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REVIEW 3 major objections 6 minor 157 references

An effective deformation parameter and f(R,T) gravity both raise the maximum mass compact stars can support for a fixed equation of state.

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2026-07-11 00:29 UTC pith:RZYXAKTP

load-bearing objection Competent thesis that systematically combines an existing one-parameter deformation with f(R,T) stellar structure; the mass shifts are real within the model but partly driven by an ad-hoc mass rescaling the author himself flags. the 3 major comments →

arxiv 2607.07736 v1 pith:RZYXAKTP submitted 2026-07-07 gr-qc

Deformed Compact Objects in General Relativity and Modified Gravity

Jonathan Tejeda Quartuccio This is my paper
classification gr-qc PACS 04.40.Dg04.50.Kd97.60.Jd
keywords compact objectsneutron starsTolman-Oppenheimer-Volkoffdeformed TOVf(R,T) gravityequation of statestrange starsmass-radius relation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This thesis asks how far the usual spherical, static picture of neutron stars and strange stars can be stretched before the predicted masses and radii change in an important way. It introduces a single dimensionless parameter D that rescales the radial part of the metric and the mass-continuity equation, recovering the ordinary Tolman–Oppenheimer–Volkoff equations when D equals one. Oblate (D less than one) sequences systematically support larger masses and radii than the spherical case; prolate (D greater than one) sequences are lighter and more compact. The same deformed structure equations are then solved inside two f(R,T) models whose extra terms couple geometry to the trace of the energy-momentum tensor. Those couplings further shift the mass–radius curves, generally allowing still higher maximum masses for positive coupling strength. The calculations are performed for three standard equations of state (GM1 hadronic, MIT bag quark matter, and a polytrope), so the geometric and gravitational effects can be compared side by side. The practical claim is that both effective shape and matter–geometry coupling can move equilibrium sequences enough to matter for the interpretation of the heaviest observed compact objects.

Core claim

Within the one-parameter D-TOV formalism, oblate configurations (D < 1) support systematically larger masses and radii than the spherical limit for a given equation of state, while prolate configurations (D > 1) produce lighter, more compact stars; when the same deformation is combined with linear or optimized f(R,T) = R + f(T) gravity, the trace-dependent couplings further raise the maximum supported mass.

What carries the argument

The deformed Tolman–Oppenheimer–Volkoff (D-TOV) system: the metric ansatz ds^{2} = e^{2Φ} dt^{2} − (1 − 2m/r)^(−D) dr^{2} − r^{2} dΩ^{2} together with the effective mass continuity dm/dr = 4π D r^{2} ε, which reduces to ordinary TOV when D = 1 and is then inserted into the hydrostatic equations of f(R,T) gravity.

Load-bearing premise

The whole construction treats a single constant D as an adequate stand-in for moderate non-spherical shape even though the angular part of the metric stays spherical and no consistent exterior vacuum solution is matched for D not equal to one.

What would settle it

Construct fully axisymmetric equilibrium models (or slow-rotation Hartle–Thorne sequences) with the same microphysical equations of state and check whether the mass–radius shifts predicted by the D-TOV sequences for D near 0.9–1.1 survive once a consistent exterior geometry is imposed.

