{"id":"64c90cc2-54a4-4a53-bdb8-448fde622ccc","arxiv_id":"2412.09306","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"Interacting quark stars in Rastall gravity reach maximum masses of 2.04 to 2.81 solar masses, exceeding general-relativistic limits and matching heavy pulsars and GW190814.","lead":"This paper models quark stars made of interacting quark matter in Rastall gravity, a modified theory where matter and geometry are coupled in a nonstandard way. It calculates that these hypothetical stars can reach 2.6 to 2.8 solar masses, which could explain the heaviest compact objects detected by gravitational-wave observatories.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper never fixes which mass—meff(rs) or m(rs)=∫4πr²ρ—is reported in Tables I–III or matched to the Schwarzschild exterior; since these can differ substantially (Eq. 23), the headline maximum masses and the GW190814 comparison are not well-defined.","rationale":"The paper's central claim is that Rastall gravity permits stable quark stars with maximum masses up to 2.81 M⊙, consistent with GW190814. This claim is only as strong as the mass actually computed and matched to the exterior spacetime. The manuscript conflates m(r) and meff(r): Eq. (17) integrates meff, Eq. (23) shows meff differs from m by a pressure integral, and Eq. (25) references m(rs)=M without stating that M should equal meff(rs). Because the numerical tables do not specify which quantity is tabulated, the reader cannot reproduce or verify the headline result. The stress-test identifies this as the single most load-bearing concern. Additional concerns—such as the uncritically imported GR stability criteria in §V, the circular parameter constraints in §IV-A, and multiple internal inconsistencies (GW190425 vs. GW190814 in the abstract, Table II caption listing η=0.8 while §IV.C uses η=0.6, Fig. 5 referenced as M−ρc when it shows the adiabatic index)—amplify the need for revision but are secondary. The reader's REJECT verdict is therefore supported; no change is needed.","tokens_in":12885,"tokens_out":5149,"duration_ms":48903,"concrete_test":"For the η=0.8, λ̄=0.6, Beff=90 MeV/fm³ case in Table I, independently integrate Eqs. (16)–(17) with the EoS (6), tabulating both meff(rs) and m(rs)=∫0^{rs}4πr²ρ dr. Check which quantity equals the reported 2.57 M⊙. Then enforce the exterior junction by setting M=meff(rs) and recompute the M–R curve; if instead m(rs) was used, correct the TOV equations and recompute. If the two masses differ by more than 10%, the claims of 2.57 M⊙ and consistency with GW190814 are not robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The TOV integration solves for meff via dmeff/dr = 4πr²ρeff (Eq. 17), but the exterior is stated as Schwarzschild with mass M = m(rs) (Eq. 25), where m is earlier defined as ∫4πr²ρ dr (Eq. 20). Eq. (23) shows meff(r) and m(r) are not equal for η≠1: meff = ½(3η−1)/(2η−1) m + (3/2)(η−1)/(2η−1) ∫4πr² p dr. For η=0.8, meff = (7/6)m − (1/2)I_p, so the difference is not negligible. The boundary condition p(rs)=0 does not identify which mass appears in the tables. If the tables list meff(rs), then exterior matching requires meff(rs)=M, which contradicts the text's definition of M as m(rs); if the tables list m(rs), the junction is wrong because the interior metric is 1−2meff/r, not 1−2m/r. The maximum masses 2.57 and 2.81 M⊙, and the claimed consistency with GW190814, are only meaningful once this identification is made. Secondary issues (uncritically imported GR stability criteria, circular parameter constraints, internal inconsistencies such as abstract GW190425 vs. body GW190814, Table II caption η=0.8 vs. §IV.C η=0.6, and Fig. 5 described as M−ρc when it shows the adiabatic index) compound the problem, but the mass ambiguity is the load-bearing flaw.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies static, spherically symmetric interacting quark stars in Rastall gravity using a pQCD-inspired equation of state. The authors derive modified TOV equations, solve them numerically for variations of the Rastall parameter η, the interaction parameter ¯λ, and the effective bag constant B_eff, and report maximum masses up to 2.57 M⊙ (η=0.8) and 2.81 M⊙ (¯λ=1.0). They compare these results with observed pulsar masses and the GW190814 secondary object, and assess stability via the static stability criterion, the adiabatic index, and the causality condition v_s²<1. The central claim is that Rastall gravity can accommodate stable quark stars