{"id":"a70284f4-a593-46f4-8dd3-d2ac25835fa4","arxiv_id":"2507.17384","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"Using four vector f(R) gravity inflation models and nine equations of state, the TOV solver finds that the MPA1 equation of state yields neutron star maximum masses around 2.75 solar masses, inside the mass gap.","lead":"This paper computes how heavy and large neutron stars can be in a family of modified gravity theories linked to the early universe. It finds that one common recipe for nuclear matter, called MPA1, gives stars near 2.75 times the Sun's mass, which could explain the mysterious objects in the mass gap observed by LIGO and Virgo.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The scalar-matter coupling α(φ) entering the TOV equations is defined inconsistently: Eq. 38 implies α = 1/(2√(6+β^2)), while Eq. 39 gives α = √(6+β^2)/2; for Model IV these differ by ~10^8, so the reported 2.75 M⊙ MPA1 maximum mass cannot be interpreted without knowing which form the solver used.","rationale":"Good-faith reading: the paper is a numerical phenomenology study whose central claim is that vector-f(R)-inspired scalar-tensor models, including cosmologically non-viable ones, can produce mass-gap neutron stars, with MPA1 singled out at about 2.749 M⊙. The strongest support would be a reproducible solver and a GR-limit sanity check; neither is supplied. The paper does give the TOV equations and the model potentials, so partial independent checks are possible, but the key input α(φ) is ambiguous. The manuscript itself contains a limitation admission in the concluding remarks: 'the present theoretical context did not reveal any new physics or curious predictions regarding NSs in modified gravity.' This does not refute the central claim but supports a low-novelty assessment. The reader's weakest assumption was the Jordan-frame ADM mass formula (Eq. 23) and the near-GR limit. I agree that the near-GR limit is the right probe, but the deeper problem is that the text does not uniquely define the coupling used to integrate the TOV equations: Eq. (39) is not the logarithmic derivative of Eq. (38). This makes the near-GR check impossible to perform from the paper alone. If the code followed Eq. (39), all four models are strongly coupled and the near-GR expectation is irrelevant; if it followed Eq. (38), Model IV should reduce to GR and Table II's MPA1 maximum mass should coincide with a standard GR TOV calculation, which is not reported. The concrete test proposed above would settle which theory was actually solved and whether the mass extraction is biased. Because the central numerical claim cannot currently be checked against either the stated equations or an external GR baseline, the appropriate status is unverdictable rather than a conditional acceptance or rejection.","tokens_in":22118,"tokens_out":10310,"duration_ms":102365,"concrete_test":"Recompute the Model IV TOV solution for the MPA1 piecewise-polytropic EoS with the code used for Table II, first as-is, then with α(φ) replaced by the logarithmic derivative of Eq. (38) (i.e. α = 1/(2√(6+β^2))) while leaving the potential unchanged, and compare both to a direct GR TOV integration of the same MPA1 table. If the near-GR variant does not recover the standard GR maximum mass at the reported precision, the discrepancy must be traced to Eq. (23) or the numerical-infinity boundary conditions; if it does, the manuscript needs to state explicitly which α was used and why Eq. (39) is not the implemented coupling.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The paper's central result — that all four vector-f(R) models, including the β=10^4 'non-viable' Model IV, produce an MPA1 maximum mass of about 2.749 M⊙ — rests entirely on the scalar-tensor TOV system Eqs. (10)–(14), in which the scalar-matter coupling α(φ) appears in Eqs. (12) and (13). The paper defines α via Eq. (8) as d ln A/dφ, and gives A(φ) in Eq. (38) as exp(φ/(2√(6+β^2))). The logarithmic derivative of Eq. (38) is α = 1/(2√(6+β^2)). Eq. (39), however, states α = (1/2)√(6+β^2). These agree only when β^2+6 = 1, which is never satisfied. The discrepancy is a factor (β^2+6); for Models II–IV this ranges from ~6 to ~10^8. This is not a cosmetic typo: the two choices produce very different TOV solutions. In particular, the near-GR check for Model IV is not well defined by the text. If the code uses Eq. (39), Model IV has α ≈ 5×10^3 and is far from GR, so the 'even non-viable models work' conclusion is being tested in a strongly coupled regime. If the code uses the derivative of Eq. (38), Model IV is near-GR and should reproduce standard GR maximum masses for the same piecewise-polytropic EoS; the paper gives no such baseline. Either way, the central mass-gap claim cannot be evaluated from the manuscript alone.