{"id":"f53e5e26-3c5f-477d-8bee-4911f7ca1b9b","arxiv_id":"2506.07623","paper_version":3,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":9,"one_line_summary":"An f(R,G,T) power-law gravity model is fitted to cosmological data and reported to be moderately favored over the standard Lambda CDM model, with a Bayes factor of about 3.6.","lead":"The paper tests a modified-gravity theory whose Lagrangian depends on curvature, the Gauss-Bonnet invariant, and the trace of the matter energy-momentum tensor, and it fits one concrete version to supernova, cosmic chronometer, and galaxy clustering data. The authors report a Bayes factor of about 3.6 favoring this model over standard dark energy, together with a dynamical-systems reconstruction of cosmic history.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The fitted Lagrangian is not real on the trajectory: G<0 during decelerating epochs but G^m with m≈1.95 is nonintegral, so Eq. (35) is complex or undefined in the past.","rationale":"The reader's weakest assumption is exactly the concern I find load-bearing. I looked for other defects—absence of released code, typos such as the undefined 'g' in Eq. (26), the β inconsistency between Table I and the figure caption, and the missing AIC/BIC—but those affect reproducibility and presentation, whereas the reality of the Lagrangian affects validity. The derivation of the specific f from the conservation law in Sec. II.A is internally coherent, and the critical-point analysis is elaborate, but neither repairs the sign problem. A real scalar Lagrangian must be real-valued; a noninteger power of a negative Gauss-Bonnet invariant is not. Since Eq. (35) is the equation actually integrated to compute the likelihood, the reported parameter constraints and the Bayes factor cannot be attributed to the stated model. This is a mathematical flaw in the central argument, so the reader's REJECT verdict remains appropriate. A conditional acceptance could follow if the authors restrict m to integers or define G^m using |G|^m with explicit justification and then refit and rerun the analysis; as written, the central claim does not stand.","tokens_in":33057,"tokens_out":5407,"duration_ms":69649,"concrete_test":"Using the best-fit values from Table I (H0=66.1, Ωm0=0.294, α1=0.510, α2=0.500, α3=0.520, β=2.544, m=1.950), integrate Eq. (35) from z=0 to z=3 with RK45 and monitor s(z)=H(z)-(1+z)H'(z) at every accepted step. If s(z)<0 for any z (expected from the ΛCDM-like behavior in Fig. 2 for z≳0.6), then G<0 and (G/24)^m is non-real for m=1.95, proving the reported fit is not a real solution. A complementary check is to rerun the nested-sampling fit with G^m replaced by |G|^m; if ln(B0i) changes substantially across the Jeffreys 'moderate' threshold, the headline evidence depends on the undefined branch.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The model Lagrangian (34) is not real on the trajectory being fitted and integrated. From Eq. (21), G=24H^2(Hdot+H^2)=-24qH^4. In the radiation and matter eras q>0, so G<0; the paper's own Fig. 10b shows q>0 in the past. Table I gives m=1.950±0.021, a noninteger exponent, with a prior m∈[1,3] that does not restrict m to integers. Eq. (35) contains [H(z)^3(H(z)-(1+z)H'(z))]^m=(G/24)^m, so wherever the bracket is negative the right-hand side is complex or undefined. The paper never states that |G|^m is intended or that m must be integer. The same issue enters f_G=mα1G^{m-1} in Eqs. (25)-(26) and the dynamical-system variables x6,x7 in Sec. IV.B. Consequently the MCMC likelihood, the reported H(z), the Bayes factor ln(B0i)=3.5994, and the critical-point analysis are not predictions of the stated real f(R,G,T) model. This is an internal inconsistency, not merely a disagreement with current consensus.