with masses exceeding those allowed in general relativity, and that the model is consistent with massive compact object observations.","tokens_in":13178,"tokens_out":3069,"duration_ms":33139,"significance":"If the central result were established, it would be a useful contribution to the literature on modified-gravity neutron-star and quark-star models: the paper combines a contemporary interacting quark matter equation of state with the Rastall formalism and provides a systematic parameter scan. The derived relations (14)-(19) are a clear extension of the standard TOV equations, and the comparison with several pulsar mass measurements is a potentially valuable falsifiable setup. However, the numerical headline masses and all astrophysical comparisons rest on an unresolved definition of the total mass, and the stability conclusions rely on general-relativistic criteria that are asserted, not derived, for Rastall gravity. These issues are central and must be fixed before the results can be evaluated.","major_comments":[{"comment":"The paper never states which mass is reported. The TOV integration solves for m_eff via Eq. (17), and the metric in Eq. (19) contains m_eff, but the exterior is matched with M=m(r_s) where m is defined in Eq. (20) as the integral of the physical density ρ. Equation (23) shows that m_eff and m differ by a pressure integral whenever η≠1; for η=0.8, m_eff = (7/6)m − (1/2)I_p, which is not a negligible correction. If Tables I-III list m_eff(r_s), then the exterior matching condition in Eq. (25) is inconsistent; if they list m(r_s), then the interior metric used in the TOV equation is not the same quantity that is matched. The maximum masses 2.57 M⊙ and 2.81 M⊙, and hence the claimed consistency with GW190814 and massive pulsars, are not well-defined until the authors specify which mass is tabulated, enforce the correct junction condition, and recompute the relevant quantities.","section":"§III Eq. (23)"},{"comment":"Stability is claimed on the basis of dM/dρ_c > 0, γ > 4/3, and v_s² < 1, but these criteria are imported from general relativity without derivation. The static stability criterion is derived in GR from the properties of equilibrium sequences and radial oscillation modes; the adiabatic-index threshold γ > 4/3 is likewise derived from the GR pulsation equation. In Rastall gravity, the field equations and the effective fluid variables differ, so the standard criteria do not automatically apply. The manuscript itself states that radial oscillations are not analyzed, but it still uses these criteria to assert dynamical stability. The authors should either derive the Rastall radial perturbation equations and show that the GR criteria remain valid, or soften the stability claim to a necessary-but-not-sufficient check that is explicitly conditioned on the validity of the GR criteria.","section":"§V"},{"comment":"The parameter constraints in Section IV.A are obtained by varying η, ¯λ, and B_eff until the predicted maximum masses match the observed masses of PSR J0348+0432, PSR J0740+6620, PSR J0952-0607, and GW190814. The abstract and Section VI then present the agreement with these same observations as confirmation that the model is viable. This is a circular validation procedure: the same data are used both to set the parameters and to test the model. The authors should reframe this as calibration, and provide an independent check—for example, comparing predicted radii or tidal deformabilities with independent measurements—or explicitly state that the observations are used only to select parameters, not to validate the model.","section":"§IV.A"},{"comment":"There are several internal inconsistencies that make the results difficult to interpret. The abstract and introduction mention GW190425 as the gravitational-wave constraint, but the numerical sections use GW190814 (§IV.B and Fig. 1). Table II has a caption stating η=0.8 while the text of §IV.C says η=0.6, and the figures are generated with η=0.6. In §V.A, the static stability discussion refers to Fig. 5 as the M−ρ_c plot, but Fig. 5 in the manuscript shows the adiabatic index and the M−ρ_c curves are in Fig. 4; §V.B also refers to Fig. 5 for the adiabatic index. These inconsistencies must be resolved because they prevent the reader from associating the tabulated maximum masses with the stated parameter values.","section":"§IV.B-§IV.C, Table II, Fig. 5"}],"minor_comments":[{"comment":"There are typographical issues: both integrals contain a stray