{"id":"c147559f-5f6c-4bcc-8090-fc305daa47f9","arxiv_id":"2607.23552","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"In WIG with a constant-density halo, shadow radii and test-field scalar QNMs are computed for mass-dependent λ values fitted to keep a horizon, showing mild GR deviations and long-lived damped modes.","lead":"The paper computes black-hole shadow sizes and scalar quasi-normal modes in Weyl-incorporated gravity with a constant-density matter halo and optional plasma. It is a theoretical exploration of a minimal GR extension that couples matter to the Weyl tensor, under assumptions the authors themselves flag as uncontrolled.","discovery_kind":"extension","skeptic_critique":null,"referee_report":{"model":"moonshotai/kimi-k3","summary":"The manuscript studies spherically symmetric static black holes in \"Weyl-incorporated gravity\" (WIG/MORD), which adds a coupling λ C_{αμβν}T^{αβ}T^{μν} to the Einstein–Hilbert action. Assuming a constant-density baryonic halo (ρ_c = 0.5) and a single-lapse ansatz ds² = −fdt² + dr²/f + r²dΩ², the authors solve the field equations iteratively to first order in λρ_c, determine λ for each black hole mass by imposing that the lapse vanish at r_s = 2m, and compute photon-sphere radii, shadow impact parameters (Table 1), energy emission using the Schwarzschild Hawking temperature, shadow modifications from a \"homogeneous plasma\" (Table 2), and massive-scalar QNM frequencies via 6th-order WKB and AIM, supplemented by a characteristic time-domain integration. The authors state all computed QNMs have ω_i < 0 with quality factors up to ~10⁴. The paper is explicitly framed as a theoretical exploration, and it is unusually candid about its own approximations — eqs. (32)–(33) quantify ε ≈ 0.5 and δ ≈ 1.9 — but the central numerical results are nonetheless presented at face value.","tokens_in":17676,"tokens_out":5387,"duration_ms":216007,"significance":"If the computations were controlled, the paper would provide the first black-hole shadow and QNM phenomenology for WIG/MORD, a framework whose galactic-scale fit (λ = 4.3125, Ref. [23]) makes a strong-field cross-check genuinely interesting. The manuscript has real strengths: it derives and reports its own consistency diagnostics (eqs. 29–33) rather than hiding the ansatz error; the QNM sector cross-checks two semi-analytic methods (6th-order WKB and AIM) against an explicit time-domain integration, a reasonable robustness standard for test-field claims; and the long-lived massive-scalar modes (Q ~ 10³–10⁴) are a falsifiable phenomenological prediction. However, the central numerical outputs currently rest on a first-order expansion at δ ≈ 1.9 and an internally inconsistent metric ansatz, so the results as tabulated cannot yet support the paper's conclusions.","major_comments":[{"comment":"§2.1, eqs. (29)–(33): The single-lapse ansatz (4) is inconsistent with the field equations for the matter model used: eq. (31) gives G^t_t − G^r_r = −κρ_c + λO(ρ_c²) ≠ 0 for pressureless dust, and the authors' own error measures at the working point are ε ≡ ρ_c r² ≈ 0.5 and δ ≡ λρ_c ≈ 1.9. Every quantitative output of the paper (Tables 1–2, Figs. 1–7) is built on f^(1), a first-order truncation whose neglected terms are the same order as those retained. The authors acknowledge this and call the exercise 'pedagogical', but the text then presents Table 1/2 numbers to three significant figures and draws trend conclusions ('positive correlation between mass and shadow size', '~7% barrier reduction'). Either the two-lapse system (A ≠ B) must be solved, or the claims must be explicitly re-scoped so that no quantitative precision is asserted beyond the uncontrolled regime.","section":"§2.1"},{"comment":"§2.1, Table 1: λ is not predicted; it is solved per-mass from the condition f^(1)(r_s) = 0. This has two load-bearing consequences. (a) The reported deviations of r_p and R_s from Schwarzschild are partially by construction, since λ is tuned to hold the horizon at r_s. (b) The text states 'for the larger λ the horizon lies slightly inside r = 2m', which contradicts the construction f^(1)(r_s)=0 with r_s = 2m — the horizon location should be re-solved and reported, not asserted. (c) The near-constancy λ ≈ 3.864–3.869 over nine decades of m followed by a jump to 3.568 at m = 10³ km