{"id":"b23a5bdb-659d-4627-9135-48ae16bb0c28","arxiv_id":"2506.21019","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A statistical strong lensing test shows that Conformal Gravity's constant gamma* under-predicts the observed lensing probability, and the gamma* value needed to match individual systems varies with stellar mass.","lead":"The authors test Conformal Gravity, an alternative to dark matter, by comparing its predicted strong lensing statistics with surveys of lensed quasars. They find that the theory's universal linear potential parameter cannot match the lensing data unless it is allowed to depend on galaxy mass, which contradicts the theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The conclusion that gamma* cannot be a universal constant rests on an unvalidated point-mass/enclosed-mass assumption; exterior-mass contributions claimed not to change the result are never computed.","rationale":"The reader's weakest_assumption is exactly the load-bearing step I identify: the point-mass lens with mass M*(theta_E) and the unsupported assertion that exterior mass only elevates gamma* without changing its order of magnitude or mass scaling. This assumption is not peripheral; it is what converts Table 1 into Eq. (35), the mass-dependent gamma*_t = 4.57e-15 (M*/M_sun)^-1.51 m^-1, and it is also what underlies the probability calculation in Section 6. If exterior mass contributes differently for low-mass and high-mass lenses, the anti-correlation in Eq. (35) disappears, and the fitted constant gamma*_c in Eq. (45) is no longer trustworthy. The paper provides no computation or error estimate for this contribution, despite explicitly acknowledging it. I do not object to the more basic result that the rotation-curve value gamma* = 5.42e-39 m^-1 underpredicts strong lensing by orders of magnitude; that conclusion appears robust even with the point-mass model. But the stronger claim—that gamma* cannot be constant and that CG's universality is falsified—depends on the unvalidated mass-dependent fit. Because the reader already reached CONDITIONAL on precisely this basis, my stress-test does not move the verdict: the concern is real, but it is a request for a specific recomputation rather than a demonstrated internal contradiction.","tokens_in":20873,"tokens_out":6965,"duration_ms":77567,"concrete_test":"Recompute every entry in Table 1 using the full Hernquist stellar profile instead of the point mass M*(theta_E). Concretely, integrate Eq. (30) with the deflection kernel obtained from Eqs. (22)-(23) for the Hernquist density (Eq. (33)), including all stellar mass out to a truncation radius, and solve for gamma* for each system. Then refit log gamma* versus log M*(theta_E) and rerun the Section 6 lensing probability with both the new mass-dependent fit and a new best-fit constant. If the slope is no longer significantly negative, or if the new constant-gamma* prediction matches the observed histogram at Delta theta > 3 arcsec, the claimed falsification of universal gamma* is a modeling artifact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central falsification claim—that no constant gamma* can fit the strong-lensing statistics—is carried by two steps. First, each system in Table 1 is reduced to a point mass equal to the stellar mass enclosed within the Einstein radius, M*(theta_E), using Eq. (34). Second, the text asserts after Eq. (31) (Section 4) that luminous mass outside theta_E contributes non-locally to the deflection, that fitting with only M*(theta_E) therefore overestimates gamma*, but that this 'does not alter the order of magnitude of gamma*.' No calculation is given for this last assertion. In conformal gravity the linear-potential parameter for an extended source is an integral over the full mass distribution (Eqs. (22)-(23)), so exterior mass can in principle enter with a different weight per unit mass than interior mass. If that weight is mass-dependent—for example, if theta_E/theta_s varies across the sample—then the per-system gamma*_t values and the -1.51 slope in Eq. (35) are artifacts of the point-mass reduction. The same reduction is used in the lensing probability of Section 6 (Eqs. (42)-(44)), so the large-separation under-prediction attributed to a constant gamma*_c=3.50e-32 m^-1 (Eq. (45)) could also be an artifact. This is not a mere choice of convention: it is the only step connecting the derived mass-dependent gamma*_t to the claim that CG's universal parameter is falsified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper tests conformal gravity (CG) by confronting its universal linear-potential parameter \\gamma^* with statistical strong lensing observations. The authors use the Oguri et al. strong-lensing sample to infer a value of \\gamma^* for each system under a point-mass lens model, with the stellar mass taken as the Hernquist-enclosed mass within the Einstein radius, and obtain a mass-dependent empirical relation \\gamma^*_t = 4.57\\times10^{-15}(M_*^{\\rm enc}/M_\\odot)^{-1.51} m^{-1} (Eq. 35). They then compute the lensing probability for