comma before dr′ (e.g., \"4πr′2ρ(r′), dr′\"), and the notation would be clearer if written as 4πr′²ρ(r′) dr′.","section":"Eqs. (21), (23)"},{"comment":"The text says \"massive QSs exits with masses 2.20 to 2.65 M⊙\"; this should be \"exist\". Also, in the same paragraph the statement that the maximum compactness \"does not affected by increasing or decreasing values of B_eff\" is grammatically awkward and should be rephrased.","section":"§IV.D"},{"comment":"The caption of Fig. 6 and the text in §V.C describe v_s² as a function of radial coordinate r, but the discussion of the \"brinjal\" (purple) line and the divergence in the outer layers would be easier to follow if the figure included a clear legend matching the parameter values used in the three panels.","section":"Fig. 6"},{"comment":"Several references are arXiv preprints without journal or DOI information (e.g., Refs. [1], [4], [5], [6], [42]). If the manuscript is intended for journal publication, these should be updated to published versions where available.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The mass ambiguity is the central technical problem, but it is in principle fixable by recomputing the tables with a consistent definition of mass and junction condition. My recommendation is major revision rather than rejection because the underlying derivation of the modified TOV equations is straightforward and the paper could be rescued by a careful revision. The circularity in the parameter constraints should be addressed explicitly, as it currently overstates the level of observational validation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe thing to know: this is a straightforward TOV shootout for quark stars in Rastall gravity, and the qualitative message—smaller η gives larger maximum masses—is probably right. But the quantitative headline is compromised because the paper never says whether the tabulated masses are meff(rs) or m(rs)=∫4πr²ρ dr, and these differ for η≠1. The junction to the Schwarzschild exterior is therefore ambiguous, which makes the 2.57 and 2.81 M⊙ numbers and the GW190814 comparison unfixed.\n\nWhat is new is the combination: the interacting quark EoS from Zhang and Mann has not been run through Rastall TOV before. The paper does that cleanly, gives M-R and compactness curves, and reports the expected trends (max mass rises as η drops, as λ¯ rises, as Beff drops). The numerical work is transparent enough that someone could reproduce it. The limitations the authors state—no crust, no radial oscillations—are acknowledged openly, which is honest.\n\nThe soft spots are real. The mass ambiguity is the load-bearing one, and it's exactly as the stress test says: Eq. (23) shows meff and m differ by a pressure integral, and the boundary condition p(rs)=0 doesn't identify which one appears in Tables I–III. If the tables list meff, the text's definition of M as m(rs) is wrong; if they list m, the interior metric 1−2meff/r is matched to the wrong exterior. Either way the central numbers need re-deriving. Secondary issues: the stability conclusions import GR criteria (γ>4/3, dM/dρc) without deriving Rastall versions; the parameter \"constraints\" are really a fit to the very pulsars and GW events used for validation, so calling it consistent is circular in a weak sense; and there are internal inconsistencies (GW190425 in the abstract vs. GW190814 in the body, Table II caption says η=0.8 while §IV.C uses η=0.6, Fig. 5 described as M−ρc when it shows the adiabatic index). None of these alone would sink the paper, but together with the mass issue they leave the viability claim unsupported.\n\nWho is this for? People working on modified-gravity neutron/quark star models will want to see the M-R curves and the qualitative trend. They should not cite the numbers until the authors fix the mass definition. I would not use this as a citable result, but it deserves a serious referee because the underlying question is legitimate and the fix is plausibly short.\n\nRecommendation: send to peer review with a request for major revision; the authors need to state which mass is integrated, enforce the junction condition consistently, and re-run the numbers. If they do, the qualitative conclusion likely survives.