{"id":"9171d232-2856-4909-be8b-718ca8829bdc","arxiv_id":"2501.19141","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"Energy-momentum squared gravity can reproduce flat galactic rotation curves only because its free parameter s is set from the observed velocities.","lead":"This paper applies energy-momentum squared gravity, a modified theory of gravity, to galaxy rotation curves and claims the theory can mimic dark matter. It finds that a specially chosen metric ansatz yields a flat rotation curve, but the flatness is built into the ansatz and the key parameter is fitted to observations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Flat rotation curve is imposed by the power-law ansatz (14), not derived from the EMSG term; the paper's own Sec. V admits alpha is only a second-order effect.","rationale":"The reader's weakest assumption correctly identifies the metric ansatz (14) as the load-bearing premise. My independent reading confirms this and strengthens it with a direct quotation from the paper's Sec. V, where the authors admit that the modified-gravity features appear due to relation (14) and that alpha is a second-order effect. This makes the concern not merely an external suspicion but an internal admission: the flat rotation curve is a consequence of the chosen l(r), not of the EMSG dynamics. The reader's REJECT verdict is therefore appropriate. I found no reason to soften the verdict; the analytic expressions are explicit and could be verified, but they do not rescue the central claim because the key result (36) is a restatement of the ansatz. The only independent content—the lensing and radar-delay formulas—depends on the same parameter s and is not tested against data with uncertainties. Thus the concern lands, and the verdict should remain unchanged.","tokens_in":9717,"tokens_out":7759,"duration_ms":81257,"concrete_test":"Recompute the rotation curve from Eq. (36) with alpha set to zero and compare with the full expression. If v^2 = s/2 at leading order independently of alpha, the flat curve does not arise from the energy-momentum squared term. Additionally, examine the exact density solutions (23)-(24) in the limit alpha -> 0; if they diverge or become singular, the 'near GR' expansion is not a valid physical limit and the alpha-independent leading-order result is an artifact of the assumed ansatz (14).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that the model 'through the role of the energy-momentum squared term' produces flat rotation curves is not supported by the derivation. The flatness is an algebraic consequence of the assumed metric ansatz e^{a+b} = (r/sigma)^s (Eq. 14), which gives a'+b' = s/r (Eq. 16). Combined with the standard weak-field relation v^2 = r a'/2 (Eq. 35), this yields v^2 ≈ s/2 with corrections that are suppressed by s^2 and multiplied by alpha (Eq. 36). Thus at leading order the rotation velocity is independent of alpha, the EMSG coupling. The paper itself states in Sec. V: 'Actually, the modified gravity features appears due to the adoption of relation (14) since the parameter alpha enters into the dynamics as a second order effect as seen in (36).' This is an explicit admission that the dark-matter-like signature is not produced by the energy-momentum squared term but by the ad hoc power-law ansatz. The density profile arising from that ansatz is rho ∝ s/(kappa r^2) (Eq. 25), i.e., the isothermal-sphere profile known to yield flat rotation curves in Newtonian gravity; it is inserted by hand, not predicted by the field equations. The later lensing and radar-delay sections inherit the same fitted s and therefore provide no independent confirmation. A further internal issue is that the exact solutions (23)-(24) contain 1/alpha terms, so the alpha -> 0 limit is singular; the expansion near GR is not a controlled perturbation, making the 'near general relativity' framing misleading. The load-bearing premise of the paper is the unjustified ansatz (14), and the paper's own remarks concede that the EMSG term is not the origin of the flat rotation curve.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies energy-momentum squared gravity (EMSG) as a possible explanation of dark matter on galactic scales. After deriving the field equations for a static, spherically symmetric dust spacetime, the authors impose the metric ansatz e^{a+b}=(r/\\sigma)^s (Eq. 14) with s\\ll 1. They solve for the density and metric functions, obtain an approximate rotation velocity v^2\\approx s/2 (Eq. 36), identify s/2 with the observed squared circular speed, and claim that the model 'effectively illustrates the presence of dark matter.' The paper then uses the same fitted metric to compute light deflection angles and radar echo delays, with figures for four galaxies.","tokens_in":10029,"tokens_out":4861,"duration_ms":51544,"significance":"If the central claim were correct, the paper would provide a concrete modified-gravity mechanism for flat galactic rotation curves without particle dark matter. The