{"id":"5628c222-1beb-4989-8166-9b7a70a80ccf","arxiv_id":"2411.13897","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"EHT shadow sizes and LIGO/Virgo/KAGRA inspiral waveforms allow only small values of the dimensionless regularity parameter ℓ in the Ghosh-Simpson-Visser regular black hole.","lead":"This paper tests a singularity-free black hole model against gravitational wave data and black hole shadow images, and finds its extra parameter must be small. It shows how two observational probes can work together to check whether real black holes are described by this alternative to general relativity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"GW constraints rest on a stationary-phase mapping that uses ν=f instead of ν=f/2, giving ℓ=-9/80 φ2δφ2 that is twice the correct -9/160; the quoted values are not yet reliable.","rationale":"Reading in good faith: the EHT analysis is a clean use of the photon-sphere condition and the shadow relation, and the accretion images are internally consistent. The central quantitative contribution is the GW bound. There the derivation has a concrete SPA error: the stationary phase condition is written correctly in the text but not used in the substitution. This is not a disagreement with external priors; it is a mismatch between two equations in the same section. The resulting factor of two flips the most prominent quoted numbers (0.041 and 0.050 to roughly half), although the qualitative conclusion of small positive ℓ survives because the EHT intervals and even the corrected GW values remain small. Since the paper itself flags that Eq. (44) is valid only for ℓ<<1 and reports one-sided limits as two-sided, a revision that redoes the SPA mapping, fixes Eq. (28), and states upper limits honestly would make the quantitative claims trustworthy. The verdict stays conditional.","tokens_in":29751,"tokens_out":33464,"duration_ms":309770,"concrete_test":"Re-derive Eq. (38) from Eq. (35) using the stationary condition ν(t0)=f/2 stated in the text. If the coefficient is -(5/12) rather than -(5/24), recompute Eq. (44) and Table I; a factor-of-two shift in all ℓ bounds confirms the error. Separately, compute the exact circular-orbit coordinate frequency Ω=˙φ/˙t for the EOS metric and compare with Eq. (28); if the 3m/r and 9m²/r² terms disappear, the Kepler relation used in Eqs. (31)-(32) must be replaced before any quantitative PPE mapping is trusted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section IV.B, the mapping from the time-domain orbital phase to the Fourier GW phase is the load-bearing step for the GW constraints. Equation (35) gives δϕ=-(5/24η)(2mπν)^{-1}ℓ. The text of Eq. (37) states that the stationary point satisfies ν(t0)=f/2, so the Fourier-phase correction is 2δϕ(f/2)=-(5/12η)(mπf)^{-1}ℓ=-(5/12)η^{-2/5}ℓ u^{-1}, not the -(5/24) coefficient used in Eq. (38). Equating to the LVK PPE form therefore yields ℓ=-(9/160)φ2δφ2, not Eq. (44)'s ℓ=-(9/80)φ2δφ2. Every entry in Table I is a factor of two too large in magnitude; for example GW191204-171526 shifts from ℓ=0.041 to about 0.020. This is an internal algebraic error, independent of the external LVK data. The reader's frequency-convention concern is related and real: Eq. (28) substitutes L=r²Ω where L=r²dφ/dτ and Ω=dφ/dt, so the 3m/r and 9m²/r² terms are not the coordinate-frequency Kepler law; that issue also needs repair, but the SPA factor alone changes the headline numbers.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper tests the regular black hole metric of Ghosh and Simpson-Visser (the 'EOS' metric), which adds a free parameter ℓ to the mass and spin, against two independent observational probes. For gravitational waves, the authors restrict to a=0, derive a leading-order correction to the binding energy and orbital phase in an effective-one-body setup, map this correction to the PPE parameter δφ2, and then convert LVK posterior samples from GWTC-1/2/3 into constraints on ℓ. For EHT, they compute the nonrotating shadow radius and angular diameter, compare the predicted values for SgrA* and M87* with the reported EHT measurements, and generate intensity profiles and images for static, infalling, and thin-disk accretion models. The headline results are EHT bounds 0≤ℓ≤0.148 (SgrA*) and 0≤ℓ≤0.212 (M87*), and GW bounds for which the most stringent events are GW191204-171526 and GW190924-021846 with ℓ≈0.04-0.05 for the SEOBNRv4 model.","tokens_in":29999,"tokens_out":9668,"duration_ms":98253,"significance":"If the GW derivation is repaired, the paper offers a useful two-channel test of a popular regular black hole family: EHT shadow-size data and LVK inspiral phase data constrain the same deformation parameter, and the authors correctly use public posterior samples rather than fitting a new waveform model. The EHT shadow part is clean and internally consistent, and the photon-ring classification into direct, lensed, and photon-ring emission is thorough and well illustrated. The GW part is the main quantitative asset, but its headline numbers are currently affected by internal algebraic errors in the stationary-phase mapping