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies static neutron stars in a class of vector f(R) gravity theories that are recast as scalar-tensor theories. The author integrates the Einstein-frame TOV equations (10)-(14) with a Python LSODA double-shooting solver for nine piecewise-polytropic equations of state, and converts the results to the Jordan frame using the conformal factor A(φ) and the ADM mass formula (23). Four inflationary models are considered: the Starobinsky R^2 model (Model I), a Starobinsky variant with β≈0.1 (Model II), a power-law f(R) model with β=1, n=1.8 (Model III), and a large-β limit with β=10^4, n=4 (Model IV), the last being cosmologically non-viable. The central claim is that the MPA1 equation of state produces maximum masses around 2.749 M⊙ for all four models, placing them in the mass-gap region while remaining below the 3 M⊙ causal limit, and that even the non-viable inflationary model produces viable neutron star phenomenology, with all models giving nearly indistinguishable mass-radius curves.","tokens_in":22417,"tokens_out":10497,"duration_ms":105917,"significance":"If the numerical results are correct, the paper would be a useful forward-prediction study: the inflationary parameters (β, n, M, m) are fixed before the neutron-star calculation, so the MPA1 maximum-mass prediction is not obtained by fitting theory parameters to neutron star data. The paper also confronts the results with multiple constraints (NICER I/II, PSR J0740+6620, CSI–CSIII) and honestly states that the model does not produce new physics beyond existing modified-gravity behavior. However, the central quantitative claim is currently unverifiable from the manuscript because of internal inconsistencies in the definitions of A(φ) and α(φ), and because no GR baseline is tabulated. The significance of the result therefore cannot be assessed until these issues are resolved.","major_comments":[{"comment":"Equations (37), (38), and (39) are mutually inconsistent. From Eq. (37), ϕ = exp(2φ/√(6+β²)), so with A = Ω^{-1/2} = ϕ^{-1/2} from Eq. (5) the correct conformal factor is A(φ) = exp(-φ/√(6+β²)) and α(φ) = d ln A/dφ = -1/√(6+β²). The manuscript instead gives A(φ) = exp(+φ/(2√(6+β²))) in Eq. (38) and α(φ) = √(6+β²)/2 in Eq. (39). The two α values differ by a factor of (6+β²), which for Model IV (β=10^4) is about 10^8. Since α enters the TOV equations (12) and (13), the computed mass-radius curves and maximum masses in Table II depend critically on which expression was used in the solver. The author must state explicitly which definitions were implemented, correct any typographical or conceptual errors, and rerun the analysis if the implemented expressions do not match the corrected ones.","section":"Sec. I.A, Eqs. (37)-(39)"},{"comment":"The near-GR limit of Model IV is not documented. For β=10^4, the potential in Eq. (43) is strongly suppressed and, if α is taken from the derivative of the (corrected) conformal factor, the model should reduce essentially to general relativity. In that case the reported MPA1 maximum mass of 2.749149 M⊙ should equal the GR maximum mass for the same piecewise-polytropic MPA1 equation of state. The paper never quotes the GR maximum mass (or GR radii at 1.4 M⊙ or 2 M⊙) for any of the nine EoSs, so the reader cannot tell whether Model IV is being computed in the weak-coupling near-GR regime or in a strongly coupled regime (if Eq. (39) with α ≈ 5×10³ was used). Without this baseline, the assertion that all four models—including the non-viable one—produce viable and nearly identical mass-radius curves is not testable. Please provide the GR M–R curve and tabulated GR maximum masses for all EoSs, and identify the actual α used in the code.","section":"Sec. I.B, Table II and Fig. 6"},{"comment":"The Jordan-frame ADM mass formula (23) depends explicitly on A(φ(r_E)) and α(φ(r_E))(dφ/dr) evaluated at numerical infinity. Given the inconsistency of Eqs. (37)–(39), both the sign and the magnitude of the scalar-field correction term are ambiguous; for Model IV the correction could either vanish (small α) or dominate (large α from Eq. (39)). In addition, the manuscript does not provide convergence tests for the choice of the numerical infinity r_E, the shooting tolerance, or the residual of the scalar field at infinity. Please include such tests, together with the asymptotic values of A(φ) and α(φ) at r_E, to demonstrate that the extracted Jordan-frame masses are stable and that the quoted 2.749 M⊙ value is well defined.","section":"Sec. I, Eq. (23)"}],"minor_comments":[{"comment":"Table VIII lists R_ENG = 1.437837 km for Model III, which is almost certainly a typo (likely 11.437837 km) and is inconsistent with the other entries for the ENG EoS; please correct it.","section":"Table VIII"},{"comment":"Table XI lists M_max = 10.866 M⊙ for the SLy EoS in