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

If this is right

  • Oblate-like sequences for a fixed equation of state can reach higher maximum masses than the corresponding spherical models, offering an additional channel for accommodating objects near or above two solar masses.
  • Positive matter–geometry coupling in linear f(R,T) systematically increases maximum mass for any fixed D, so the combined effect of deformation and modified gravity can exceed either correction alone.
  • Radial-mode frequencies extracted on the deformed backgrounds are sensitive to D, supplying an independent dynamical diagnostic of the same equilibrium sequences.
  • Canonical-mass radius constraints can already limit how far D may deviate from unity for a given equation of state before the predicted radius becomes observationally implausible.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the mass–radius degeneracy between D and the f(R,T) coupling is not broken by additional observables (tidal deformability, moment of inertia, or oscillation frequencies), future multimessenger data may still leave the two effects entangled.
  • The same effective D-framework could be used as a cheap prior when scanning hybrid or hyperonic equations of state to estimate how much of a mass excess might be absorbed by moderate global deformation before invoking new microphysics.
  • Because the exterior matching is left incomplete, gravitational-wave templates built on these interiors would need a separate matching calculation before they could be compared with actual ringdown or continuous-wave signals.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This doctoral thesis investigates hydrostatic equilibrium of neutron stars and strange stars under an effective one-parameter deformation (D-TOV) in General Relativity and in f(R,T)=R+f(T) gravity (optimized nonlinear and linear 2λT models). After reviewing dense-matter microphysics and deriving the standard TOV, D-TOV, and modified-gravity structure equations (with full algebra in Appendices A–D), the author integrates sequences for GM1, MIT bag, and polytropic EoSs. The main reported result is that oblate-like D<1 configurations support larger masses and radii than the spherical case, prolate D>1 configurations are lighter and more compact, and positive λ (or the optimized f(T) sector) further raises maximum masses; a preliminary hybrid-star and radial-oscillation analysis is also included.

Significance. If the D-TOV framework is accepted as a controlled phenomenological probe of moderate non-sphericity, the work provides a useful unified survey of how geometric deformation and trace-dependent matter–geometry coupling separately and jointly shift mass–radius sequences across standard EoSs. Strengths include complete algebraic derivations of the equilibrium equations, systematic numerical sequences (mass–radius, central-density, and internal profiles), and explicit recovery of known limits (D→1, f(T)→0). The radial-mode table and hybrid Maxwell construction are secondary but add breadth. The significance is primarily methodological and exploratory rather than a definitive prediction for real deformed stars, because the deformation scheme is not a self-consistent axisymmetric GR solution.

major comments (3)
  1. §3.3.1 and Eqs. (3.63)–(3.64) [also (5.1)–(5.2)]: Φ′ is obtained from the rr Einstein equation of the deformed metric (3.58), but the mass continuity dm/dr=4π D r² ε is introduced by hand as a ‘deformed volume element’ and is explicitly not derived from the tt component (Appendix C). The gravitational mass that enters a(r)=1−2m/r is therefore not the mass consistent with the same metric that generates hydrostatic balance. Because both the pressure gradient and the enclosed mass are rescaled by the free parameter D, the large reported Mmax shifts (e.g. GM1 ~3.06 M⊙ at D=0.8 to ~1.88 M⊙ at D=1.2 in §5.1.1) may be driven largely by the artificial mass rescaling rather than by a genuine geometric effect. The same construction is inherited by the f(R,T) mass equation (4.34). The manuscript acknowledges the phenomenological character of the ansatz but still presents the sequences as evidence t
  2. §§3.3.1–3.3.2: For D≠1 the metric is not an exact exterior vacuum solution, and no matching to a consistent exterior (e.g. Hartle–Thorne or axisymmetric vacuum) is performed; the surface is defined solely by p(R)=0. The reported ‘mass’ and ‘radius’ are therefore interior-sequence labels, not necessarily the asymptotic gravitational mass and circumferential radius that observers would measure. The abstract and conclusions should state this limitation more sharply when comparing to pulsar mass–radius constraints and to GW190814, and any observational language should be restricted to qualitative trends within the effective model.
  3. §5.2 and Table 5.1: The optimized f(T) functional is taken from a cosmological Gaussian-process reconstruction, then A, β, γ are strongly re-tuned for stellar applications while α, λ, T0 are kept fixed. The resulting NS sequences are only slightly more massive/extended than GR. The claim that the ‘optimized’ model modifies stellar structure is therefore largely a statement about a new phenomenological parameter set, not a prediction of the cosmologically reconstructed f(T). The text should separate (a) the cosmological functional as motivation from (b) the stellar re-fit, and avoid implying that the same optimized model simultaneously describes cosmology and compact stars without further justification.
minor comments (6)
  1. §5.5 / Table 5.5: Radial frequencies are shown only for D=0.9 and D=1.0 and for selected modes; the surface residual (5.22) uses the spherical factor (1−2M/R). Clarify whether this boundary condition is consistent with the deformed background for D≠1, and extend or qualify the stability discussion accordingly.
  2. §5.4: The hybrid Maxwell construction is performed once at the EoS level and then fed into D-TOV; the transition is not re-matched for each D. Label Fig. 5.25 more clearly as exploratory and avoid over-interpreting the ‘stable’ solid segments.
  3. Notation: the radial metric potential is called Λ(r) in Ch. 3 and ω(r) in Ch. 4; the cosmological constant is also Λ. A short notation table would reduce confusion.
  4. Several figures (e.g. 5.1–5.8) would benefit from explicit numerical Mmax and Req values in the captions or a summary table for the pure GR D-TOV sequences, analogous to Tables 5.2–5.4.
  5. Typos and language: occasional missing spaces in Portuguese/English front matter and compound words (e.g. ‘doutorado-sanduíche’, ‘Porfim’); standardize ‘D-TOV’ vs ‘D–TOV’ and ‘f(R,T)’ spacing throughout.
  6. Related literature: when discussing effective deformation, a brief comparison to existing anisotropic-fluid TOV models (Bowers–Liang and later works) and to slow-rotation Hartle–Thorne would help place the D ansatz relative to standard alternatives.