manuscript is clearly written in structure and includes explicit field equations, a transparent parameter count (s, \\sigma, \\alpha), and a candid closing admission about the role of \\alpha. However, the derivation does not supply what it claims: the flat rotation curve is effectively inserted by the power-law ansatz, and the EMSG coupling \\alpha only appears at second order. Because the paper's own Section V concedes this, the result, as stated, does not establish a dark-matter effect of EMSG. The lensing and radar-delay sections inherit the same fitted parameter s and do not provide an independent test of the model.","major_comments":[{"comment":"The flat rotation curve is not derived from the EMSG field equations; it is imposed by the ansatz e^{a+b}=(r/\\sigma)^s. This ansatz immediately gives a'+b'=s/r (Eq. 16). Combined with the weak-field formula v^2=ra'/2 (Eq. 35) and the solution (26) for e^a, the result v^2\\simeq s/2 follows at leading order. The accompanying density profile (25), rho\\simeq s/(\\kappa r^2), is the isothermal-sphere profile that is already known to give flat rotation curves in Newtonian gravity. Thus the flat curve is a consequence of choosing a logarithmic time-time metric component, not a prediction of the energy-momentum squared term.","section":"Sec. III, Eqs. (14)-(16), (35)-(36)"},{"comment":"The central claim stated after Eq. (36) is that the model, 'through the role of the energy-momentum squared term in the field equations for pressureless matter,' illustrates dark matter. This is directly contradicted by the paper's own closing remark in Sec. V: 'the modified gravity features appears due to the adoption of relation (14) since the parameter \\alpha enters into the dynamics as a second order effect as seen in (36).' Since the leading-order flat rotation velocity is independent of \\alpha, the dark-matter-like signature is not attributable to the energy-momentum squared correction.","section":"Sec. V and Sec. III after Eq. (36)"},{"comment":"The 'near general relativity' expansion is not a controlled perturbative limit. The exact density solutions (23) and (24) contain 1/\\alpha terms, so the \\alpha\\to 0 limit is singular, while the later expansions in equations (26)-(29) treat \\alpha s^2 as a small correction without defining the dimensionless small parameter. This makes the claim that the model stays 'in the vicinity of general relativity' (Eq. 15 and surrounding text) insufficiently supported. A proper perturbative derivation in which the EMSG coupling is demonstrably small and subdominant is needed before the flat-curve result can be attributed to the modified-gravity sector.","section":"Sec. III, Eqs. (23)-(29)"}],"minor_comments":[{"comment":"The conserved angular momentum should be J=r^2 d\\phi/d\\tau, not J=r^2(d\\phi/d\\tau)^2 as written; the subsequent use in Eq. (33) is only consistent with the former definition.","section":"Eq. (32)"},{"comment":"The radar echo delay expression appears dimensionally inconsistent: the first term contains a factor s/\\sigma^2 multiplied by r_d^{s-2}, which does not have dimensions of length in geometric units. Please check the derivation and the displayed formula.","section":"Eq. (43)"},{"comment":"The split of the integration domain at r_d introduces a sharp transition between the 'dark matter' region and the exterior Schwarzschild region, but no matching or continuity condition for the metric or its derivative is stated. This weakens the quantitative status of the deflection-angle result.","section":"Sec. IV A, Eq. (39)"},{"comment":"The text says Eq. (22) is obtained by eliminating \\rho^2 from Eq. (19), but Eq. (22) also involves \\rho'' and \\rho'^2 and appears to require a separate combination of equations. The derivation should be made explicit.","section":"Sec. III, Eqs. (19)-(22)"}],"recommendation":"reject","confidential_remarks":"The paper's central claim is undermined by its own Section V: the flat rotation curve follows from the power-law ansatz (14), with the EMSG coupling \\alpha entering only at second order. I do not see a way to repair this within the manuscript's current scope, because the advertised dark-matter effect of the energy-momentum squared term is not actually demonstrated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this paper doesn't demonstrate what its abstract claims. The flat rotation curve follows directly from the metric ansatz e^(a+b) = (r/sigma)^s (Eq. 14), which gives a'+b' = s/r; combined with the standard weak-field formula v^2 = r a'/2, you get v^2 ≈ s/2 at leading order, independent of the EMSG coupling alpha. The authors basically admit this in Sec. V: \"the modified gravity features appears due to the adoption of relation (14) since the parameter alpha enters into the dynamics as a second order effect.\" So the headline result—dark matter as an effect of the energy-momentum squared term—is not derived; it's installed by hand.