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies a four-dimensional f(R,G,T) gravity with Lagrangian f(R,G,T)=α1G^m+α2R^β−2α3√(−T), where G is the Gauss-Bonnet invariant and T is the trace of the matter energy-momentum tensor. It derives the generalized Friedmann equations, specializes them to a pressureless dust universe, and obtains a second-order ODE for H(z) (Eq. 35). The model is fitted to cosmic-chronometer, Type Ia supernova, and DESI BAO data with the PolyChord nested sampler; the authors report a Bayes factor ln(B0i)=3.5994, which they interpret as moderate evidence in favor of the f(R,G,T) model over ΛCDM. The paper also rewrites the background equations as a four-dimensional autonomous dynamical system, identifies eight critical points, and uses linear stability analysis to associate them with radiation, matter, late-time acceleration, and phantom eras. The central claim is that this model is a viable background-level alternative to dark energy.","tokens_in":33307,"tokens_out":5816,"duration_ms":72700,"significance":"If the central claim were sound, the paper would be a useful contribution to modified-gravity cosmology: it provides explicit field equations, reports priors and data sets, uses the full cosmic-chronometer covariance matrix, and makes a concrete model comparison through Bayesian evidence. The derivation of the T-dependent term from the conservation law (Eqs. 31-34) is a clear and transparent step, and the use of external cosmological data is appropriate. However, the main quantitative results are undermined by an internal inconsistency: the fitted Lagrangian is not real on the decelerating part of the trajectory, so the ODE being integrated and the dynamical system being analyzed are not the ones stated. This is not a disagreement with current consensus but a mathematical flaw in the model as presented, and it invalidates the reported statistical and dynamical conclusions.","major_comments":[{"comment":"The fitted Lagrangian is not real on the trajectory over which Eq. (35) is integrated. From Eq. (21), G=24H^2(Ḣ+H^2)=-24qH^4 on an FLRW background. The paper's own Fig. 10b shows q>0 in the past, with the transition at N≈-0.29, so G<0 throughout the radiation- and matter-dominated epochs that a fit spanning z=0 to z=3 must include. Table I gives m=1.950±0.021 with a prior m∈[1,3] that does not restrict m to integers, so G^m and G^{m-1} are complex or undefined for those epochs. The term [H(z)^3(H(z)-(1+z)H'(z))]^m in Eq. (35) is (G/24)^m, and the same nonintegral power enters f_G in Eqs. (25)-(26) and the variables x6,x7 in Eqs. (39),(59). The manuscript nowhere states that |G| is used or that m must be an integer. Consequently the MCMC likelihood, the reported H(z), the Bayes factor ln(B0i)=3.5994, and the critical-point analysis are not predictions of the stated real f(R,G,T) model.","section":"II.A, III.B, IV.B (Eqs. 21, 34, 35; Table I; Fig. 10b)"},{"comment":"The reported posterior for β is internally inconsistent. Table I lists β=2.544±0.021, while Fig. 1 gives β≈2.449 with asymmetric errors (+0.021/−0.025). Since β multiplies R^β in the Lagrangian and appears throughout Eq. (35), the dynamical-system equations, and the eigenvalue expressions, the two central values cannot both describe the same fit. This makes the statistical results irreproducible even if the reality issue in the previous comment were resolved.","section":"III.B (Table I vs Fig. 1)"}],"minor_comments":[{"comment":"Equations (26) and (30) contain notation errors that obstruct verification: the terms \"R fr\" and \"g\" should presumably read \"R f_R\" and \"f\", and \"¨f R\" in Eq. (30) should read \"\\ddot f_R\". The authors should correct these typos and re-check the subsequent derivation of Eq. (35).","section":"II.A (Eqs. 26 and 30)"},{"comment":"The text oscillates between describing the sampler as MCMC and as nested sampling; PolyChord is a nested-sampling algorithm, so the first sentence of Section III.A and the later statements about MCMC samples should be made consistent.","section":"III.A"},{"comment":"The concluding section states that the statistical relevance of the results was estimated using AIC, BIC, and p values, but no AIC, BIC, or p values are reported anywhere in the paper; either these quantities should be reported or the sentence should be removed.","section":"VI (Conclusions)"},{"comment":"The symbol G is used for Newton's gravitational constant in Eq. (1) and for the Gauss-Bonnet invariant throughout the rest of the paper; this notational conflict should be resolved, for example by writing G_N for Newton's constant.","section":"Eq. (1) and throughout"}],"recommendation":"reject","confidential_remarks":"The central problem is an internal inconsistency in the model definition, not a disagreement with current consensus. The non-real Lagrangian issue is decisive: because G<0 during the decelerating eras and the fitted m is nonintegral, the