Circularity Check

1 steps flagged

No load-bearing circularity: D and λ are free phenomenological parameters scanned over modest ranges; EoSs are external inputs; mass continuity is an openly admitted effective ansatz (not derived from the deformed metric's tt equation); self-citations are normal thesis-related papers, not uniqueness theorems forcing the result.

specific steps
  1. ansatz smuggled in via citation [§3.3.1, eqs. (3.58), (3.64); also inherited by (4.34)]
    "Finally, within the same effective spheroidal prescription of Zubairi et al. (2017), the enclosed mass is modeled by introducing a deformed volume element. It is important to emphasize that this mass-continuity relation is not derived directly from the tt component of the field equations for the metric (3.58). Rather, it should be understood as an effective prescription that rescales the standard spherical mass integral... dm(r)/dr = 4π D r^{2} ε(r)"

    The central numerical claim (D significantly shifts Mmax and R) rests on the coupled system that uses this mass equation. The paper itself states the mass equation is not obtained from the Einstein equations of the deformed metric but is imported as an effective volume-element rescaling from the cited prior work. Because both the hydrostatic balance and the enclosed mass that enters a(r) are rescaled by the same free D, part of the reported mass-radius effect is built into the ansatz rather than derived from the geometry alone. The paper is transparent about the phenomenological status, so this is only a mild (not load-bearing) circularity of the 'effective deformation' interpretation.

full rationale

The derivation chain is: Einstein equations (or f(R,T) field equations) + phenomenological metric ansatz (3.58)/(4.29) with free D + effective (not field-equation-derived) mass continuity dm/dr = 4π D r^{2} ε (explicitly flagged in §3.3.1) + external EoS (GM1 table, MIT bag, polytrope) → numerical integration of the structure ODEs → M(R) sequences. The reported shifts with D or λ are simply the solutions of those ODEs; nothing is fitted to data and then re-predicted, no uniqueness theorem is imported from the authors to forbid alternatives, and the optimized f(T) parameters are taken from an external cosmological reconstruction then openly re-tuned for stellar densities (Table 5.1). Self-citations are to the author's own prior papers that develop the same framework (normal for a thesis) and are not load-bearing for a claimed first-principles result. The skeptic concern that the mass rescaling is ad-hoc is a correctness/consistency issue of the model, not circularity of the derivation. Score 2 only for the minor self-citation chain and the transparent phenomenological character of the mass equation.

