7.56$ keV (Cornell) or 15.9 keV (Coulomb), $\Gamma(\eta_t\to gg)=1.69$ MeV or 3.54 MeV, and $\Gamma(\Theta\to\ell^+\ell^-)=6.09$ keV or 12.9 keV. The large spread between the Cornell and Coulomb results is attributed to the linear potential substantially reshaping the wave function, and the paper argues that future experiments can discriminate between the two potentials.

Load-bearing premise

The computation treats toponium as a stable bound state and never includes the top quark's decay width, even though that width (1.42 GeV) is comparable to the computed binding energy; if the top quark decays before a bound state forms, the whole spectrum and width predictions lose physical meaning.

Editorial extensions

If this is right

  • If the spectrum is correct, only the $1S$ toponium state lies near 343.6 GeV, matching the 343–344 GeV peak region reported by threshold-production studies of $t\bar t$ pairs.
  • The claimed spin degeneracy means that spin-dependent relativistic corrections can be dropped in toponium calculations, simplifying future wave-function and production studies.
  • The predicted two-photon, two-gluon, and dilepton widths provide concrete targets: a ground-state pseudoscalar toponium near 343.6 GeV should have a two-photon partial width of roughly 7.6–16 keV depending on the potential.
  • Because the Cornell and Coulomb results differ by nearly a factor of two for the ground-state decays, a sufficiently precise measurement of any one of these channels would distinguish which potential better describes the toponium system.

Reading between the lines

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

  • The paper treats toponium as stable, but the top-quark width (1.42 GeV) is comparable to the binding energy of about 1.8 GeV; a natural extension is to solve the Salpeter equation with a complex mass or width term and see how much the masses and widths shift, or whether the bound state dissolves.
  • If the spin degeneracy holds, the production amplitudes for $^1S_0$ and $^3S_1$ toponia near threshold become essentially equal at leading order, which could be used as a consistency check in $t\bar t$ threshold scans.
  • The wave functions provided could be plugged into production cross-section simulations to estimate whether the predicted narrow $1S$ state is visible above the $t\bar t$ continuum at the LHC and future colliders.
  • A decisive test would be a dedicated search for the diphoton final state from a 343.6 GeV resonance; a measured width outside the 7.6–15.9 keV band would force a change in the potential or the bound-state formalism.
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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 / 4 minor

Summary. The paper solves the instantaneous Bethe-Salpeter equation for toponium using the Cornell potential and, for comparison, a pure Coulomb potential, with mt = 172.7 GeV. It reports mass spectra for S-, P-, D-, F-, and G-wave states, finds that singlet-triplet and triplet spin splittings vanish exactly, and provides nonrelativistic wave functions for S-, P-, and D-wave toponia. It then uses these wave functions to compute the two-photon and two-gluon widths of the pseudoscalar eta_t and the dilepton width of the vector Theta. The main numerical outputs are masses in the 343-345 GeV range, with the 1S state around 343.6 GeV, and ground-state widths that differ by roughly a factor of two between the Cornell and Coulomb potentials.

Significance. If the central results are correct, the paper provides a useful complementary study of toponium spectroscopy and decays in a regime where the top quark is extremely heavy and the Coulomb interaction dominates. The comparison between Cornell and Coulomb potentials is transparent, and the Coulomb-potential results for the 1S mass and the decay widths are broadly consistent with earlier threshold and NRQCD estimates. The explicit normalization conditions and wave function ansaetze are helpful for reproducibility. However, the paper does not provide code or numerical details of the Salpeter solver, and the printed decay formulas have a serious dimensional inconsistency that prevents the reader from reproducing the quoted widths. The significance of the decay predictions is therefore conditional on correcting the formulas and re-deriving the numbers.