ODE integrated for the MCMC analysis is not the real f(R,G,T) model stated in the paper. A routine revision that merely corrects typos or rewords claims would not suffice; the authors would need to reformulate the model (for example, with |G|^m or with m restricted to integers) and rerun all fits and the dynamical-system analysis. The additional discrepancy between the Table I and Fig. 1 values for β reinforces my view that the current manuscript is not reliable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plainly: this is a workmanlike f(R,G,T) paper. The specific power-law Lagrangian f=α1G^m+α2R^β−2α3√(−T) is new for this theory, and the authors do a real job: they derive the generalized Friedmann equations, fit to CC+SNe+DESI DR2 with PolyChord, report ln B=3.6 over ΛCDM, then reduce the system to a 4D dynamical system and classify eight critical points. The DESI DR2 constraints are current, and the derivation of the √(−T) term from the matter-conservation condition is a nice touch. The phase-space analysis is thorough, and the paper is honest that H0 comes out near 66, so the Hubble tension is not solved.\n\nThe soft spot is not minor. The Lagrangian is not real on the trajectory being fitted. From Eq. (21), G=24H^2(Ḣ+H^2)=−24qH^4. During the radiation and matter eras q>0 — their own Fig. 10b shows this — so G<0. The fitted value m≈1.95 is non-integer, so G^m is complex or undefined in the past. The same quantity enters f_G=mα1G^{m−1} and the dynamical-system variables x6, x7. The paper never states that |G|^m is meant or that m is restricted to integers. So the ODE being integrated, the likelihood, the Bayes factor, and the physical interpretation of the fixed points are not predictions of the stated real f(R,G,T) model. That is a load-bearing internal inconsistency.\n\nSmaller issues: Table I gives β=2.544±0.021 while Fig. 1 shows 2.449 with asymmetric errors; the conclusions cite AIC/BIC but the paper does not report them; Eqs. (26) and (30) have apparent typos. No code is released, which makes reproduction harder.\n\nI do not think the circularity concern is the real problem: the √(−T) term is fixed by conservation and the same conservation gives ρ(z), but that is model construction, not fitting to the same data. The reality issue is what sinks the current version.\n\nThe work deserves a serious referee: the core idea is salvageable by restricting m to integers or using |G|^m with justification, then rerunning the fit. I would not cite it in this form, but I would want to see the revised version. If you have a student in modified gravity, this is a good case study in checking the sign of G before integrating.","headline":"The model is new and the fitting is careful, but the stated Lagrangian is complex on the fitted trajectory, so the main statistical claim does not hold.","tokens_in":33911,"tokens_out":4477,"would_cite":false,"duration_ms":51875,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83D05","83F05"],"pacs":["04.50.Kd","98.80.-k"],"model":"deepseek-v4-flash","headline":"The paper claims that a specific f(R,G,T) gravity model—a mix of Gauss-Bonnet, Ricci-scalar, and matter-trace powers—fits combined cosmic data with moderate Bayesian evidence over ΛCDM and its dynamical system reproduces all standard…","keywords":["f(R,G,T) gravity","Gauss-Bonnet invariant","modified gravity","dark energy","dynamical system analysis","Bayesian model comparison","cosmic chronometers","baryon acoustic oscillations"],"falsifier":"Evaluate $G(z)=24H^2(\\dot H+H^2)$ along the best-fit $H(z)$ from the paper; wherever $G<0$, check whether the numerical integration evaluated $(G)^m$ or $|G|^m$. If it integrated the former, the fitted ODE is complex-valued in the matter and radiation epochs; if the latter, the model's Lagrangian differs from the stated one. Either way, the reported Bayes factor and critical-point analysis depend on which branch was chosen.","tokens_in":32801,"feed_emoji":"🌌","tokens_out":7730,"duration_ms":84874,"temperature":0.7,"pith_summary":"This paper tries to establish that a particular higher-curvature theory of gravity—an action that mixes powers of the Gauss-Bonnet invariant, the Ricci scalar, and the trace of the matter stress-energy tensor—can serve as a background-level alternative to the cosmological constant. The chosen Lagrangian $f(R,G,T)=\\alpha_1 G^m+\\alpha_2 