Model II, which is physically implausible and inconsistent with Model II's other rows and with the M–R figures; all tables should be carefully regenerated from the corrected numerical output.","section":"Table XI"},{"comment":"The formula for the parameter m is garbled: 'm = 5.1 × 10−4pc−1/2 n (2pN )−(p+2)/4' is not readable. Please write the expression with clear notation for the Planck mass, the e-folding number N, and the indices p and n.","section":"Sec. I.A, after Eq. (42)"},{"comment":"The causal-limit formula (1) uses ρ_u and P_u without defining the transition density or the matching procedure; please clarify what ρ_u and P_u refer to in the causal EoS construction.","section":"Eq. (1)"},{"comment":"The maximum masses in Table II are quoted to as many as seven significant figures (e.g., 2.749149112 M⊙). This precision exceeds what is meaningful for piecewise-polytropic parametrizations and should be rounded consistently.","section":"Table II"},{"comment":"Figure 1 is described as presenting the CSI, CSII, and CSIII constraints, but the figure appears to be an edited astronomical image without visible constraint bands; please overlay the actual mass-radius constraint regions or replace the figure with one that conveys the relevant information.","section":"Fig. 1"},{"comment":"The abstract says 'we solve the TOV equations', but the paper actually solves the Einstein-frame scalar-tensor TOV equations and then converts the results to the Jordan frame; this distinction should be stated explicitly in the abstract or in the opening of Section I.","section":"Abstract and Sec. I"},{"comment":"The header of Table XII contains a broken phrase ('the and the correspondent') and the table formatting of the EoS names is inconsistent; please correct the formatting.","section":"Table XII"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the central numerical claim is currently unsupported because of the internal inconsistency in the definitions of A(φ) and α(φ) and the absence of a documented GR baseline for Model IV. However, the flaw appears to be fixable: the author can correct the conformal factor and α, rerun the solver, and provide GR reference values for all EoSs. I therefore recommend major revision rather than rejection. The paper's stated conclusion that no new physics is revealed is a commendable limitation, but the 'surprise' regarding non-viable models working may need to be tempered once the near-GR limit is clarified. There are no concerns about citation norms or novelty."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the paper is a straightforward numerical application: it solves the scalar-tensor TOV equations for four vector-f(R) inflationary potentials and nine piecewise-polytropic EoSs, and reports that MPA1 gives maximum masses around 2.75 solar masses for all four models, inside the mass gap and below the 3 solar mass causal limit. Second, the central numbers cannot be reproduced from the manuscript because the coupling alpha(phi) is defined two incompatible ways. Eq. (38) gives A(phi)=exp(phi/(2 sqrt(6+beta^2))); Eq. (8) then implies alpha = 1/(2 sqrt(6+beta^2)). Eq. (39) states alpha = sqrt(6+beta^2)/2. These differ by a factor (6+beta^2) and never agree. The TOV system (10)-(14) depends directly on alpha, so the reported maximum masses cannot be interpreted without knowing which form the solver used. For Model IV, with beta=10^4, the two choices differ by ~10^8, turning a supposedly near-GR limit into a strongly coupled problem. The paper also provides no GR baseline for the same EoSs, which is the obvious check.\n\nWhat is new here is the specific numerical output for these four potentials. The framework is standard scalar-tensor TOV applied to potentials from Ozkan-Pang-Tsujikawa, and the conclusion that MPA1 fits best repeats the earlier Odintsov-Oikonomou result. The author is honest about this in the conclusion and about the model-dependence of the 'non-viable models also work' finding. The double-shooting LSODA method and the Jordan/Einstein frame mass extraction are described clearly.\n\nThe other issues are secondary but real: no code or data, several table typos (Table XI lists a 10.866 solar mass SLy maximum for Model II; Table VIII gives an ENG radius of 1.437 km for Model III), and a reference list with malformed entries. Annoying, fixable. The alpha inconsistency is the load-bearing problem: it undermines the paper's main claim as written, not just its presentation.\n\nI would still send this to peer review. The question is relevant, the framework is standard, and the flaw can be repaired by stating the alpha definition, providing a GR baseline, and releasing the code. But I would not cite the numerical results until that is done.","headline":"A standard scalar-tensor TOV scan for four vector-f(R) inflationary models whose central mass-gap claim is currently unreproducible because the coupling alpha is defined two incompatible ways.","tokens_in":23092,"tokens_out":4367,"would_cite":false,"duration_ms":41553,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C55","83D05","85A15"],"pacs":["04.50.Kd","95.36.