is unexplained; it suggests either numerical conditioning of the root-finding or a branch change, and must be diagnosed.","section":"Table 1"},{"comment":"Table 1, columns 3–5: For m = 1 km the table gives r_p = 1.504 km while r_s = 2m = 2 km; for m = 10³ km it gives r_p = 1.873 km while 2m = 2000 km. The 'photon sphere' thus lies inside the nominal horizon for the two largest masses, and r_p is nearly mass-independent (~0.36–1.87 km) while m spans 10⁻⁶–10³ km. If r_p < r_h, the maximum of V_eff is not causally accessible to distant observers and R_s = r_p/√f(r_p) (eq. 42) is not an observable shadow radius. Either the units/normalization of Table 1 are mislabeled, the horizon is not at ~2m (in which case r_h must be tabulated), or the reported configurations do not describe shadows. This must be resolved before any conclusion from Table 1 stands.","section":"Table 1"},{"comment":"§4, eq. (46): the 'homogeneous plasma' refractive index n(r) = √(1 − ρ_p/r) is (i) explicitly r-dependent and therefore not homogeneous; (ii) dimensionally inconsistent if ρ_p is a density, since ρ_p/r must be dimensionless; and (iii) not the standard cold non-magnetized plasma dispersion n² = 1 − ω_p²/ω² with constant ω_p used in the cited formalism (Synge; Perlick/Tsupko). No derivation or reference is given. Since all of Table 2 and Fig. 4 follow from eq. (46), the plasma section must either derive this index from a stated microphysical model or be reworked with the standard plasma dispersion, including a clear definition of ρ_p and its units.","section":"§4"},{"comment":"§3, eq. (45): the energy-emission estimate uses the Schwarzschild temperature T = 1/(8πm) while the background lapse is explicitly non-Schwarzschild. The surface gravity T = f^(1)′(r_h)/(4π) is directly computable from the f^(1) the authors already construct, so the approximation is not forced by complexity. As it stands, §3 combines a WIG-corrected σ_lim with an uncorrected T, which is internally inconsistent and removes precisely the WIG effect the section is meant to estimate. At minimum the exact T should be computed or the induced error quantified.","section":"§3"}],"minor_comments":[{"comment":"Units of the densities are never stated: ρ_c = 0.5 and ρ_p = 0.4/0.2 are given as pure numbers, but in geometric units with m in km a density has units km⁻². Please state units for ρ_c, ρ_p, and λ, and note how far ρ_c = 0.5 km⁻² sits from realistic halo densities (~10⁻²⁴ kg/m³).","section":"§2.1"},{"comment":"Eq. (42) and Fig. 2: the Carter constant is set to K = 1 without justification. For spherical symmetry K is fixed by the orbit, not chosen; please clarify the role of K and why K = 1 is consistent with the critical orbit used for R_s.","section":"§2.1"},{"comment":"Eq. (35): the conserved energy E is written with an overall factor 1/2 multiplying the g_tt ṫ term; as written this is not the standard normalization E = f(r) ṫ. Please check and correct, since eqs. (40)–(42) inherit this normalization.","section":"§2.1"},{"comment":"§5: the ~30% WKB–AIM discrepancy for (ℓ=0, µ=0) is reported only in words. A small table of representative ω values for both methods would make the claimed '<1% agreement for µ > 1' verifiable and properly delimit the WKB results.","section":"§5.2"},{"comment":"Fig. 2 caption: '3600' should be '360°'; 'homogenous' in Fig. 4 caption should be 'homogeneous'; 'yeild' above eq. (43); author name spelled 'Khokhar' on p.1 but 'Khookhar' in running heads.","section":"Figures"},{"comment":"The claimed galactic value λ = 4.3125 ± 0.0012 from Ref. [23] and the black-hole values λ ≈ 3.5–3.9 differ by ~20% in a theory whose motivating hypothesis (§1) is that λ is unique. Even granting the different halo models, this tension should be stated explicitly and its implications for the uniqueness hypothesis discussed, including whether the units of λ in the two fits coincide.","section":"§6"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is an exploratory exercise from the group that has been developing the WIG/MORD framework; the citation pattern draws heavily on the authors' own prior work ([15,16,18,20,23,57]). This is not improper, but the editor may wish to note that the paper's results cannot, even in principle, test the framework's central \"unique λ\" claim, and the cross-sector tension with Ref. [23] (λ=4.3125 vs 3.5–3.9) is currently under-acknowledged. The manuscript is candid about most of its limitations, which speaks in its favor, but the numerical tables are presented at a precision the underlying approximations do not support."