\n\nThat's my take.","headline":"The interacting-quark EoS is new to Rastall TOV and the qualitative trend is plausible, but the paper never fixes which mass it reports, so the headline maximum masses and the GW190814 match are not well-defined.","tokens_in":13802,"tokens_out":3323,"would_cite":false,"duration_ms":32940,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C55","83D05"],"pacs":["04.40.Dg","04.50.Kd"],"model":"deepseek-v4-flash","headline":"Quark stars in Rastall gravity can reach 2.81 solar masses, exceeding the general-relativistic limit and matching the GW190814 companion's mass.","keywords":["quark stars","Rastall gravity","interacting quark matter","modified TOV equations","mass-radius relation","GW190814","color superconductivity","compact stars"],"falsifier":"Recompute the mass-radius relation using the Schwarzschild mass m(r_s) obtained from Eq. (23), enforcing the exterior junction condition at p(r_s)=0; if the maximum mass at η=0.8 falls below the 2.5-solar-mass threshold or the λ̄=1.0 case drops below the GW190814 2.6-solar-mass bound, the central claim fails. Additionally, solving the full radial-oscillation equations could show an instability before the maximum mass, which would contradict the stability conclusion.","tokens_in":12557,"feed_emoji":"⭐","tokens_out":3195,"duration_ms":32575,"temperature":0.7,"pith_summary":"This paper claims that quark stars built from a QCD-inspired interacting quark matter equation of state can be significantly more massive in Rastall gravity than in general relativity, reaching maximum masses up to 2.81 solar masses. If true, this would offer a modified-gravity explanation for massive compact objects like the GW190814 secondary without invoking exotic or unusually stiff matter. The authors derive the structure of these stars from modified Tolman-Oppenheimer-Volkoff equations, survey how the Rastall parameter, interaction parameter, and bag constant shift the mass-radius relation, and argue that the configurations satisfy standard stability and causality criteria.","feed_headline":"Rastall gravity builds quark stars up to 2.81 solar masses","feed_subtitle":"A modified gravity theory lets quark stars exceed the GR mass limit, matching the GW190814 companion and massive pulsars.","key_machinery":"The load-bearing structure is the Rastall-gravity modification of the Tolman-Oppenheimer-Volkoff equations, in which the effective energy density ρ_eff and pressure p_eff are linear combinations of the ordinary fluid density and pressure weighted by the Rastall parameter η. The integration is performed for an effective mass m_eff defined through ρ_eff, while the exterior spacetime is taken as Schwarzschild with a mass m(r_s) that the paper treats as equal to the total mass; an explicit relation (Eq. 23) shows m_eff and m differ by a pressure integral, yet the paper does not clarify which mass is reported in the tables. The equation of state is the interacting quark matter model of Zhang and Mann, expressed in dimensionless form with parameters λ̄ and Beff, and used for the color-flavor-locked superconducting phase.","core_discovery":"The paper's central claim is that inside Rastall gravity, quark stars described by an interacting quark matter equation of state can achieve higher maximum masses than their general-relativistic counterparts. Concretely, for an effective bag constant of 90 MeV/$fm^{3}$ and interaction parameter 0.6, lowering the Rastall parameter to 0.8 raises the maximum mass to 2.57 solar masses with a radius near 12 km, while the general-relativistic case (η = 1) gives only 2.22 solar masses. Raising the interaction parameter to 1.0 pushes the maximum to 2.81 solar masses at a radius of 12.91 km. The paper interprets these results as consistent with the ~2.6-solar-mass compact object in GW190814 and with massive pulsar measurements, and reports that static stability, adiabatic index, and sound-speed checks all support stability. The authors also constrain the model parameters η, λ̄, and Beff by comparing predicted masses with observed pulsars and the GW190814 event.","pith_inferences":["The reported maximum masses may be sensitive to which mass definition is tabulated: if the effective mass m_eff is used instead of the Schwarzschild mass m(r_s), the GW190814 compatibility could weaken, because Eq. (23) shows m_eff differs from m by a pressure-volume term.","A full radial-oscillation analysis (Sturm-Liouville eigenmode problem) is needed to confirm the claimed stability; the static criteria alone are necessary but not sufficient for dynamical stability in modified gravity.","The same interacting quark matter EoS could be tested in other modified gravity theories, such as f(R) or scalar-tensor models, to see