Axiom & Free-Parameter Ledger

5 free parameters · 4 axioms · 1 invented entities

The central claim rests on a phenomenological metric deformation, standard perfect-fluid GR and f(R,T) field equations, three external EoSs, and a handful of free coupling/deformation parameters that are scanned rather than derived. No new fundamental fields or particles are introduced; the only invented construct is the effective D-rescaling itself.

free parameters (5)
  • D (deformation parameter) = 0.8–1.2 (exploratory)
    Scanned over 0.8–1.2; controls both the radial metric power and the mass-continuity prefactor. Not fixed by any first-principles calculation.
  • λ (linear f(R,T) coupling) = 0–0.16
    Scanned over 0, 0.08, 0.16; multiplies the trace term. Chosen to keep f_T ≪ 8π.
  • optimized f(T) coefficients (A, β, γ) = A=5e-4, β=γ=1.05e-3 (stellar)
    Cosmological values strongly suppressed for stellar densities (Table 5.1); phenomenological retuning required for acceptable NS sequences.
  • MIT bag constant B = 60 MeV/fm³
    Fixed at 60 MeV fm^{-3}; standard but free within the usual 50–90 range.
  • polytropic K, Γ = K=100–150, Γ=2
    K=100 or 150 km², Γ=2 (or 5/3 for comparison); pure phenomenological baseline.
axioms (4)
  • domain assumption Einstein field equations (or their f(R,T)=R+f(T) generalization) with perfect-fluid energy-momentum tensor and Λ=0.
    Standard starting point of Chapters 3–4; no alternative gravity or imperfect-fluid stress is considered.
  • ad hoc to paper The effective line element with radial power −D and the rescaled mass continuity dm/dr=4π D r² ε adequately capture moderate global deformation.
    Introduced in §3.3 following Zubairi et al.; explicitly not a solution of the full axisymmetric Einstein equations.
  • domain assumption Barotropic EoS p=p(ε) closes the system; temperature, magnetic fields and rotation are neglected except through the effective D.
    Standard cold-star approximation used throughout Chapter 5.
  • ad hoc to paper Surface defined by p(R)=0; no exterior matching required for the interior sequences.
    Stated in §§3.3.1 and 5.1; Birkhoff’s theorem is abandoned for D≠1.
invented entities (1)
  • D-TOV effective deformation scheme no independent evidence
    purpose: Encode global oblate/prolate shape into a one-dimensional hydrostatic equation without solving 2-D Einstein equations.
    Phenomenological construct; independent evidence would require comparison with full numerical-relativity or Hartle–Thorne sequences, which is not provided.

pith-pipeline@v1.1.0-grok45 · 59144 in / 3538 out tokens · 40615 ms · 2026-07-11T00:29:36.602914+00:00 · methodology

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read the original abstract

Neutron stars and related compact objects are unique laboratories for probing matter at supranuclear densities and gravity in the strong-field regime. In this thesis, we investigate the hydrostatic equilibrium of compact stars in different geometric and gravitational scenarios, combining relativistic stellar structure, dense-matter microphysics, and numerical modeling. We derive the standard Tolman-Oppenheimer-Volkoff equation in General Relativity, introduce an effective one-parameter deformation scheme leading to the deformed TOV (D-TOV) formalism, and extend the hydrostatic equilibrium framework to f(R,T) gravity. Using these structure equations, we compute equilibrium sequences for neutron stars and strange stars described by the GM1 equation of state, the MIT Bag Model, and a polytropic equation of state. In General Relativity, the deformation parameter D significantly affects the global stellar properties: oblate configurations generally support larger masses and radii than the spherical case, while prolate configurations lead to less massive and more compact stars. In optimized and linear f(R,T) models, trace-dependent matter-geometry couplings further modify the mass-radius relation and the maximum supported mass. Overall, the results indicate that effective deformation and trace-dependent matter-geometry coupling can significantly affect the equilibrium structure of compact stars.