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new piece here is straightforward: first explicit shadow radii, plasma corrections, WKB/AIM scalar QNMs, and time-domain profiles on the WIG/MORD background with a constant-density halo. That is real, if narrow. The tables and figures are not in the earlier Lee–Qadir or galactic-halo papers.\n\nThey do several things cleanly. The vacuum limit recovers Schwarzschild–de Sitter. They write down the non-conservation of Tμν and the exchange with Iμν. They flag that the A=B ansatz is inconsistent for pressureless dust, quantify ε∼ρc r²∼0.5 and δ≡λρc∼1.9 at their working point, and state that ρc=0.5 is pedagogical and the λ∼3.5–3.9 values are conditional on the toy halo, not observational constraints. The dual WKB/AIM check and the time-domain comparison to Schwarzschild are competent standard tools. Test-field ωi<0 with high quality factors for massive scalars is a concrete phenomenological note.\n\nThe soft spots are real and load-bearing for the quoted decimals. Because δ is O(1), the first-order lapse is not a controlled expansion; photon spheres and QNMs to three places sit outside the regime they define. λ is solved so f(1)(rs)=0 for each mass—consistency fit, not prediction—and is a different number from the galactic λ≈4.31. Energy emission uses the Schwarzschild temperature. QNM “stability” is only for a probe scalar. None of this is hidden, but it means Table 1 and the shadow/QNM plots are exploratory numerics on an uncontrolled background, not strong-field predictions of unique λ.\n\nThis is for people already tracking minimal Weyl-matter couplings or building a catalog of modified-gravity shadows/QNMs. A serious referee should see it; the structural caveats are visible in the text and fixable by a better metric or a clearer “strong-coupling toy model” framing. I would not cite it for numbers in the next year, but I would not desk-reject it either. Engage if you care about the WIG program; otherwise skim the limitations section and move on.","headline":"First WIG shadows and scalar QNMs, but the numbers sit outside the controlled expansion the authors themselves define.","tokens_in":18482,"tokens_out":550,"would_cite":false,"duration_ms":15223,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83D05","83C56"],"pacs":[],"model":"grok-4.5","headline":"Weyl-matter coupling yields mass-dependent black-hole shadows and long-lived scalar ringdowns under a constant-density halo.","keywords":["black hole shadows","modified gravity","quasi-normal modes","dark matter halos","Weyl-incorporated gravity","plasma","test-field approximation"],"falsifier":"Recompute the horizon condition and shadow radius with a realistic, radially falling halo density that vanishes at the horizon and a fully consistent two-function metric; if no unique λ near 3.5–3.9 survives, or if the resulting shadows become indistinguishable from Schwarzschild, the reported phenomenology collapses.","tokens_in":18473,"feed_emoji":"🌑","tokens_out":986,"duration_ms":25397,"temperature":0.7,"pith_summary":"This paper explores Weyl-incorporated gravity, a minimal extension of general relativity that adds a single coupling λ between matter and the free gravitational field (the Weyl tensor). The authors place static spherical black holes inside a constant-density baryonic halo, solve for the corrected lapse function, and extract photon-sphere radii and shadow sizes for masses spanning many orders of magnitude. They find unique λ values near 3.5–3.9 that keep a horizon near the Schwarzschild radius; the resulting shadows grow with mass and shrink when a homogeneous plasma is added. Under the test-field approximation they also compute massive-scalar quasi-normal frequencies and time-domain waveforms, all of which decay (ω_i < 0) yet can ring for thousands of cycles. A sympathetic reader cares because the same coupling already claimed to explain galactic rotation curves is here shown to leave measurable imprints on strong-field observables that future shadow and gravitational-wave data