{"id":"9b618b35-38ff-416e-b9a1-8b35160c1844","arxiv_id":"2506.21646","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":6,"one_line_summary":"A fixed-point stability catalog for six f(Q) gravity models is presented, but the catalog tables contradict the paper's own equations.","lead":"This paper builds autonomous dynamical systems for six f(Q) gravity models and classifies their fixed points as stable, unstable, or saddle. The classification is meant to map which cosmic epochs each modified-gravity model can produce, but the tables do not match the equations in the same paper.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Table 2's point (1,-2,1) has Jacobian eigenvalues 3,3,-5/2 from the paper's own equations, not (-3,-3,-5/2); the central stability classification is reversed and the comparative claim fails.","rationale":"The reader's rationale already contains the Model II trace check, and my independent Jacobian calculation confirms it. I therefore agree with the rejection, though the reader's formal 'weakest_assumption' field points to non-hyperbolic classification rather than to the direct eigenvalue error; I count that as partial agreement. The direct eigenvalue mismatch is more load-bearing than the non-hyperbolic caveat because it invalidates a hyperbolic row on which linearization is reliable, and it indicates a systematic problem across the tables. A further compounding issue is that eq. (25) silently drops the factor (1+ω) while eqs. (14)/(21) keep ω arbitrary, so even a corrected eigenvalue computation would need to specify whether the analysis is dust-only. Finally, many tabulated points lie outside the phase space Ψ defined by 0≤Ω_m≤1 yet are interpreted as physical cosmological phases; this would also need to be addressed in any revised classification. None of these points require rejecting the authors' motivations; they show that the central comparative claim as printed is not supported.","tokens_in":17168,"tokens_out":11525,"duration_ms":120844,"concrete_test":"Symbolically differentiate eqs. (34)–(36) at the point (1,-2,1). If the Jacobian eigenvalues are 3,3,-5/2 rather than (-3,-3,-5/2) as in Table 2, the stability classification is reversed. Extend this symbolic recomputation to every row of Tables 1–6; the central claim stands only if every eigenvalue and stability label matches the printed autonomous systems.","verdict_should_be":"REJECT","load_bearing_attack":"Tables 1–6, the paper's central deliverable, do not follow from the autonomous systems printed in Section 3. A decisive check is Model II, F(Q)=Q+η log_e(αQ): at the point (1,-2,1) in Table 2, the Jacobian of eqs. (34)–(36) is [[3,3/2,0],[0,3,0],[-3/2,-3/2,-5/2]], whose eigenvalues are 3,3,-5/2. Table 2 reports (-3,-3,-5/2) and labels the point stable; the actual eigenvalues contain two positive real parts, so the point is not stable. This is a sign error in the stability classification, not a matter of convention or of non-hyperbolic center-manifold subtleties. Since all six tables use the same construction, one verified mismatch undermines the paper's comparative classification. The paper's own caveat (page 3) that linear analysis cannot classify non-hyperbolic points is also not honored in the zero-eigenvalue rows of Tables 1–6, because no complete center-manifold treatment is given for Models II–VI. The central claim of 'exactly those listed in Tables 1–6' is therefore unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies six f(Q) models in symmetric teleparallel gravity by constructing three-dimensional autonomous systems in the variables (x, y, u) defined in Eq. (18), listing fixed points and linearized stability in Tables 1-6, and interpreting the points as matter, radiation, de Sitter, or dark-energy phases. The central claim is that the fixed points and stability properties are exactly those listed in the tables, and that these points describe cosmological phases when the initial state is perturbed. The work uses the quasi-static matter-perturbation equation (17), carries free parameters (n, eta, alpha) from earlier f(Q) literature, and does not fit data.","tokens_in":17456,"tokens_out":11219,"duration_ms":112558,"significance":"The comparative classification would be a convenient reference if the tables were correct, and the inclusion of the perturbation variable u in the phase space is a useful feature. The paper also provides a systematic treatment and phase portraits for all six models, plus explicit center-manifold sketches for two non-hyperbolic cases. However, the central deliverable is not reproducible from the printed equations: the stability signs in several tables are reversed, and at least one table lists points that are not fixed points of the stated system. The analysis is derivative of existing f(Q) dynamical-system literature and offers no falsifiable new prediction; its value would rest entirely on the correctness of the classification.","major_comments":[{"comment":"At