Figures

Figures reproduced from arXiv: 2607.07736 by Jonathan Tejeda Quartuccio.

Figure 1.1
Figure 1.1. Figure 1.1: Observed masses of selected pulsars, highlighting the canonical mass scale [PITH_FULL_IMAGE:figures/full_fig_p022_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Representative masses of compact objects inferred from selected gravitational [PITH_FULL_IMAGE:figures/full_fig_p023_1_2.png] view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: Schematic representation of the deformation parameter: spherical configura [PITH_FULL_IMAGE:figures/full_fig_p025_1_3.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Relative particle abundances in neutron-star matter as a function of the [PITH_FULL_IMAGE:figures/full_fig_p030_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Schematic structures of a strange quark star (left) and a neutron star (right). [PITH_FULL_IMAGE:figures/full_fig_p032_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Pressure as a function of energy density for representative equations of state. [PITH_FULL_IMAGE:figures/full_fig_p034_2_3.png] view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: Energy per baryon E/A for different forms of matter (see Weber, 2005). some of the u and d quarks into s quarks, consistent with beta equilibrium. Heavier quark flavors (c, b, and t) are not expected to be populated in NS interiors because of their large current masses and the corresponding high threshold chemical potentials. For example, producing charm quarks would require densities of order ≳ 1017 g c… view at source ↗
Figure 5
Figure 5. Figure 5: shows the stellar mass as a function of the normalized central energy den [PITH_FULL_IMAGE:figures/full_fig_p067_5.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Mass–radius relations obtained from the D-TOV equations in General Rela￾tivity using the GM1 equation of state, for different values of the deformation parameter D. The markers indicate the maximum-mass configuration of each sequence. Another relevant aspect of [PITH_FULL_IMAGE:figures/full_fig_p068_5_1.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the normalized energy-density profiles corresponding to the maximum [PITH_FULL_IMAGE:figures/full_fig_p068_5.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Stellar mass as a function of the normalized central energy density for the [PITH_FULL_IMAGE:figures/full_fig_p069_5_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the normalized pressure profiles for the maximum-mass configurations. [PITH_FULL_IMAGE:figures/full_fig_p069_5.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Normalized energy-density profiles for the maximum-mass configurations ob [PITH_FULL_IMAGE:figures/full_fig_p070_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Normalized pressure profiles for the maximum-mass configurations obtained [PITH_FULL_IMAGE:figures/full_fig_p071_5_4.png] view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Mass–radius relations obtained from the D-TOV equations in General Relativ￾ity using the massless MIT Bag Model, for different values of the deformation parameter D. The markers indicate the maximum-mass configuration of each sequence [PITH_FULL_IMAGE:figures/full_fig_p072_5_5.png] view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Stellar mass as a function of the normalized central energy density for the [PITH_FULL_IMAGE:figures/full_fig_p073_5_6.png] view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: Normalized pressure profiles for the maximum-mass configurations obtained [PITH_FULL_IMAGE:figures/full_fig_p074_5_7.png] view at source ↗
Figure 5.8
Figure 5.8. Figure 5.8: Normalized energy-density profiles for the maximum-mass configurations ob [PITH_FULL_IMAGE:figures/full_fig_p075_5_8.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the stellar mass as a function of the normalized central energy density. [PITH_FULL_IMAGE:figures/full_fig_p076_5.png] view at source ↗