R^\\beta-2\\alpha_3\\sqrt{-T}$ is fitted to combined cosmic-chronometer, Type Ia supernova, and DESI BAO data; the authors report $\\ln(B_{0i})=3.5994$, which they read as moderate evidence for this model over $\\Lambda$CDM. They also recast the Friedmann equations as an autonomous dynamical system and identify eight critical points whose stability reproduces radiation, matter, late acceleration, and phantom eras. If right, the model shows that geometry–matter coupling can mimic the standard cosmological history without a cosmological constant, at the price of a richer parameter space.","feed_headline":"Modified gravity model edges past ΛCDM in cosmic data fit","feed_subtitle":"A Gauss-Bonnet–Ricci–matter Lagrangian fits CC, supernova, and BAO data with ln(B)=3.60 and reproduces every cosmic epoch.","key_machinery":"The load-bearing object is the explicit Lagrangian $f(R,G,T)=\\alpha_1 G^m+\\alpha_2 R^\\beta-2\\alpha_3\\sqrt{-T}$, where $G=R^2-4R_{\\mu\\nu}R^{\\mu\\nu}+R_{\\mu\\nu\\xi\\eta}R^{\\mu\\nu\\xi\\eta}$ is the Gauss-Bonnet topological invariant. The $\\sqrt{-T}$ term is chosen so that dust matter is covariantly conserved, and the full form feeds into generalized Friedmann equations whose Hubble evolution is a second-order nonlinear ODE solved numerically for the fit. For the stability analysis, seven dimensionless variables $x_1,\\dots,x_7$, built from $f$ and its derivatives, reduce the system to four first-order ODEs; the critical points $P_1,\\dots,P_8$ and their Lyapunov linear stability supply the cosmological epochs.","core_discovery":"The central claim is that the Lagrangian density $f(R,G,T)=\\alpha_1 G^m+\\alpha_2 R^\\beta-2\\alpha_3\\sqrt{-T}$ is both observationally viable and dynamically complete. A nested-sampling Bayesian fit to cosmic chronometers, Type Ia supernovae without the SHOES calibration, and DESI DR2 BAO data yields $H_0=66.1\\pm3.8$ km/s/Mpc, $\\Omega_{m0}=0.294\\pm0.022$, and a Bayes factor $\\ln(B_{0i})=3.5994$, which the paper interprets as moderate support over $\\Lambda$CDM. The same model, when recast in terms of seven dimensionless phase-space variables, reduces to a four-dimensional autonomous system whose eight critical points include a radiation epoch, a matter saddle, a transition at $N\\approx-0.29$, and a stable late-time accelerating attractor that can reach the phantom regime. The paper also reports the background observables $q_0\\approx-0.57$, $\\Omega_{m0}\\approx0.24$, $j_0\\approx1.05$, and $s_0\\approx0$.","pith_inferences":["A reader should note that the fitted $m\\approx1.95$ is not an integer and the Gauss-Bonnet invariant changes sign during decelerating epochs, so the reported solution depends on an unstated branch choice for $G^m$; checking whether $|G|^m$ was used would alter the likelihood surface and possibly the Bayes factor.","The same Lagrangian could be tested at the perturbation level: the paper analyzes only background dynamics, so whether the extra $f_G$ terms introduce ghost or gradient instabilities, as in earlier $f(G)$ models, remains open.","Adding a radiation pressure term would close the loop with early-universe epochs and could sharpen the predicted transition redshift and phantom crossing, which the present dust-only treatment leaves as future work.","If the model is correct, the $\\sqrt{-T}$ matter coupling implies that baryonic matter is not described by a purely metric-compatible conservation law at the perturbative level; a fifth-force-like signature in structure formation would be a distinctive test."],"forward_implications":["The combined CC+SNe+BAO fit gives $H_0=66.1\\pm3.8$ km/s/Mpc and $\\Omega_{m0}=0.294\\pm0.022$, consistent with Planck and DESI DR2 values, so the model does not obviously worsen the Hubble tension.","The Bayes factor $\\ln(B_{0i})=3.5994$ places the model in the moderate-evidence band relative to $\\Lambda$CDM; the paper reads this as marginal preference, not a ruling out.","The dynamical-system analysis yields a complete background history: radiation-dominated critical points, a matter-dominated saddle, transition at $N\\approx-0.29$, and a late-time de Sitter or phantom attractor, so the model can in principle reproduce the full expansion history without a scalar field.","The statefinder pair $(j_0,s_0)\\approx(1.05,0)$ lies near the $\\Lambda$CDM values, making the model difficult