the fixed point (1,-2,1), the Jacobian of the printed system is [[3,3/2,0],[0,3,0],[-3/2,-3/2,-5/2]], whose eigenvalues are 3, 3, -5/2 (trace +7/2). Table 2 lists (-3,-3,-5/2) and calls the point stable. With two positive real parts, the point is not stable; this is a sign error in the central stability classification, not a non-hyperbolic subtlety. The same sign pattern appears in Tables 3, 5, and 6 (for example, for Model V at (0,0,1), Eqs. (49)-(51) give a Jacobian eigenvalue +3, while Table 5 lists -3). Since the paper's central claim is that the fixed points have exactly the properties listed in Tables 1 to 6, this mismatch is load-bearing.","section":"§3.2, Table 2, Eqs. (34)-(36)"},{"comment":"Setting x=1 and y=-2 in Eq. (46) gives u' = -u^2 - 3u + 3/2, whose roots are (-3 +/- sqrt(15))/2, not the values (-1 +/- sqrt(7))/2 reported in Table 4. Therefore the last two rows of Table 4 are not equilibrium points of the autonomous system (44)-(46); the sign of the final term in Eq. (46) appears inconsistent with the table. This directly affects the claimed stable point in Model IV.","section":"§3.4, Table 4, Eq. (46)"},{"comment":"The paper explicitly states that linear analysis is insufficient for non-hyperbolic equilibria, yet most non-hyperbolic rows (e.g., (1-y,y,-2) and (1-y,y,0) in Tables 2-6) are classified as stable, unstable, or saddle directly from the linearized eigenvalues. Center-manifold calculations are sketched only for Model I and Model V, in Sections 3.1 and 3.5; no analogous calculation is provided for Models II, III, IV, or VI. Hence the non-hyperbolic stability claims in those tables are not established.","section":"Section 1, page 3, and Tables 1-6"}],"minor_comments":[{"comment":"For the critical point (0,0,1), the text says 'Since u=-2, thus the matter density delta varies as a^{-2}', but the point has u=1; the sentence should state u=1 and delta proportional to a.","section":"§3.5, paragraph after Table 5"},{"comment":"The statement that n=1 recovers GR should be qualified, because f(Q)=alpha(-Q)^n with n=1 equals -alpha Q, which coincides with Q only for alpha=-1.","section":"§3.6"},{"comment":"Equation (17) is introduced as a quasi-static approximation, but the paper does not discuss the scales at which this approximation applies or compare it with the sub-horizon limit; a brief justification would improve the perturbation interpretation.","section":"Eq. (17)"},{"comment":"There are numerous typographical errors, e.g., 'Leimatre' in Section 3.5 and 'eads' in Section 3.3; a thorough proofread is needed.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The stress-test concern is confirmed by direct computation: Table 2's point (1,-2,1) has eigenvalues 3, 3, -5/2 from Eqs. (34)-(36), and Table 4's last two rows are not fixed points of Eq. (46). The errors are systematic and affect multiple central tables, so I do not see a local fix that would preserve the paper's comparative claim; the manuscript would need a full recomputation of the fixed points and Jacobians before it could be considered publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper's central tables don't survive contact with its own equations. On Model II, the point (1,-2,1) is listed in Table 2 with eigenvalues (-3,-3,-5/2) and labelled stable. But the Jacobian of eqs. (34)-(36) at that point is [[3,3/2,0],[0,3,0],[-3/2,-3/2,-5/2]], which has eigenvalues 3,3,-5/2. Two positive real parts means the point is not stable. That is not a sign convention; it is a reversed stability classification, and it appears in the paper's central deliverable.\n\nTo give credit where it's due: the paper does a clean job of setting up the x,y,u variables from the unified f(Q) dynamical system of ref [28], and it derives the autonomous equations for six known f(Q) forms without obvious algebra mistakes in the derivatives. The discussion of center manifolds, while incomplete, shows the authors know hyperbolic vs non-hyperbolic matters.\n\nThe soft spots are large. Table 3 and Table 4 have similar trace mismatches; Model V confuses F(Q) with f(Q); and several fixed points have x+y outside [0,1], the paper's own physical region, yet are interpreted as matter or dark-energy dominated. The paper also labels non-hyperbolic points as stable or saddle without the center-manifold analysis it says is required. Because all six tables are built the same way, the one verified error undermines the comparative claim. The novelty is thin: the variables and system come from ref [28], and the models have all been studied before, so the only possible value is a reliable catalog, and it is not reliable.