the SQLS/Inada et al. quasar-lens sample using the galaxy stellar mass function, finding that the rotation-curve value \\gamma^*=5.42\\times10^{-39} m^{-1} severely underpredicts lensing, that the mass-dependent relation overpredicts at small separations and underpredicts at large separations, and that even a best-fit constant value \\gamma^*_c=3.50\\times10^{-32} m^{-1} (Eq. 45) fails at large separations. The paper concludes that CG cannot self-consistently explain both galactic rotation curves and the observed statistics of strong lensing.","tokens_in":21164,"tokens_out":9553,"duration_ms":108559,"significance":"If the result is correct, it is a significant multi-scale test of conformal gravity: the same parameter that fits rotation curves would be ruled out by strong lensing statistics, and the apparent mass dependence of \\gamma^* would contradict the theory's foundational premise of a universal constant. The paper is valuable in using larger, well-defined samples than earlier single-cluster studies, in including the second-order \\sim GM\\gamma term in the deflection angle, and in making its fitted formulas explicit and reproducible from the published sample. However, the central falsification claim is currently carried by an unvalidated point-mass/exterior-mass reduction, and the probability calculation contains a mismatch between the enclosed mass used in the fitted relation and the total stellar mass sampled by the mass function. The conclusion may survive those issues, but the manuscript does not yet demonstrate that it does.","major_comments":[{"comment":"The manuscript asserts that luminous mass outside the Einstein radius contributes non-locally to the deflection and that fitting with only M_*(\\theta_E) therefore overestimates \\gamma^*, but that this overestimation 'does not alter the order of magnitude of \\gamma^*.' No calculation is provided for this assertion. This is load-bearing: the per-system values in Table 1, the mass-dependent fit Eq. (35), and the constant-\\gamma^*_c comparison in Eq. (45) all inherit the point-mass reduction. A concrete estimate should be given, e.g., by recomputing the inferred \\gamma^*_t under alternative assumptions such as using the total stellar mass M_* rather than M_*(\\theta_E), or by a genuine extended-source calculation in CG. Without this, the paper's strongest claim—that no constant \\gamma^* can fit the strong-lensing statistics—is not established.","section":"Section 4, paragraph after Eq. (31)"},{"comment":"There is a mismatch in the mass variable between the fitted relation and the probability integral. Equation (35) is a function of the enclosed stellar mass M_*(\\theta_E), while Eq. (44) integrates over the total stellar mass M drawn from the galaxy stellar mass function. The text never specifies how M_*(\\theta_E) is obtained from M and \\theta_E in the probability calculation; no effective-radius or size-mass relation appears in Section 6. If the authors set M = M_*(\\theta_E), this contradicts Eq. (34) and overestimates the linear-potential contribution for low-mass systems; if they use total M, then Eq. (35) is misapplied. The dash-dotted curve in Figure 3 is therefore not a well-defined prediction of the model, and the large-separation underprediction attributed to \\gamma^*_t needs to be recomputed with a consistent mass prescription.","section":"Section 6, Eq. (44)"},{"comment":"The claim that the theoretical predictions fall 'significantly below' the observed distribution at large image separations is not quantified. The observed probability is built from only 26 lensed quasars, so the large-separation bins carry substantial Poisson uncertainties, yet Figure 3 shows no error bars, and no goodness-of-fit statistic (e.g., a Kolmogorov-Smirnov test or a likelihood) is reported. In addition, the procedure used to obtain the best-fit constant value \\gamma^*_c=3.50\\times10^{-32} m^{-1} is not described: there is no statement of the fitted quantity, the binning, the treatment of magnification bias, or the confidence interval. The central falsification claim should be supported by a quantitative statistical comparison.","section":"Section 6 / Figure 3"},{"comment":"The mass-dependent relation Eq. (35) is a least-squares fit to values of \\gamma^*_t that are, by construction, forced to reproduce the observed Einstein radius of each system through the lens equation. The scatter in \\gamma^*_t is therefore not an independent prediction of conformal gravity; the apparent anti-correlation with mass may reflect the assumed Hernquist profile and the mass-concentration relation rather than the theory. This should be stated explicitly, and the relation should be presented as an empirical parametrization, not as a 'derived' formula, in the abstract and conclusions.","section":"Section 4, Eq. (35)"}],"minor_comments":[{"comment":"The cosmological Hubble parameter in Eq. (14) contains \\bar\\Omega_{M0}, but the text specifies only \\bar\\Omega_{K0}=0.67 and \\bar\\Omega_{\\Lambda0}=0.33; the adopted value of \\bar\\Omega_{M0} should be stated