+x","98.80.-k","98.80.Cq","11.25.-w"],"model":"deepseek-v4-flash","headline":"The paper argues that vector f(R) gravity inflationary models, solved as scalar-tensor theories, support static neutron stars up to about 2.75 solar masses with the MPA1 equation of state, placing them in the mass-gap region below the…","keywords":["neutron stars","mass-gap region","vector f(R) gravity","scalar-tensor gravity","Tolman-Oppenheimer-Volkoff equations","MPA1 equation of state","modified gravity","inflationary models"],"falsifier":"Recompute the mass-radius curve for the near-general-relativity model ($\\beta = 10^4$, coupling about $5\\times10^{-5}$) with the MPA1 equation of state and compare its maximum mass with the general-relativistic result for the same equation of state; the two must agree if the mass conversion is correct, yet the paper reports about $2.749\\,M_\\odot$ for that model and gives no GR baseline.","tokens_in":21753,"feed_emoji":"🌟","tokens_out":14311,"duration_ms":133925,"temperature":0.7,"pith_summary":"Neutron stars are observed up to about 2.35 solar masses, while gravitational-wave observatories have detected compact objects in the 2.5-5 solar-mass gap without knowing whether they are heavy neutron stars or small black holes. This paper tries to show that modified gravity can populate that gap without breaking the general-relativistic ceiling of 3 solar masses for static stars. It solves the Tolman-Oppenheimer-Volkoff equations for four vector f(R) gravity inflationary models and nine piecewise-polytropic equations of state. The central result is that the MPA1 equation of state produces maximum masses near 2.749 solar masses for every model, inside the mass-gap region, and that even cosmologically non-viable inflationary models give acceptable neutron-star phenomenology. If this is right, the identity of mass-gap objects could be heavy static neutron stars, and the equation of state rather than the inflationary model would decide which scenarios survive.","feed_headline":"Modified gravity reaches 2.75-solar-mass neutron stars","feed_subtitle":"Four inflationary models agree: MPA1 supports stars near the mass-gap ceiling, under the 3-solar-mass limit.","key_machinery":"The load-bearing machinery is the conformally transformed scalar-tensor system together with the physical-mass conversion formula. After the transformation $\\tilde{g}_{\\mu\\nu} = A^{-2}g_{\\mu\\nu}$ with $A(\\phi) = e^{\\phi/(2\\sqrt{6+\\beta^2})}$, each vector $f(R)$ model supplies a potential $V(\\phi)$ that enters the TOV equations along with the coupling $\\alpha(\\phi) = \\frac{1}{2}\\sqrt{6+\\beta^2}$. A double-shooting LSODA solver tunes the central values $\\nu_c$ and $\\phi_c$ so that the scalar field vanishes at numerical infinity, where the metric becomes Schwarzschild; Eq. (23) then converts the Einstein-frame ADM mass into the physical Jordan-frame mass, which receives contributions from outside the star because the scalar field does not vanish at the stellar surface. The four models share the same $A(\\phi)$ and $\\alpha(\\phi)$ and differ mainly in $V(\\phi)$, which is why their mass-radius curves nearly coincide.","core_discovery":"Vector $f(R)$ gravity, built by replacing the Ricci scalar $R$ with $R + A^\\mu A_\\mu + \\beta\\nabla_\\mu A^\\mu$ in the Lagrangian, becomes a scalar-tensor theory on shell, and its four inflationary models are solved numerically for static neutron stars. The paper reports that the four models produce almost indistinguishable Jordan-frame mass-radius curves. Confronted with the NICER bounds, a refined version of NICER, the PSR J0740+6620 constraints, and the CSI, CSII, and CSIII radius constraints, the MPA1 equation of state is the only one compatible with all of them for all four models, with maximum masses around 2.749 solar masses for each model, below the 3-solar-mass causal limit. The WFF1, MS1, and MS1b equations of state are excluded by the constraints used. The paper also asserts that a cosmologically non-viable inflationary model can still give viable neutron-star phenomenology, that this is a model-dependent feature, and closes by noting that the theoretical context complies with the general behavior of viable modified-gravity models rather than producing a new physics prediction.","pith_inferences":["The near-identical curves suggest the result is controlled by the shared conformal coupling functions rather than by the specific potential, so other scalar-tensor theories with the same $A(\\phi)$ and $\\alpha(\\phi)$ would likely give