major comments (3)
  1. [III, Eq. (24) and Eq. (27)] Equation (24) is dimensionally inconsistent as printed. From the normalization condition Eq. (12), phi_S has dimension GeV^-2. The integral in Eq. (24) then has dimension GeV^3 * GeV^-2 * GeV^-2 = GeV^-1, so its square is GeV^-2. The prefactor 12 pi alpha^2 e_q^4 / M^3 has dimension GeV^-3, making the right-hand side dimension GeV^-5 rather than GeV. As written, Eq. (24) cannot produce the keV values in Table III. Equation (27) inherits the same problem. In addition, the replacement e_q^4 alpha^2 -> (2/9) alpha_s^2 stated in the text would convert the prefactor of Eq. (24) to 8 pi/3 alpha_s^2 / M^3, not 8/(3 pi) alpha_s^2 / M^3 as printed in Eq. (27); the printed coefficient is smaller by a factor pi^2. The authors should provide a full derivation of the decay formulas, correct the prefactors and dimensions, and recompute or verify every entry in Table III.
  2. [Sec. I and Sec. IV] The calculation treats toponium as a stable bound state: the Salpeter equation is solved without any top-quark width term, although Sec. I quotes the top width as 1.42 GeV. The computed 1S binding energy is 2 mt - 343.62 GeV = 1.78 GeV, comparable to the top width, and the 2S-1S splitting is only about 0.97 GeV. A state whose constituent decays on a timescale comparable to the binding dynamics cannot be described as a stationary eigenstate without additional justification or a complex-mass/optical-potential treatment. This issue is load-bearing for the entire mass spectrum and for the decay widths, which would in practice be smeared or suppressed. The paper should either include the width in the formalism or clearly state that all results are for a hypothetical stable top, with a quantitative discussion of the expected effect of the finite width.
  3. [Sec. II, Eqs. (5)-(9)] The exact degeneracies M(n1S0)=M(n3S1), M(n3P0)=M(n3P1)=M(n3P2)=M(n1P1), and their analogues in Eqs. (5)-(9) are not dynamical predictions of the model; they follow by construction because the input Cornell and Coulomb potentials contain no spin-dependent or spin-orbit terms. In QCD, spin splittings are nonzero, albeit small, with a parametric size of order alpha_s^4 m_t. The paper should present these equalities as approximations of a central-potential model and estimate the size of the omitted spin-dependent corrections, rather than stating that the splittings vanish. This is important because the abstract and Sec. IV present the vanishing splittings as one of the main findings.
minor comments (4)
  1. [Sec. III, after Eq. (31)] The text refers to the 'color-triplet ground state toponium' when discussing Theta -> l+ l-; the vector toponium is a color singlet. This appears to be a typo and should be corrected.
  2. [Figs. 1 and 2] The figure legends label the curves phi1_S(q) and phi2_S(q), but the text defines a single radial function phi_S(q). The relation between phi1, phi2 and phi_S should be stated explicitly.
  3. [Throughout] There are several typographical errors, including 'exce edingly' in the abstract, 'dileton' instead of 'dilepton' in Sec. III, and inconsistent formatting such as 'm t¯t'. A careful proofreading pass is needed.
  4. [Tables I-III] The masses and widths are quoted without uncertainties or sensitivity estimates. Given that the Cornell and Coulomb potentials differ by roughly a factor of two in the ground-state decay widths, a discussion of the dependence on lambda, Lambda, and the scale of alpha_s would help the reader assess the robustness of the predictions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: masses and decay widths are obtained by solving a fixed-potential Salpeter equation and checked against independent literature values.

full rationale

The paper's derivation chain is an eigenvalue problem with externally fixed inputs: mt = 172.7 GeV, lambda = 0.18 GeV^2, and alpha_s determined from a one-loop running-coupling formula with Lambda = 0.10 GeV. Solving the instantaneous Bethe-Salpeter/Salpeter equation with these inputs yields the mass eigenvalues in Tables I-II, and the resulting wave functions are inserted into decay formulae (Eqs. 24, 27, 30) to produce the widths in Table III. No toponium observable is fitted to obtain any of these outputs; the comparisons to Refs [9,10], [46], [22], and [25] are external benchmarks, not fitting targets. The self-citations (Refs [41,42,47-51]) supply the general wave-function ansatze and the numerical Salpeter solver used in prior applications; they do not contain the toponium masses or widths, so citing them is methodological rather than circular. The spin degeneracies (Eqs. 5-9) follow from the spin-independent Cornell/Coulomb kernel, and are a model consequence rather than a fitted input. The dimensional inconsistency of Eqs. 24/27 noted by the skeptic is an internal correctness issue; if real, it means the printed formulas cannot reproduce Table III, not that the table was used to construct the formulas. Hence no circular step is identifiable.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central calculation rests on four unproven model assumptions: the instantaneous BS kernel, the Cornell/Coulomb potential with inherited parameters, the stability of toponium despite the top width, and the nonrelativistic reduction. No new particles or forces are proposed. The two free parameters, lambda and Lambda, are external inputs rather than derived quantities.