whether the mass enhancement is a generic feature of non-minimal matter-geometry coupling.","If future gravitational-wave observations refine the mass of the GW190814 secondary or discover a heavier compact object, the Rastall parameter η would be more tightly constrained than the current range [0.8, 1.2] allows."],"forward_implications":["If the maximum-mass results hold, Rastall gravity offers a way to accommodate compact objects above 2.5 solar masses without resorting to unusually stiff hadronic equations of state.","The study provides a parameter-constraint scheme (η, λ̄, Beff) that can be tested against future pulsar mass measurements and gravitational-wave detections of compact object mergers.","The modified TOV framework could be applied to other equations of state, such as hybrid stars or hyperon-inclusive matter, to see whether the mass enhancement persists.","The stability analysis suggests that static stability, adiabatic index, and causality checks are satisfied in the Rastall setting, supporting the viability of these configurations.","The mass-radius relations for different η values indicate that the deviation from general relativity is most pronounced at high central densities, where quark matter is stiffest."],"supporting_citations":[{"why":"Provides the interacting quark matter equation of state with color superconductivity and pQCD corrections that is used throughout the paper.","marker":"[44]"},{"why":"Original Rastall gravity formulation with the modified conservation law, the basis for the field equations.","marker":"[46,47]"},{"why":"Defines the mass function and the relation between effective and standard density, central to the TOV integration.","marker":"[48]"},{"why":"Gives the Rastall TOV equations and effective mass formulation used to compute mass-radius curves.","marker":"[49]"},{"why":"Shows energy conditions restrict the Rastall parameter η for positive η, used to justify the parameter range.","marker":"[54]"},{"why":"Reports the GW190814 event with a 2.6-solar-mass secondary, the observational benchmark for the maximum mass claim.","marker":"[53]"}],"fun_headline_variants":["Quark stars beat GR limit in Rastall gravity to 2.81 Msun","Rastall gravity pushes quark stars past 2.81 Msun","Quark stars hit 2.81 Msun in Rastall gravity, beating GR","Modified gravity enables quark stars beyond GR mass limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes that the mass it tabulates is the Schwarzschild mass that appears in the exterior spacetime, but the TOV integration computes an effective mass that differs from it by a pressure integral, and the paper never states which mass is actually reported.","fun_headline_variants_meta":{"raw":{"variants":["Quark stars beat GR limit in Rastall gravity to 2.81 Msun","Rastall gravity pushes quark stars past 2.81 Msun","Quark stars hit 2.81 Msun in Rastall gravity, beating GR","Modified gravity enables quark stars beyond GR mass limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000474,"raw_usage":{"total_tokens":2318,"prompt_tokens":872,"completion_tokens":1446,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":488,"completion_tokens_details":{"reasoning_tokens":1364}},"tokens_in":488,"tokens_out":1446,"duration_ms":9171,"temperature":1.0,"reasoning_tokens":1364,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T17:06:18.737740+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the mass-radius relation using the Schwarzschild mass m(r_s) obtained from Eq. (23), enforcing the exterior junction condition at p(r_s)=0; if the maximum mass at η=0.8 falls below the 2.5-solar-mass threshold or the λ̄=1.0 case drops below the GW190814 2.6-solar-mass bound, the central claim fails. Additionally, solving the full radial-oscillation equations could show an instability before the maximum mass, which would contradict the stability conclusion.","supporting_citations":[{"cited_title":"Wu and C","cited_arxiv_id":null,"evidence_quote":"Provides the interacting quark matter equation of state with color superconductivity and pQCD corrections that is used throughout the paper."},{"cited_title":"Maulana, and A","cited_arxiv_id":null,"evidence_quote":"Shows energy conditions restrict the Rastall parameter η for positive η, used to justify the parameter range."},{"cited_title":"Velten, A","cited_arxiv_id":null,"evidence_quote":"Reports the GW190814 event with a 2.6-solar-mass secondary, the observational benchmark for the maximum mass claim."}],"review_version":1}