\n\nThis is not a paper I would send to a serious referee in its current form. The central classification is internally inconsistent, and the derivative setup does not compensate. If the authors correct the Jacobians and redo the tables, a shorter, accurate catalog might be worth a look. As it stands, I would skip it.","headline":"The stability tables contradict the paper's own Jacobian equations; the central comparative classification is unsound.","tokens_in":18057,"tokens_out":4714,"would_cite":false,"duration_ms":47008,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.50.Kd","98.80.Jk","02.40.Yy","64.60.F"],"model":"deepseek-v4-flash","headline":"For six f(Q) gravity models, cosmic evolution reduces to a three-variable autonomous system whose fixed points—stable, saddle, or unstable—are the matter, radiation, de Sitter, and dark-energy eras.","keywords":["symmetric teleparallel gravity","f(Q) gravity","non-metricity scalar","dynamical system analysis","fixed points","phase portraits","matter perturbations","dark energy"],"falsifier":"A center-manifold reduction at the non-hyperbolic point $(1+2n,-2n,0)$ in the exponential model would settle whether the linear 'stable' label survives; observationally, the stable exponential-model point predicts $\\delta\\propto a^{(-7+\\sqrt{73})/4}$, so a late-time growth-rate measurement inconsistent with that scaling while the model sits in that phase would falsify the assignment.","tokens_in":16920,"feed_emoji":"🌌","tokens_out":10240,"duration_ms":112251,"temperature":0.7,"pith_summary":"The paper tries to establish that six different modified-gravity theories built from the non-metricity scalar $Q$, a measure of how much the connection fails to preserve lengths, can be compared on one footing by converting cosmology into a three-variable autonomous system. For each model it locates the fixed points of that system and classifies them as stable, unstable, or saddle, then reads every point as a cosmic epoch such as matter domination, radiation, cosmological-constant-like acceleration, or an era with growing or decaying matter perturbations. A sympathetic reader would care because this turns the question of whether an $f(Q)$ model produces the observed late-time acceleration into a check of whether a suitable stable fixed point exists, without solving the full nonlinear field equations.","feed_headline":"Stable cosmic endpoints found for five of six f(Q) models","feed_subtitle":"A three-variable dynamical system maps each modified-gravity model to the era that wins at late times.","key_machinery":"The carrying object is the three-dimensional autonomous system with variables $x=F/(6H^2)$, $y=-2F_Q$, and $u=\\mathrm{d}\\ln\\delta/\\mathrm{d}\\ln a$, where $F(Q)=f(Q)-Q$, $F_Q=\\mathrm{d}F/\\mathrm{d}Q$, and derivatives are taken with respect to $\\ln a$. The physical phase space is the strip $0\\le x+y\\le1$ with $\\Omega_m=1-x-y$; fixed points solve $(x',y',u')=(0,0,0)$, stability is read from the eigenvalues of the Jacobian matrix, and for non-hyperbolic points the paper supplements linear labels with a center-manifold approximation in the exponential and pure-quadratic models. This machinery turns each $f(Q)$ theory into a phase portrait whose attractors, repellers, and saddles are interpreted as cosmological epochs.","core_discovery":"The paper's central claim is that for each of the six functional forms $F(Q)=e^{nQ}$, $Q+\\eta\\ln(\\alpha Q)$, $Q+\\eta Q^{-1}$, $Q+\\eta Q^2$, $\\eta Q^2$, and $\\alpha(-Q)^n$, the autonomous system in $x$, $y$, $u$ has fixed points whose stability properties are exactly those listed in Tables 1 to 6. Those fixed points include cosmological-constant-like curves with $\\Omega_Q=1$, matter-dominated points with $\\Omega_m=1$, and hyperbolic points whose equation-of-state parameter matches the model's effective dark-energy behavior. The paper interprets stable fixed points as late-time attractors, unstable points as repellers or early stages, and saddle points as transitions between epochs, and it uses the $u$ variable to say whether matter overdensities grow or decay at each phase.","pith_inferences":["The linear labels attached to non-hyperbolic points are provisional, since the paper itself notes that center-manifold analysis is required; completing that reduction for the exponential and pure-quadratic models is the direct next step.","Because several fixed points are curves parametrized by $y$, the observationally relevant question is not which entire curve is stable but which segment of it is allowed by the matter-density constraint $0\\le\\Omega_m\\le1$; observational priors would select the physically realized part of each family.","The parameter-dependent stability of the $\\alpha(-Q)^n$ and $Q+\\eta Q^2$ models suggests that a bifurcation diagram in the model-parameter plane could organize all six cases into a single map of cosmic endpoints.","The quasi-static growth rates in the tables, such as $\\delta\\propto a^{(-7+\\sqrt{73})/4}$ for the stable exponential-model point, are direct predictions that redshift-space distortion surveys could test once the model parameters are fixed by other observations."],"forward_implications":["For the exponential model $F(Q)=e^{nQ}$, a hyperbolic stable