free parameters (2)
  • Cornell string tension lambda = 0.18 GeV^2
    Taken from charmonium and bottomonium phenomenology; controls the linear confining term and affects the wave functions and decay widths. Not derived in this paper.
  • QCD scale Lambda in running alpha_s = 0.10 GeV
    The one-loop running coupling with Nf=5 gives alpha_s(mt)=0.11. This scale is an input inherited from prior fits and directly changes the decay widths.
assumptions (4)
  • domain assumption The instantaneous approximation to the Bethe-Salpeter kernel is valid for toponium.
    Appendix A replaces the full interaction kernel V(P,k,q) with an instantaneous kernel V(q_perp - k_perp). No explicit justification for toponium is given beyond heavy quark mass.
  • domain assumption The Cornell or Coulomb potential with the stated parameters describes the t tbar interaction.
    The potential model with lambda=0.18 GeV^2 and Lambda=0.10 GeV is inherited from lower-mass quarkonium fits and is not derived from QCD for toponium.
  • domain assumption Toponium can be treated as a stable bound state despite the large top quark width.
    The Salpeter equation is solved without a width term, although the top width (1.42 GeV) is comparable to the binding energy (~1.8 GeV). This is the most fragile physical assumption.
  • domain assumption Negative-energy wave function components are negligible for toponium.
    The paper reduces the relativistic Salpeter wave functions to nonrelativistic forms based on the heavy top mass. This is a motivated approximation, not an exact result.

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Cite this review

Pith. "Pith review of Mass spectra and wave functions of toponia." pith.science (2026). https://pith.science/paper/F2OFZMO5

@misc{pith2026241117955,
  author       = {Pith},
  title        = {Pith review of: Mass spectra and wave functions of toponia},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F2OFZMO5}},
  note         = {Machine review of arXiv:2411.17955}
}
abstract

In this article, {we solve the instantaneous Bethe-Salpeter equation with Cornell potential and Coulomb potential} and conduct a meticulous study of the mass spectrum and wave function of toponium. Our investigation reveals that, owing to the exceedingly heavy mass of the top quark, the mass splitting between singlet and triplet states, as well as within the triplet states, is negligible. Consequently, relativistic corrections can be safely disregarded in the study of toponium. As such, we present the nonrelativistic wave functions for $S$-wave, $P$-wave, and $D$-wave toponia and study the decays $\eta_t\to \gamma\gamma$, $\eta_t\to gg$, and $\Theta\to \ell^+\ell^-$.

Figures

Figures reproduced from arXiv: 2411.17955 by the authors.

Figure 1
Figure 1. FIG. 1: The radial wave functions of the ground state and the fi [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The radial wave functions of the ground state and the fi [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 5 Pith papers

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

  1. Topped baryons from QCD sum rules

    hep-ph 2025-07 conditional novelty 6.0 of 10

    QCD sum rules in HQET predict ground-state singly topped baryon masses near 174 GeV, some 1.1-1.5 GeV above the top quark pole mass.

  2. Ensuring that toponium is glued, not nailed

    hep-ph 2024-11 conditional novelty 6.0 of 10

    QCD toponium and its excited states can explain the LHC near-threshold cross-section excess, while a short-range new-physics bound state would look different in the same data.

  3. Toponium Spectrum in the Complex-Energy Plane

    nucl-th 2026-07 conditional novelty 5.0 of 10

    Toponium's T-matrix poles survive at realistic top width, supporting a quasi-bound-state interpretation despite the spectrum appearing as a single broad peak.

  4. Phenomenology of Hypothetical Single-Top Hadronic States

    hep-ph 2026-05 unverdicted novelty 5.0 of 10

    QCD sum-rule calculations yield single-top baryon and meson masses near the top-quark mass, with a few channels slightly below the naive quark-sum threshold.