the same neutron-star phenomenology.","Because no general-relativistic baseline for the MPA1 equation of state is reported, an independent integration should reproduce the known GR maximum mass before the $2.75\\,M_\\odot$ value is taken at face value.","Tidal deformability, moment of inertia, and oscillation spectra are more sensitive to the scalar field than mass-radius curves are, so those observables are the most promising way to break the degeneracy among the four models.","The MPA1 scenario predicts a specific radius near 11.3 km at maximum mass, so a future radius measurement of a heavy neutron star candidate could test the scenario independently of mass alone."],"forward_implications":["The MPA1 equation of state in these vector $f(R)$ models supports static neutron stars up to about $2.749\\,M_\\odot$, inside the mass-gap region and below the $3\\,M_\\odot$ causal limit.","The WFF1, MS1, and MS1b equations of state are ruled out, while MPA1 satisfies all the observational constraints used in the paper.","A single mass-gap neutron star observation would not distinguish the four inflationary models, because their mass-radius curves are nearly identical.","Cosmological viability of the inflationary model is not a prerequisite for viable neutron-star phenomenology in this theory class.","The paper's conclusion implies that mass-gap objects could be static, non-rotating neutron stars without requiring masses above the $3\\,M_\\odot$ limit."],"supporting_citations":[{"why":"Supplies the vector f(R) gravity inflationary models and their Einstein-frame potentials, the theories under study.","marker":"[20]"},{"why":"Provides the piecewise-polytropic parameterization used to generate the equations of state.","marker":"[91]"},{"why":"Provides the MPA1 equation of state, which the paper finds compatible with all constraints.","marker":"[97]"},{"why":"Supplies the CSI radius constraints for 1.4 and 2 solar-mass neutron stars.","marker":"[78]"},{"why":"Supplies the CSII radius constraint for 1.4 solar-mass neutron stars.","marker":"[87]"},{"why":"Supplies the CSIII constraints on the 1.6 solar-mass radius and the maximum-mass radius.","marker":"[82]"},{"why":"Supplies the NICER and PSR J0740+6620 constraints used to judge the mass-radius curves.","marker":"[105]"},{"why":"Provides the LSODA double-shooting code on which the TOV integration is based.","marker":"[109]"},{"why":"Earlier scalar-tensor study of inflationary attractors that already singled out MPA1 as the best-fitting equation of state.","marker":"[11]"},{"why":"Defines the ADM mass used in the formula converting computed masses into physical Jordan-frame masses.","marker":"[102]"}],"fun_headline_variants":["MPA1 equation of state fits all neutron star data in vector f(R) gravity","Mass-gap neutron stars yielded by vector f(R) gravity","Vector f(R) gravity reaches mass-gap neutron stars","2.75-solar-mass neutron stars from vector f(R) gravity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's mass values rest on the assumption that the numerical integration and the formula converting the computed mass into the physical mass are correct, including the boundary condition that the scalar field vanishes at very large radius; the near-general-relativity model is never checked against the known GR maximum mass for the MPA1 equation of state, so a systematic offset in the reported masses cannot be ruled out.","fun_headline_variants_meta":{"raw":{"variants":["MPA1 equation of state fits all neutron star data in vector f(R) gravity","Mass-gap neutron stars yielded by vector f(R) gravity","Vector f(R) gravity reaches mass-gap neutron stars","2.75-solar-mass neutron stars from vector f(R) gravity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001183,"raw_usage":{"total_tokens":4987,"prompt_tokens":1147,"completion_tokens":3840,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":763,"completion_tokens_details":{"reasoning_tokens":3763}},"tokens_in":763,"tokens_out":3840,"duration_ms":25002,"temperature":1.0,"reasoning_tokens":3763,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:52:02.159329+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the mass-radius curve for the near-general-relativity model ($\\beta = 10^4$, coupling about $5\\times10^{-5}$) with the MPA1 equation of state and compare its maximum mass with the general-relativistic result for the same equation of state; the two must agree if the mass conversion is correct, yet the paper reports about $2.749\\,M_\\odot$ for that model and gives no GR baseline.","supporting_citations":[{"cited_title":"Neutron star equation of state: identifying hadronic matter characteristics","cited_arxiv_id":"2307.05086","evidence_quote":"Supplies the NICER and PSR J0740+6620 constraints used to judge the mass-radius curves."}],"review_version":1}