fixed point exists with $u=(-7+\\sqrt{73})/4>0$, so matter perturbations grow while the universe approaches a dark-energy-like late-time phase.","For the logarithmic model $F(Q)=Q+\\eta\\ln(\\alpha Q)$, the fixed point $(1,-2,1)$ is hyperbolic stable with growing matter perturbations, while the cosmological-constant-like curve $(1-y,y,0)$ is a non-hyperbolic saddle.","For the inverse model $F(Q)=Q+\\eta Q^{-1}$, Table 3 shows no stable fixed point: all four critical points are unstable or saddle, so this form lacks a late-time attractor within the present analysis.","For the quadratic-correction and pure-quadratic models, stability of the non-hyperbolic cosmological-constant-like curves depends on the value of $y$, and the stable hyperbolic point of $Q+\\eta Q^2$ has $u=(\\sqrt{7}-1)/2$, meaning growing matter perturbations.","For the power-law model $F(Q)=\\alpha(-Q)^n$, the fixed point $(0,0,1)$ is stable for $n>1$ and unstable for $n<1$, making $n$ a switch between a stable and an unstable matter-growth phase."],"supporting_citations":[{"why":"Supplies the f(Q) field equations, the F(Q) split, and the standard model forms on which the six cases are based.","marker":"[27]"},{"why":"Gives the unified background-plus-perturbation dynamical-system treatment that motivates the x, y, u variables.","marker":"[28]"},{"why":"Provides the quasi-static matter perturbation equation from which the u growth-rate variable is derived.","marker":"[30]"},{"why":"Supports dropping time derivatives in the quasi-static limit that the perturbation equation and the u dynamics assume.","marker":"[32]"},{"why":"Supplies the exponential f(Q) model whose late-time acceleration and perturbation behavior the first fixed-point analysis extends.","marker":"[34]"}],"fun_headline_variants":["f(Q) gravity models mapped to stable cosmic endpoints","Six f(Q) models, one dynamical system: fixed points found","Late-time attractors emerge in f(Q) cosmology","Dynamical system reveals stability of f(Q) cosmologies","Which f(Q) model wins at late times? Fixed points say"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole stability map rests on the quasi-static approximation to the matter perturbation equation (time derivatives of the fluctuation are dropped) and on taking the linearized eigenvalues as decisive even at non-hyperbolic fixed points; if either assumption fails at the scales considered, the table classifications are not reliable.","fun_headline_variants_meta":{"raw":{"variants":["f(Q) gravity models mapped to stable cosmic endpoints","Six f(Q) models, one dynamical system: fixed points found","Late-time attractors emerge in f(Q) cosmology","Dynamical system reveals stability of f(Q) cosmologies","Which f(Q) model wins at late times? Fixed points say"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000485,"raw_usage":{"total_tokens":2363,"prompt_tokens":887,"completion_tokens":1476,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":503,"completion_tokens_details":{"reasoning_tokens":1392}},"tokens_in":503,"tokens_out":1476,"duration_ms":12201,"temperature":1.0,"reasoning_tokens":1392,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:36:28.073322+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A center-manifold reduction at the non-hyperbolic point $(1+2n,-2n,0)$ in the exponential model would settle whether the linear 'stable' label survives; observationally, the stable exponential-model point predicts $\\delta\\propto a^{(-7+\\sqrt{73})/4}$, so a late-time growth-rate measurement inconsistent with that scaling while the model sits in that phase would falsify the assignment.","supporting_citations":[{"cited_title":"Cosmology inf(q) geometry,","cited_arxiv_id":null,"evidence_quote":"Supplies the f(Q) field equations, the F(Q) split, and the standard model forms on which the six cases are based."},{"cited_title":"Cosmology inf(Q) gravity: A unified dynamical systems analysis of the background and perturbations,","cited_arxiv_id":null,"evidence_quote":"Gives the unified background-plus-perturbation dynamical-system treatment that motivates the x, y, u variables."},{"cited_title":"First evidence that non-metricityf(q) gravity could challenge λcdm,","cited_arxiv_id":null,"evidence_quote":"Provides the quasi-static matter perturbation equation from which the u growth-rate variable is derived."},{"cited_title":"New models and big bang nucleosynthesis constraints in f(q) gravity,","cited_arxiv_id":null,"evidence_quote":"Supports dropping time derivatives in the quasi-static limit that the perturbation equation and the u dynamics assume."},{"cited_title":"Analysis of the cosmological evolution parameters, energy conditions, and linear matter perturbations of an exponential-type model in f(q) gravity,","cited_arxiv_id":null,"evidence_quote":"Supplies the exponential f(Q) model whose late-time acceleration and perturbation behavior the first fixed-point analysis extends."}],"review_version":1}