\n\nWhat's actually new: the explicit metric components (Eqs. 26-29), the density profile rho ≈ s/(kappa r^2) (Eq. 25), and the lensing/delay expressions are concrete and could be checked. The density profile is the familiar isothermal-sphere form that Newtonian gravity already knows produces flat curves, and s is then fixed from observed velocities. That's a fit, not a prediction.\n\nSoft spots, roughly in order of weight: (1) the ansatz (14) is presented as \"plausible\" with no independent justification; the result is circular. (2) Alpha enters only at second order, so the claimed mechanism isn't the mechanism. (3) The exact solutions (23)-(24) have 1/alpha terms, making the alpha→0 limit singular; the \"near GR\" expansion isn't controlled. (4) The radar-delay expression (43) appears to have a dimensional inconsistency, and no junction conditions are stated for matching to Schwarzschild at rd. (5) The lensing/delay plots have no uncertainties and no actual fit to data—different s values are just shown.\n\nWho's it for? Someone cataloging EMSG solutions might find the analytic forms useful, but as a dark matter explanation it doesn't hold up. I wouldn't cite it, and I wouldn't bring it to a reading group. If it crossed my desk, I'd reject; if I were generous, I'd suggest the authors reframe it as a fitting exercise in EMSG and do the data comparison properly, but that's a different paper.\n\nRecommendation: don't send to review; the load-bearing premise is conceded in the authors' own remarks.","headline":"The flat rotation curve is imposed by the power-law ansatz, not produced by EMSG—despite the abstract's claim, and the paper's own Sec. V says as much.","tokens_in":10652,"tokens_out":4309,"would_cite":false,"duration_ms":40041,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.50.Kd","95.35.+d"],"model":"deepseek-v4-flash","headline":"Energy-momentum squared gravity can flatten galaxy rotation curves without dark matter, the paper argues, while predicting testable lensing and radar-delay effects.","keywords":["energy-momentum squared gravity","dark matter","galactic rotation curves","flat rotation curve","modified gravity","light deflection","radar echo delay","spherically symmetric spacetime"],"falsifier":"Take a galaxy with both a measured rotation curve and measured lensing deflection: fit $s$ from $v^2\\simeq s/2$, then check whether the predicted deflection (Eq. 40) and the $\\rho\\propto r^{-2}$ density profile match the data; a substantial mismatch would refute the claim. A second decisive check is to solve the EMSG field equations for dust without imposing Eq. (14) and see whether the resulting rotation curve is still flat.","tokens_in":9434,"feed_emoji":"🌀","tokens_out":12067,"duration_ms":104780,"temperature":0.7,"pith_summary":"The paper aims to establish that energy-momentum squared gravity (EMSG), an extension of general relativity obtained by adding the scalar $T_{\\mu\\nu}T^{\\mu\\nu}$ to the Lagrangian, can account for the flat rotation curves of galaxies without invoking particle dark matter. Restricting attention to static, spherically symmetric, pressureless dust and to metrics close to general relativity, the authors derive a circular orbital speed whose square is approximately $s/2$, independent of the radial coordinate, with the small parameter $s$ entering through $e^{a+b}=(r/\\sigma)^s$. Identifying $s/2$ with the observed $v^2\\sim10^{-6}$ for halo speeds of roughly $200$–$500$ km/s gives a parameter-light modified-gravity route to the dark-matter phenomenology of galaxy rotation, and the same spacetime predicts modified light deflection and radar echo delays that reduce to the general-relativistic values when the halo contribution is switched off.","feed_headline":"Squared-matter term in gravity flattens galaxy rotation curves","feed_subtitle":"A dust-filled halo then moves at constant speed set by s/2, with testable lensing and radar-delay effects.","key_machinery":"The load-bearing object is the ansatz $e^{a+b}=(r/\\sigma)^s$, equivalently $a'+b'=s/r$ with $s\\ll 1$, imposed on the combination of metric functions $a(r)$ and $b(r)$ in the static spherical line element. Its work is to convert the constant parameter $s$ into a constant orbital speed through the weak-field formula $v^2=ra'/2$, while the EMSG source term $T_{\\mu\\nu}T^{\\mu\\nu}$ supplies an effective $\\rho^2$ pressure that makes the halo density fall as $r^{-2}$; the coupling $\\alpha$ enters the velocity only at order $s^2$, so the flat-curve prediction is carried by the ansatz rather than by the strength of the squared-matter term.","core_discovery":"On the paper's own terms, the discovery is that a dust-filled static spherically symmetric spacetime in EMSG, taken in the vicinity of general relativity, has a halo rotation velocity that is essentially independent of radius: $v^2\\approx s/2$ up to subleading terms involving the coupling $\\alpha$, so the tangential speed is flat and the energy-momentum squared term effectively plays the role of dark matter. The supporting density profile is $\\rho\\approx