and by a frequency-convention issue in the modified Kepler law; these need to be corrected before the quoted constraints can be used.","major_comments":[{"comment":"The stationary-phase factor is off by a factor of two. Since ΨGW(f)=2ϕ(t0) and the stationary point satisfies ν(t0)=f/2, the ℓ-correction in Eq. (35), which scales as (2πν)^{-1}, must be evaluated at ν=f/2 and then multiplied by 2. This yields δΨ=-(5/12)η^{-2/5}ℓu^{-1}, not the coefficient -(5/24) shown in Eq. (38). Equating this corrected coefficient with the LVK PPE form β=(3/128)φ2δφ2η^{-2/5} gives ℓ=-(9/160)φ2δφ2, not Eq. (44)'s -(9/80)φ2δφ2. Consequently every entry in Table I is a factor of two too large in magnitude; for example GW191204-171526 would shift from ℓ≈0.041 to ℓ≈0.020. This is an internal algebraic error, independent of the LVK data, and must be fixed before the GW constraints are quoted.","section":"Sec. IV B, Eqs. (37)-(44)"},{"comment":"The modified Kepler law is derived using L=r²Ω, but L in Eqs. (19)-(24) is the proper-time specific angular momentum, L=r²dφ/dτ, while Ω=dφ/dt is the coordinate angular velocity. For a static, spherically symmetric metric the correct relation is Ω=L(1-2M(r)/r)/(E r²), so replacing L by r²Ω is not valid at the order retained. Indeed, setting ℓ=0 in Eq. (28) gives Ω²=(m/r³)(1+3m/r+9m²/r²+...), which disagrees with the exact Schwarzschild coordinate-frequency result Ω²=m/r³. Since the ℓ-dependent term in Eq. (28) enters at the same 1PN order at which the proper-time/coordinate-time distinction first appears, the mapping from ℓ to the orbital phase in Eqs. (31)-(35) is built on an incorrect intermediate quantity. This needs to be rederived from dφ/dt obtained directly from the geodesic equations.","section":"Sec. IV A, Eq. (28)"},{"comment":"The EHT constraints are computed only for the nonrotating subfamily (a=0), but the abstract and conclusion present them as constraints on ℓ of the EOS spacetime without this qualification. Since the underlying model includes spin and the observed sources are not known to be nonrotating, the bounds should be explicitly labeled as applying to the a=0 slice of the parameter space, or accompanied by a quantitative estimate of how the shadow diameter depends on spin.","section":"Abstract, Sec. V, and Sec. VII"}],"minor_comments":[{"comment":"The collaboration name appears as 'L VK' with a space in the title, abstract, and several section headings; it should be 'LVK'.","section":"Throughout"},{"comment":"The waveform model is written as 'IMPRPhenomPv2' in the text and table; this should be 'IMRPhenomPv2'.","section":"Sec. IV B and Table I"},{"comment":"The table caption reports 90% confidence while the discussion section reports 95% confidence; the two should be made consistent.","section":"Table I caption and Sec. VII"},{"comment":"The orbital phase integral as written has no explicit lower limit and the change of variables leading to ∫(1/˙E)(dE/dΩ)ΩdΩ is not shown; adding the frequency limits would make the SPA step easier to verify.","section":"Eq. (33)"},{"comment":"The caption says 'Central and left panels' but the figure contains central and right panels; the wording should be corrected.","section":"Sec. V A, Fig. 5 caption"},{"comment":"There are numerous typographical errors, including 'panle', 'colunms', 'usign', 'sourroundings', 'ETH observations' in the Sec. V heading, and an incomplete bibliographic entry in reference [19]; a careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"I see no circularity in the use of external LVK posterior samples or EHT angular diameters; the ℓ-to-δφ2 mapping is derived from the metric and the comparison is standard Bayesian post-processing. My recommendation is driven entirely by the internal GW derivation: the stationary-phase factor error changes all quoted GW constraints by a factor of two, and the modified Kepler law in Eq. (28) uses a questionable frequency identification. Both issues are fixable within the scope of the paper, so revision rather than rejection seems appropriate. The EHT shadow calculation can stand largely as is."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the EHT side of this paper is clean, standard, and probably correct. The GW side has a load-bearing algebraic error that changes the quoted constraints by a factor of two, plus a related ambiguity about which angular frequency enters the Kepler law. Treat Table I as preliminary until that is fixed.\n\nWhat is actually new: a PPE mapping specific to the Ghosh-Simpson-Visser metric, applied to GWTC-1/2/3, plus EHT shadow and photon-ring constraints for the same parameter. The photon-ring and image sections are routine but complete, covering static, infalling, and thin-disk accretion. The authors are also transparent about the effective-ℓ limitation and about not combining events because each binary could have a different ℓ. That honesty is a real plus.\n\nThe EHT angular-diameter argument is robust: bph decreases monotonically with ℓ, so the measured Sgr A* and M87* diameters give upper limits 0≤ℓ≤0.148 and 0≤ℓ≤0.212. I would trust those numbers. The ring-interval tables are internally consistent with the metric.