explicitly.","section":"Section 2, Eq. (14)"},{"comment":"The axis label 'log M* (log M_\\odot)' is ambiguous; it should be written as log10[M_*/M_\\odot] for clarity.","section":"Figure 1"},{"comment":"The fit in Eq. (35) should be reported in log-space with uncertainties on the slope and normalization, since the displayed form implies a precision in the prefactor that the least-squares fit almost certainly does not warrant.","section":"Eq. (35)"},{"comment":"The abstract states that the authors 'derived a formula for \\gamma^*' as a function of stellar mass; because Eq. (35) is an empirical fit to lensing data, the wording should be softened to 'fit' or 'parameterized.'","section":"Abstract and Conclusions"},{"comment":"The subscript in \\gamma^*_t is not defined in the table header; the text should specify that 't' denotes the value fitted from strong lensing, as opposed to the rotation-curve value \\gamma^*.","section":"Table 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is already published in Universe, so this report is for the record or for a revised submission. The huge discrepancy between the fitted lensing \\gamma^* and the rotation-curve value is unlikely to be erased by the exterior luminous mass, whose contribution is at most an order-of-magnitude effect unless the CG linear potential weights exterior mass very differently. But the manuscript currently asserts rather than shows that the point-mass reduction is harmless, and the mass-mismatch in Section 6 is a genuine logical gap. I would not reject the paper, because the core tension with rotation-curve \\gamma^* is important and is partially independent of the mass-dependent fit; however, the authors should either supply the extended-mass calculation or substantially soften the falsification claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—read this one if you want a concrete statistical test of conformal gravity on lensing. The headline: the standard rotation-curve gamma* fails against strong lensing statistics by orders of magnitude, and that part is probably solid. But the paper's additional claim that gamma* cannot be constant—that it must be mass-dependent—does not yet hold. The per-system gamma* values are forced from a point-mass lens model, and the exterior-mass contribution is never actually computed.\n\nWhat's genuinely new: they use the corrected deflection angle that includes the 2Mγ term and re-fit gamma* on the large Oguri sample, not just the few clusters used in earlier work. That gives gamma*_t(M*) ∝ M*^-1.51, and they then compare the predicted lensing probability against the SDSS SQLS sample. The comparison is straightforward, and the conclusion that the rotation-curve gamma* under-predicts the observed lensing frequency is consistent with earlier studies and visible by eye in Figure 3. They are also honest in the conclusions: they call it \"difficulties\" rather than a refutation.\n\nThe soft spots are real. The per-system gamma*_t values in Table 1 are solutions of the lens equation given the Einstein radius, so their scatter is not an independent test. The exterior-mass effect is asserted after Eq. (31) to \"not alter the order of magnitude,\" but no calculation is shown. In conformal gravity the linear-potential parameter is an integral over the full mass distribution, so if exterior mass enters with a different weight per unit mass for low- and high-mass lenses, the -1.51 slope could be an artifact. Second, Eq. (35) is an empirical fit to the same sample used later to predict the lensing probability at small separations, so that part of the curve is inherited, not predicted. Third, there are no error bars on the fitted gamma* values, and the probability fits have no uncertainties, so it is hard to know how much to weight the large-separation shortfall. Finally, the paper notes that GR with SIS+NFW also under-predicts large-separation lenses, which makes one wonder whether the shortfall is a missing-mass issue rather than a specific CG failure.\n\nThe citation pattern is fine. The paper builds on Cutajar and Zarb Adami, uses standard lensing samples, and is careful to distinguish their deflection angle from earlier ones.\n\nWho this is for: anyone following modified-gravity tests with lensing statistics. It deserves a serious referee, but a referee should ask for a quantitative treatment of the exterior-mass contribution and error propagation before the mass-dependence claim is accepted. The central rotation-curve tension is probably robust.","headline":"The rotation-curve gamma* fails against strong lensing statistics, but the paper's mass-dependent gamma* claim rests on an unquantified point-mass approximation.","tokens_in":21734,"tokens_out":2523,"would_cite":true,"duration_ms":28944,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Conformal gravity's key constant fails strong-lensing data","keywords":["conformal gravity","statistical strong lensing","lensing probability","linear potential","galaxy stellar mass function","dark matter alternative","quasar lens search"],"falsifier":"Measure $\\gamma^*$ for the same lens systems using an extended-mass lens model that includes the full stellar and gas distribution instead of only the mass inside the Einstein radius; if the apparent decrease of $\\gamma^*$ with $M_*$ disappears or inverts, the claimed mass dependence is an artifact of the point-mass shortcut.","tokens_in":20595,"feed_emoji":"🔭","tokens_out":7164,"duration_ms":75195,"temperature":0.7,"pith_summary":"Conformal Gravity replaces dark matter with an extra linear term in the gravitational potential, whose strength is set by a single universal constant calibrated on spiral-galaxy rotation curves. This paper asks whether that same constant can reproduce the observed statistics of strong gravitational lensing by galaxies and clusters. Using a large sample of lens systems and the SDSS Quasar Lens Search, the authors find that the rotation-curve value predicts far too few lenses. Refitting the constant to lensing data yields values roughly seven orders of magnitude larger, and a best-fit constant still underpredicts wide-separation lenses. The paper concludes that Conformal Gravity cannot simultaneously explain rotation curves and strong lensing statistics unless its linear-potential parameter is allowed to depend on mass, which contradicts its foundational premise.","feed_headline":"Conformal gravity's key constant fails strong-lensing data","feed_subtitle":"Statistical lensing requires a mass-dependent linear-potential parameter, contradicting the theory's universal premise.","key_machinery":"The engine of the test is the conformal-gravity metric $B(r) = 1 - 2\\beta/r + \\gamma r$, whose extra linear term $\\gamma r$ adds a $\\gamma$-dependent piece to the photon deflection angle, $\\hat{\\alpha} = 4GM/c^2 r_0 + 2GM\\gamma/c^2 + \\cdots$. The universal parameter $\\gamma^*$ enters through $\\gamma = (M/M_\\odot)\\gamma^* + \\gamma_0$, with $\\gamma_0$ fixed from the cosmological background. The authors combine this deflection angle with a Hernquist luminous-mass profile, a Schechter or double-Schechter galaxy stellar mass function, and the magnification-bias-weighted cross section to predict the probability that a quasar at $z_S = 1.57$ is split into images separated by more than $\\Delta\\theta$ with flux ratio below $3.16$; matching that prediction to the observed sample yields both the mass-dependent fit and the best-fit constant.","core_discovery":"The central claim is that the universal linear-potential parameter of Conformal Gravity, $\\gamma^*$, fails the statistical strong-lensing test. Fitting $\\gamma^*$ system-by-system to the Einstein radii of a compiled galaxy and cluster lens sample gives a mass-dependent relation $\\gamma^*_t = 4.57 \\times 10^{-15} (M_*/M_\\odot)^{-1.51}\\,\\mathrm{m}^{-1}$, decreasing with stellar mass, rather than the constant value required by the theory. When this mass-dependent form is used in the lensing probability calculation, the prediction overshoots at separations below about 3 arcseconds and undershoots badly at larger separations; a best-fit constant $\\gamma^*_c = 3.50 \\times 10^{-32}\\,\\mathrm{m}^{-1}$ still falls below the observed lensing probability at large separations. The conclusion is that the current formulation of Conformal Gravity cannot provide a self-consistent account of both galactic rotation curves and strong lensing statistics.","pith_inferences":["A natural next step would be to fit the same lens sample with an extended-mass model that includes stellar mass outside the Einstein radius; if the apparent decrease of $\\gamma^*$ with $M_*$ persists, the mass dependence is physical, but if it flattens, it is an artifact of the point-mass shortcut.","The mass dependence $\\gamma^*_t \\propto M_*^{-1.51}$ has the same functional flavor as a dark-matter halo profile, so the linear-potential term may be mimicking the missing-mass scaling that Conformal Gravity was intended to remove.","Strong-lensing time delays, which depend on the absolute gravitational potential rather than only the deflection angle, could break the degeneracy between $\\gamma^*$ and the assumed mass profile and provide a sharper test than image separations alone."],"forward_implications":["The rotation-curve value of $\\gamma^*$ predicts a strong-lensing probability far below the observed one at all separations, so the same parameter cannot serve both galaxy dynamics and lensing.","The best-fit constant $\\gamma^*_c = 3.50 \\times 10^{-32}\\,\\mathrm{m}^{-1}$ improves the small-separation prediction but still leaves a deficit at separations above about 3 arcseconds, indicating the failure is not cured by simple renormalization.","If the mass-dependent $\\gamma^*_t$ is used, low-mass galaxies receive a much larger linear potential, boosting the predicted number of small-separation lenses and suppressing wide-separation lenses.","The same SDSS lensed-quasar sample, when modeled with SIS+NFW dark halos in General Relativity, also underpredicts large-separation lenses, so the large-separation deficit is not unique to Conformal Gravity.","The results imply