s/(\\kappa r^2)$, and the same metric yields deflection angles and radar echo delays that grow with the halo scales of the model, returning to the general-relativistic $4GM/r_0$ deflection and the standard radar echo delay when the dark-matter region vanishes.","pith_inferences":["The flat curve is in effect installed by the assumed $a'+b'=s/r$; a decisive test would be to derive this relation from a microphysical Lagrangian or to solve the EMSG field equations numerically without the ansatz and see whether flatness survives.","Because the leading velocity depends only on $s$ and not on $\\alpha$, the mimicry may be generic to any modified gravity that enforces a slowly varying $a+b$; comparing theories on this point would show what is unique to EMSG.","A falsifiable cross-check is to use one galaxy's fitted $s$ to predict both its lensing deflection and its radar echo delay, since all three observables are tied to the same parameter.","The $\\rho\\propto r^{-2}$ halo implies a total mass that keeps growing with radius until the halo edge, a consequence that puts additional constraints on galaxy masses and satellite dynamics beyond the rotation curve itself."],"forward_implications":["Galactic rotation curves are flat in the halo with $v^2\\simeq s/2$, so fitting observed speeds of $200$–$500$ km/s fixes $s\\sim10^{-6}$, independent of $\\alpha$ at leading order.","The halo density follows $\\rho\\propto r^{-2}$, a definite mass-profile prediction inside the dark-matter region.","The light deflection angle grows with $s$, so a galaxy's fitted rotation speed gives a predicted gravitational-lensing signature from the same parameter.","The radar echo delay increases with the halo radius $r_d$, with the general-relativistic result recovered as $r_d\\to r_0$.","The analysis is restricted to weak-field, pressureless matter, so the model's claim of dark-matter mimicry is so far limited to that regime."],"supporting_citations":[{"why":"It defines energy-momentum squared gravity and its effective field equations, the theory being tested.","marker":"[35]"},{"why":"It supplies the device $e^{a(r)+b(r)}=l(r)$ that motivates the ansatz (14) producing the flat rotation curve.","marker":"[42]"},{"why":"It provides the geodesic, deflection-angle, and radar-delay formulas used in Sections III and IV.","marker":"[43]"},{"why":"It gives the weak-field circular orbital speed expression behind $v^2=ra'/2$.","marker":"[44]"},{"why":"It supplies the observed 200–500 km/s halo speed range used to fix $s/2\\simeq v^2$.","marker":"[45]"},{"why":"It provides the four-galaxy dataset used for the light-deflection angle plots.","marker":"[46]"},{"why":"It introduces the radar echo delay test that the model's second signature builds on.","marker":"[47]"}],"fun_headline_variants":["Squared stress-energy flattens galaxy rotation without dark matter","Modified gravity's T-squared term mimics dark matter in halos","T-squared gravity yields flat rotation curves and lensing clues","Energy-momentum squared gravity: dark matter effect from geometry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole result rests on the assumed form $e^{a+b}=(r/\\sigma)^s$ with a constant small $s$; if that form is not independently justified, the flat rotation curve is placed in by hand rather than predicted by the theory.","fun_headline_variants_meta":{"raw":{"variants":["Squared stress-energy flattens galaxy rotation without dark matter","Modified gravity's T-squared term mimics dark matter in halos","T-squared gravity yields flat rotation curves and lensing clues","Energy-momentum squared gravity: dark matter effect from geometry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000164,"raw_usage":{"total_tokens":1165,"prompt_tokens":783,"completion_tokens":382,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":399,"completion_tokens_details":{"reasoning_tokens":312}},"tokens_in":399,"tokens_out":382,"duration_ms":4299,"temperature":1.0,"reasoning_tokens":312,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T21:10:48.855022+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a galaxy with both a measured rotation curve and measured lensing deflection: fit $s$ from $v^2\\simeq s/2$, then check whether the predicted deflection (Eq. 40) and the $\\rho\\propto r^{-2}$ density profile match the data; a substantial mismatch would refute the claim. A second decisive check is to solve the EMSG field equations for dust without imposing Eq. (14) and see whether the resulting rotation curve is still flat.","supporting_citations":[{"cited_title":"Weinberg, John Wiley and Sons, 1972, ISBN 978-0- 471-92567-5, 978-0-471-92567-5","cited_arxiv_id":null,"evidence_quote":"It provides the geodesic, deflection-angle, and radar-delay formulas used in Sections III and IV."},{"cited_title":"Errata in Binney and Tremaine, \"Galactic Dynamics\"","cited_arxiv_id":"astro-ph/9304010","evidence_quote":"It supplies the observed 200–500 km/s halo speed range used to fix $s/2\\simeq v^2$."}],"review_version":1}