  5. Examining possible doubly topped baryon configurations

    hep-ph 2026-01 reject novelty 4.0 of 10

    QCD sum rules give doubly topped baryon masses of 345-350 GeV, essentially the sums of the constituent quark masses, with no sign of genuine binding.

Reference graph

Works this paper leans on

52 extracted references · 45 canonical work pages · cited by 5 Pith papers

  1. [1]

    Navas et al

    S. Navas et al. (Particle Data Group), Phys. Rev. D 110 (2024) 030001

    2024

  2. [2]

    Pancheri, J.-P

    G. Pancheri, J.-P. Revol and C. Rubbia, Phys. Lett. B 277 (1992) 518

    work page 1992

  3. [3]

    Sumino and H

    Y. Sumino and H. Yokoya, JHEP 09 (2010), 034; JHEP 06 (2016), 037 (erratum)

    work page 2010

  4. [4]

    Kawabata and H

    S. Kawabata and H. Yokoya, Eur. Phys. J. C 77 (2017) 323

    work page 2017

  5. [5]

    W.-L. Ju, G. Wang, X. Wang, X. Xu, Y. Xu and L.-L. Yang, JHEP 06 (2020) 158

    work page 2020

  6. [6]

    Maltoni, C

    F. Maltoni, C. Severi, S. Tentori and E. Vryonidou, JHEP 03 (2024) 099

    work page 2024

  7. [7]

    V. S. Fadin and V. A. Khoze, JETP. Lett. 46 (1987) 525

    work page 1987

  8. [8]

    Hagiwara, K

    K. Hagiwara, K. Ma and H. Yokoya, JHEP 06 (2016) 048

    work page 2016

Show all 52 references
  1. [9]

    Hagiwara, Y

    K. Hagiwara, Y. Sumino and H. Yokoya, Phys. Lett. B 666 (2008) 71

    2008

  2. [10]

    Y. Kiyo, J. H. Kuhn, S. Moch, M. Steinhauser and P. Uwer, E ur. Phys. J. C 60 (2009) 375

    2009

  3. [11]

    A. A. Penin and M. Steinhauser, Phys. Lett. B 538 (2002) 335

    2002

  4. [12]

    Kiyo and Y

    Y. Kiyo and Y. Sumino, Phys. Rev. D 67 (2003) 071501

    2003

  5. [13]

    Buchmuller and S

    W. Buchmuller and S. H. H. Tye, Phys. Rev. D 24 (1981) 132

    1981

  6. [14]

    Moxhay and J

    P. Moxhay and J. L. Rosner, Phys. Rev. D 31 (1985) 1762

    1985

  7. [15]

    Fabiano, A

    N. Fabiano, A. Grau and G. Pancheri, Phys. Rev. D 50 (1994) 3173

    1994

  8. [16]

    Beneke, Y

    M. Beneke, Y. Kiyo and K. Schuller, Nucl. Phys. B 714 (2005) 67

    2005

  9. [17]

    J. H. Kuhn and P. M. Zerwas, Phys. Rept. 167 (1988) 321

    1988

  10. [18]

    J. H. Kuhn and E. Mirkes, Phys. Rev. D 48 (1993) 179

    1993

  11. [19]

    Fabiano, A

    N. Fabiano, A. Grau and G. Pancheri, Nuovo Cim. A 107 (1994) 2789

    1994

  12. [20]

    Fabiano, Eur

    N. Fabiano, Eur. Phys. J. C 26 (2003) 441. 13

    2003

  13. [21]

    Cakir, R

    O. Cakir, R. Ciftci, E. Recepoglu and S. Sultansoy, Acta Phys. Polon. B 35 (2004) 2103

    2004

  14. [22]

    Kats and M

    Y. Kats and M. D. Schwartz, JHEP 04 (2010) 016

    2010

  15. [23]

    I. I. Y. Bigi and H. Krasemann, Z. Phys. C 7 (1981) 127

    1981

  16. [24]

    Artymowicz, Acta Phys

    P. Artymowicz, Acta Phys. Polon. B 15 (1984) 505

    1984

  17. [25]