Figure 5.9
Figure 5.9. Figure 5.9: Mass–radius relation for neutron stars in General Relativity and optimized [PITH_FULL_IMAGE:figures/full_fig_p077_5_9.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the normalized pressure profiles. The pressure decreases smoothly [PITH_FULL_IMAGE:figures/full_fig_p077_5.png] view at source ↗
Figure 5.10
Figure 5.10. Figure 5.10: Stellar mass as a function of the normalized central energy density for neutron [PITH_FULL_IMAGE:figures/full_fig_p078_5_10.png] view at source ↗
Figure 5.11
Figure 5.11. Figure 5.11: Normalized pressure profiles for neutron stars in General Relativity and [PITH_FULL_IMAGE:figures/full_fig_p079_5_11.png] view at source ↗
Figure 5.12
Figure 5.12. Figure 5.12: Normalized energy-density profiles for neutron stars in General Relativity [PITH_FULL_IMAGE:figures/full_fig_p080_5_12.png] view at source ↗
Figure 5.13
Figure 5.13. Figure 5.13: Mass–radius relations for neutron stars in General Relativity and optimized [PITH_FULL_IMAGE:figures/full_fig_p081_5_13.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the mass–radius relation for NSs described by the [PITH_FULL_IMAGE:figures/full_fig_p081_5.png] view at source ↗
Figure 5.14
Figure 5.14. Figure 5.14: Mass–radius relations obtained for neutron stars described by the [PITH_FULL_IMAGE:figures/full_fig_p082_5_14.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the stellar mass as a function of the normalized central energy density. [PITH_FULL_IMAGE:figures/full_fig_p083_5.png] view at source ↗
Figure 5.15
Figure 5.15. Figure 5.15: Stellar mass as a function of the normalized central energy density for neutron [PITH_FULL_IMAGE:figures/full_fig_p084_5_15.png] view at source ↗
Figure 5.16
Figure 5.16. Figure 5.16: Normalized pressure profiles for the maximum-mass configurations obtained [PITH_FULL_IMAGE:figures/full_fig_p085_5_16.png] view at source ↗
Figure 5.17
Figure 5.17. Figure 5.17: Normalized energy-density profiles for the maximum-mass configurations [PITH_FULL_IMAGE:figures/full_fig_p085_5_17.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows a behavior qualitatively similar to that already observed in Fig. 5.5, [PITH_FULL_IMAGE:figures/full_fig_p086_5.png] view at source ↗
Figure 5.18
Figure 5.18. Figure 5.18: Mass–radius relations obtained for strange stars described by the [PITH_FULL_IMAGE:figures/full_fig_p086_5_18.png] view at source ↗
Figure 5.19
Figure 5.19. Figure 5.19: Stellar mass as a function of the normalized central energy density for strange [PITH_FULL_IMAGE:figures/full_fig_p087_5_19.png] view at source ↗
Figure 5.20
Figure 5.20. Figure 5.20: Normalized pressure profiles for the maximum-mass configurations obtained [PITH_FULL_IMAGE:figures/full_fig_p087_5_20.png] view at source ↗
Figure 5.21
Figure 5.21. Figure 5.21: Normalized energy-density profiles for the maximum-mass configurations [PITH_FULL_IMAGE:figures/full_fig_p088_5_21.png] view at source ↗
Figure 5.22
Figure 5.22. Figure 5.22: Mass–radius relations for strange stars in the [PITH_FULL_IMAGE:figures/full_fig_p089_5_22.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the solutions obtained for the MIT Bag Model when the strange-quark [PITH_FULL_IMAGE:figures/full_fig_p089_5.png] view at source ↗
Figure 5.23
Figure 5.23. Figure 5.23: Mass–radius relations obtained for neutron stars described by the [PITH_FULL_IMAGE:figures/full_fig_p091_5_23.png] view at source ↗
Figure 5.24
Figure 5.24. Figure 5.24: Stellar mass as a function of the normalized central energy density for neutron [PITH_FULL_IMAGE:figures/full_fig_p092_5_24.png] view at source ↗
Figure 5.25
Figure 5.25. Figure 5.25: Preliminary mass–radius relations for hybrid stars obtained by employing a [PITH_FULL_IMAGE:figures/full_fig_p094_5_25.png] view at source ↗

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