that strong-lensing statistics can discriminate between Conformal Gravity and dark-matter-based models, at least for the current generation of lens surveys."],"supporting_citations":[{"why":"Supplies the rotation-curve-calibrated value $\\gamma^* = 5.42 \\times 10^{-39}\\,\\mathrm{m}^{-1}$ and the premise that $\\gamma^*$ is a universal constant.","marker":"[3]"},{"why":"Derives the second-order deflection angle in conformal gravity that the paper adopts for the lensing analysis.","marker":"[17]"},{"why":"Shows that the standard $\\gamma^*$ fails for the galaxy clusters Abell 370 and Abell 2390, motivating the statistical test.","marker":"[22]"},{"why":"Provides the previous strong-lensing fit that the paper corrects by including the $2GM\\gamma$ term in the deflection angle.","marker":"[23]"},{"why":"Supplies the lens sample with Einstein radii and stellar masses used to fit the mass-dependent $\\gamma^*_t$.","marker":"[24]"},{"why":"Provides the Hernquist profile used to compute the stellar mass enclosed within the Einstein radius.","marker":"[25]"},{"why":"Supplies the SDSS Quasar Lens Search sample and selection criteria used for the lensing probability comparison.","marker":"[30]"},{"why":"Provides the high-redshift galaxy stellar mass function used in the probability integral.","marker":"[28]"}],"fun_headline_variants":["Lensing test breaks conformal gravity's universal constant","Conformal gravity's constant fails lensing; needs mass dependence","Lensing data shows conformal gravity's constant is mass-dependent","Strong lensing refutes conformal gravity's universal parameter"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The analysis assumes that each lens can be treated as a point mass whose visible mass inside the Einstein radius is the only source of the linear potential, so that mass outside that radius merely raises the fitted value of $\\gamma^*$ uniformly, without changing how it scales with lens mass.","fun_headline_variants_meta":{"raw":{"variants":["Lensing test breaks conformal gravity's universal constant","Conformal gravity's constant fails lensing; needs mass dependence","Lensing data shows conformal gravity's constant is mass-dependent","Strong lensing refutes conformal gravity's universal parameter"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001141,"raw_usage":{"total_tokens":4800,"prompt_tokens":1071,"completion_tokens":3729,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":687,"completion_tokens_details":{"reasoning_tokens":3661}},"tokens_in":687,"tokens_out":3729,"duration_ms":30654,"temperature":1.0,"reasoning_tokens":3661,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:36:39.479015+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure $\\gamma^*$ for the same lens systems using an extended-mass lens model that includes the full stellar and gas distribution instead of only the mass inside the Einstein radius; if the apparent decrease of $\\gamma^*$ with $M_*$ disappears or inverts, the claimed mass dependence is an artifact of the point-mass shortcut.","supporting_citations":[{"cited_title":"Alternatives to dark matter and dark energy","cited_arxiv_id":null,"evidence_quote":"Supplies the rotation-curve-calibrated value $\\gamma^* = 5.42 \\times 10^{-39}\\,\\mathrm{m}^{-1}$ and the premise that $\\gamma^*$ is a universal constant."},{"cited_title":"Gravitational lensing in Weyl gravity","cited_arxiv_id":null,"evidence_quote":"Derives the second-order deflection angle in conformal gravity that the paper adopts for the lensing analysis."},{"cited_title":"Test of conformal theory of gravity as an alternative paradigm to dark matter hypothesis from gravitational lensing studies","cited_arxiv_id":null,"evidence_quote":"Shows that the standard $\\gamma^*$ fails for the galaxy clusters Abell 370 and Abell 2390, motivating the statistical test."},{"cited_title":"Strong lensing as a test for conformal Weyl gravity","cited_arxiv_id":null,"evidence_quote":"Provides the previous strong-lensing fit that the paper corrects by including the $2GM\\gamma$ term in the deflection angle."},{"cited_title":"The stellar and dark matter distributions in elliptical galaxies from the ensemble of strong gravitational lenses","cited_arxiv_id":null,"evidence_quote":"Supplies the lens sample with Einstein radii and stellar masses used to fit the mass-dependent $\\gamma^*_t$."},{"cited_title":"An Analytical Model for Spherical Galaxies and Bulges","cited_arxiv_id":null,"evidence_quote":"Provides the Hernquist profile used to compute the stellar mass enclosed within the Einstein radius."},{"cited_title":"The Sloan Digital Sky Survey Quasar Lens Search","cited_arxiv_id":null,"evidence_quote":"Supplies the SDSS Quasar Lens Search sample and selection criteria used for the lensing probability comparison."},{"cited_title":"The COSMOS2015 galaxy stellar mass function: Thirteen billion years of stellar mass assembly in ten snapshots","cited_arxiv_id":null,"evidence_quote":"Provides the high-redshift galaxy stellar mass function used in the probability integral."}],"review_version":1}