    F. J. Yndurain, Nucl. Phys. B Proc. Suppl. 93 (2001) 196

    2001

  18. [26]

    Aad, et al

    G. Aad, et al. (ATLAS Collaboration), Nature 633 (2024) 8030, 542

    2024

  19. [27]

    Hayrapetyan, et al

    A. Hayrapetyan, et al. (CMS Collaboration), Rept. Prog. Phys. 87 (2024) 117801

    2024

  20. [28]

    T. Han, M. Low and T. A. Wu, JHEP 07 (2024) 192

    2024

  21. [29]

    Z. Dong, D. Goncalves, K. Kong and A. Navarro, Phys. Rev. D 109 (2024) 115023

    2024

  22. [30]

    J. A. Aguilar-Saavedra and J. A. Casas, Phys. Rev. Lett. 133 (2024) 111801

    2024

  23. [31]

    Cheng, T

    K. Cheng, T. Han and M. Low, Phys. Rev. D 111 (2025) 033004

    2025

  24. [32]

    Aaboud, et al

    M. Aaboud, et al. (ATLAS Collaboration), Phys. Rev. D 98 (2018) 012003

    2018

  25. [33]

    Khachatryan, et al

    V. Khachatryan, et al. (CMS Collaboration), Phys. Rev. D 95 (2017) 092001

    2017

  26. [34]

    B. Fuks, K. Hagiwara, K. Ma and Y.-J. Zheng, Phys. Rev. D 104 (2021) 034023

    2021

  27. [35]

    J. A. Aguilar-Saavedra, Phys. Rev. D 110 (2024) 054032

    2024

  28. [36]

    F. J. Llanes-Estrada, e-Print: 2411.19180

    arXiv

  29. [37]

    Francener, V

    R. Francener, V. P. Goncalves and D. E. Martins, e-Print : 2502.03295

  30. [38]

    Jeppe, (CMS Collaboration), e-Print: 2411.18414; C ontribution to: TOP2024

    L. Jeppe, (CMS Collaboration), e-Print: 2411.18414; C ontribution to: TOP2024

  31. [39]

    E. E. Salpeter and H. A. Bethe, Phys. Rev. 84 (1951), 1232

    1951

  32. [40]

    E. E. Salpeter, Phys. Rev. 87 (1952), 328

    1952

  33. [41]

    C. S. Kim and G.-L. Wang, Phys. Lett. B 584 (2004) 285; Phys. Lett. B 634 (2006) 564 (erratum)

    2004

  34. [42]

    Wang, Phys

    G.-L. Wang, Phys. Lett. B 633 (2006) 492

    2006

  35. [43]

    Eichten, K

    E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane and T. M. Yan, Phys. Rev. D 21 (1980) 203

    1980

  36. [44]

    J. L. Richardson, Phys. Lett. B 82 (1979) 272

    1979

  37. [45]

    Tang, J.-H

    J. Tang, J.-H. Liu and K.-T. Chao, Phys. Rev. D 51 (1995) 3501

    1995

  38. [46]

    Y. P. Goncharov, Nucl. Phys. A 808 (2008) 73

    2008

  39. [47]

    G.-L. Wang, T. Wang, Q. Li and C.-H. Chang, JHEP 05 (2022) 006. 14

    2022

  40. [48]

    Wang, Phys

    G.-L. Wang, Phys. Lett. B 650 (2007) 15

    2007

  41. [49]

    Wang, Phys

    G.-L. Wang, Phys. Lett. B 674 (2009) 172

    2009

  42. [50]

    Wang, G.-L

    T. Wang, G.-L. Wang, W.-L. Ju and Y. Jiang, JHEP 03 (2013) 110

    2013

  43. [51]

    Wang, H.-F

    T. Wang, H.-F. Fu, Y. Jiang, Q. Li and G.-L. Wang, Int. J. M od. Phys. A 32 (2017) 06&07, 1750035

    2017

  44. [52]

    d’Enterria and K

    D. d’Enterria and K. Kang, e-Print: 2503.10952. 15

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Reviewed August 12, 2026 · model on record in the stance chip above.