could test.","feed_headline":"Weyl gravity gives mass-dependent shadows and long ringdowns","feed_subtitle":"A single coupling that fits galaxy curves also shifts black-hole photon spheres and slows scalar decay","key_machinery":"The first-order corrected lapse function f^(1)(r) obtained by expanding the Weyl-incorporated field equations for constant density ρ_c; photon-sphere and shadow radii are read from its effective potential, and the same f^(1) supplies the potential barrier for WKB/AIM quasi-normal frequencies and characteristic time-domain integration.","core_discovery":"For spherically symmetric static black holes embedded in a constant-density baryonic halo within Weyl-incorporated gravity, there exist unique values of the coupling λ ≈ 3.5–3.9 that preserve a horizon near r = 2m; the associated photon-sphere impact parameters produce shadow radii that increase with black-hole mass, decrease under homogeneous plasma, and support test-field massive scalar modes that all decay while reaching quality factors of order 10^3–10^4.","pith_inferences":["Because the reported λ values sit outside the controlled weak-coupling regime, any observational claim will first require a non-perturbative or fully consistent two-metric-function solution.","If the long-lived massive-scalar modes survive a full gravitational perturbation analysis, they could appear as slowly damped echoes or prolonged ringdowns distinguishable from Kerr templates.","Joint EHT-plus-galaxy-rotation constraints on a single λ would constitute a sharp, mass-scale-spanning test of whether the Weyl-matter coupling is universal."],"forward_implications":["Shadow size in this framework is predicted to grow with black-hole mass once λ is fixed by the halo model.","A homogeneous plasma background systematically shrinks the apparent shadow relative to the pure baryonic-halo case.","Massive scalar test-field ringdowns on these backgrounds are long-lived (quality factors 10^3–10^4), offering a potential gravitational-wave signature.","Time-domain waveforms show slower late-time decay than Schwarzschild for low multipoles, consistent with enhanced wave trapping.","The same λ that was previously used for galactic rotation curves is here shown to leave strong-field imprints that can be compared with EHT and LIGO/Virgo data once realistic halos are included."],"fun_headline_variants":["Weyl gravity yields mass-dependent BH shadows via λ coupling","MORD shadows grow with mass, shrink in plasma, slow scalar ringdowns","λ≈3.5–3.9 preserves horizons; photon spheres set mass-linked shadows","Weyl-incorporated BHs: bigger mass, larger shadows, high-Q QNM decay","Constant-density halo in WIG shifts shadows and lengthens ringdowns"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The calculation assumes a single lapse function even though pressureless dust plus the Weyl interaction make the two metric potentials unequal, and the authors’ own error estimates show the expansion is not under control at the working densities.","fun_headline_variants_meta":{"raw":{"variants":["Weyl gravity yields mass-dependent BH shadows via λ coupling","MORD shadows grow with mass, shrink in plasma, slow scalar ringdowns","λ≈3.5–3.9 preserves horizons; photon spheres set mass-linked shadows","Weyl-incorporated BHs: bigger mass, larger shadows, high-Q QNM decay","Constant-density halo in WIG shifts shadows and lengthens ringdowns"]},"model":"grok-4.5","effort":"low","cost_usd":0.004928,"raw_usage":{"total_tokens":1367,"prompt_tokens":764,"num_sources_used":0,"completion_tokens":88,"cost_in_usd_ticks":49284000,"prompt_tokens_details":{"text_tokens":764,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":515,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":764,"tokens_out":88,"duration_ms":10907,"temperature":1.0,"reasoning_tokens":515,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T19:18:06.087749+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Recompute the horizon condition and shadow radius with a realistic, radially falling halo density that vanishes at the horizon and a fully consistent two-function metric; if no unique λ near 3.5–3.9 survives, or if the resulting shadows become indistinguishable from Schwarzschild, the reported phenomenology collapses.","supporting_citations":[],"review_version":1}