to distinguish from $\\Lambda$CDM on background data alone.","The model predicts a richer set of possible phases, including Chaplygin-gas, quintessence, and phantom eras, depending on the fitted parameters $m$ and $\\beta$."],"supporting_citations":[{"why":"It defines the f(R,G,T) action and field equations whose Friedmann limit the paper fits and analyzes.","marker":"[153]"},{"why":"It supplies the Planck values of $H_0$, $\\Omega_{m0}$, and the sound horizon $r_d$ used as the ΛCDM baseline and BAO calibration.","marker":"[12]"},{"why":"It supplies the 15 cosmic-chronometer Hubble measurements in the redshift range used for the joint likelihood.","marker":"[160]"},{"why":"It adds the high-redshift cosmic-chronometer Hubble measurement near $z\\sim2$ used in the fit.","marker":"[161]"},{"why":"It contributes the $z\\sim0.45$ cosmic-chronometer measurement that sharpens the low-redshift Hubble constraints.","marker":"[162]"},{"why":"It supplies the Type Ia supernova distances without the SHOES calibration used in the combined statistical analysis.","marker":"[166]"},{"why":"It supplies the DESI DR2 BAO distance ratios used to constrain the expansion history at high redshift.","marker":"[170]"}],"fun_headline_variants":["f(R,G,T) gravity: fits data, stable late-time attractor","Modified gravity with GB, Ricci, matter passes cosmic tests","Bayesian fit: f(R,G,T) model edges out ΛCDM","Dynamical analysis of Gauss-Bonnet–Ricci–matter gravity","New gravity theory fits cosmic data, yields stable epochs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The fit assumes $\\alpha_1 G^m$ is a real function along the whole trajectory, but the fitted non-integer $m\\approx1.95$ makes $G^m$ undefined when the Gauss-Bonnet invariant $G=24H^2(\\dot H+H^2)$ turns negative during the decelerating matter and radiation epochs.","fun_headline_variants_meta":{"raw":{"variants":["f(R,G,T) gravity: fits data, stable late-time attractor","Modified gravity with GB, Ricci, matter passes cosmic tests","Bayesian fit: f(R,G,T) model edges out ΛCDM","Dynamical analysis of Gauss-Bonnet–Ricci–matter gravity","New gravity theory fits cosmic data, yields stable epochs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000255,"raw_usage":{"total_tokens":1691,"prompt_tokens":1181,"completion_tokens":510,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":797,"completion_tokens_details":{"reasoning_tokens":417}},"tokens_in":797,"tokens_out":510,"duration_ms":6902,"temperature":1.0,"reasoning_tokens":417,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:31:31.892656+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate $G(z)=24H^2(\\dot H+H^2)$ along the best-fit $H(z)$ from the paper; wherever $G<0$, check whether the numerical integration evaluated $(G)^m$ or $|G|^m$. If it integrated the former, the fitted ODE is complex-valued in the matter and radiation epochs; if the latter, the model's Lagrangian differs from the stated one. Either way, the reported Bayes factor and critical-point analysis depend on which branch was chosen.","supporting_citations":[{"cited_title":"Kawai and J","cited_arxiv_id":null,"evidence_quote":"It defines the f(R,G,T) action and field equations whose Friedmann limit the paper fits and analyzes."},{"cited_title":"Jawad, M","cited_arxiv_id":null,"evidence_quote":"It supplies the 15 cosmic-chronometer Hubble measurements in the redshift range used for the joint likelihood."},{"cited_title":"Capozziello, C","cited_arxiv_id":null,"evidence_quote":"It adds the high-redshift cosmic-chronometer Hubble measurement near $z\\sim2$ used in the fit."},{"cited_title":"Bahamonde, C","cited_arxiv_id":null,"evidence_quote":"It contributes the $z\\sim0.45$ cosmic-chronometer measurement that sharpens the low-redshift Hubble constraints."},{"cited_title":"”Raising the bar: new constraints on the Hubble parameter with cosmic chronometers at z ∼ 2.” Monthly Notices of the Royal Astronomical Society: Letters 450.1 (2015): L16–L20","cited_arxiv_id":null,"evidence_quote":"It supplies the Type Ia supernova distances without the SHOES calibration used in the combined statistical analysis."},{"cited_title":"”Setting the stage for cosmic chronometers","cited_arxiv_id":null,"evidence_quote":"It supplies the DESI DR2 BAO distance ratios used to constrain the expansion history at high redshift."}],"review_version":1}