\n\nThe GW derivation is where I part ways. The stationary-phase mapping in Eqs. (37)-(38) is off by a factor of two: since ν(t0)=f/2, the Fourier-phase correction is 2δϕ(f/2), giving ℓ=-(9/160)φ2δφ2, not -(9/80) as in Eq. (44). This is an internal algebraic error independent of the LVK data, and it changes every entry in Table I by a factor of two. The frequency-convention issue flagged by the reader is also real: writing L=r²Ω with Ω=dφ/dt mixes proper-time angular velocity with coordinate-time angular velocity, so the 3m/r and 9m²/r² terms in the Kepler law come from that substitution and need a careful rewrite before the PPE mapping can be trusted. One more thing: the table reports one-sided upper limits as symmetric two-sided intervals, for example ℓ=0.041^{+0.106}_{-0.041}; after truncating negative samples the lower error bar is an artifact. Those should be presented as upper limits.\n\nBottom line: the EHT part is worth a serious referee, and the paper is a fair observational test of one regular-BH model. The GW part needs a corrected derivation before its quantitative constraints are usable. Send to review, but require the SPA factor, the frequency convention, and the interval reporting to be fixed first.","headline":"EHT half is a clean standard analysis and the shadow bounds are likely right; the GW half has a factor-of-two stationary-phase error and a frequency-convention muddle, so the quoted LVK constraints need repair before use.","tokens_in":30571,"tokens_out":2207,"would_cite":false,"duration_ms":64984,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83C10","83C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"EHT shadow sizes and LVK inspiral phases tie the eye-of-the-storm black hole to near-Schwarzschild geometry.","keywords":["regular black holes","eye of the storm metric","Ghosh-Simpson-Visser spacetime","no-hair theorem","gravitational wave constraints","black hole shadow","Event Horizon Telescope","parameterized post-Einsteinian framework"],"falsifier":"Recompute Kepler's law for circular equatorial orbits in the EOS metric from the coordinate angular velocity $\\Omega = d\\varphi/dt$ via the metric components $g^{\\mu\\nu}$; if the $\\ell$ term appears with a different coefficient or at a different order than in Eq. (28), the identification $\\ell = -\\frac{9}{80}\\varphi_2\\delta\\varphi_2$ is wrong and the quoted gravitational-wave constraints do not apply to this model.","tokens_in":29515,"feed_emoji":"🕳️","tokens_out":7471,"duration_ms":69821,"temperature":0.7,"pith_summary":"This paper asks how far a singularity-free black hole can deviate from the Schwarzschild solution before current observations notice. The model tested is the 'eye of the storm' metric, a rotating regular black hole whose only extra freedom is a dimensionless parameter $\\ell$ that softens the mass near the center. Using shadow angular diameters from the Event Horizon Telescope, the paper finds $0\\leq \\ell \\leq 0.148$ for Sgr A* and $0\\leq \\ell \\leq 0.212$ for M87*. Using inspiral gravitational waves from the LVK catalogs, the tightest bounds are $\\ell=0.041^{+0.106}_{-0.041}$ and $\\ell=0.050^{+0.165}_{-0.050}$ for two events under the SEOB waveform model. The takeaway is that the exterior of this regular black hole must look nearly Schwarzschild, so a singularity-free core is not excluded but leaves almost no observable trace in current data.","feed_headline":"Shadows and waves squeeze regular black hole parameter to 0.21","feed_subtitle":"M87*, Sgr A* shadow sizes and merger inspiral phases leave the eye-of-the-storm metric almost Schwarzschild.","key_machinery":"The central object is the eye-of-the-storm line element, obtained from the Kerr metric by the mass replacement $M \\to M e^{-\\ell M/r}$, which removes the central singularity while preserving separability of the Hamilton-Jacobi equations. The identity that carries the gravitational-wave argument is $\\ell = -\\frac{9}{80}\\varphi_2\\delta\\varphi_2$, obtained by matching the leading $\\ell$ correction in the Fourier phase to the PPE phase parameter $\\beta u^{-1}$; for shadows, the operative relations are the photon-sphere condition $\\partial V_{\\mathrm{eff}}/\\partial r = 0$, the shadow radius $b_{\\mathrm{ph}} = r_{\\mathrm{ph}}/\\sqrt{1 - 2M(r_{\\mathrm{ph}})/r_{\\mathrm{ph}}}$, and the angular-diameter formula $\\Theta = 2b_{\\mathrm{ph}}/D$.","core_discovery":"The paper establishes that the Ghosh-Simpson-Visser 'eye of the storm' regular black hole is observationally pinned close to Schwarzschild. The EHT angular-diameter measurements give $0\\leq \\ell \\leq 0.148$ for Sgr A* and $0\\leq \\ell \\leq 0.212$ for M87*, while the most stringent gravitational-wave constraints from the inspiral phase are $\\ell=0.041^{+0.106}_{-0.041}$ (GW191204-171526) and $\\ell=0.050^{+0.165}_{-0.050}$ (GW190924-021846) for the SEOB model. The argument maps the deformed Schwarzschild effective-one-body Hamiltonian into the parameterized post-Einsteinian phase correction, yielding the identification $\\ell = -\\frac{9}{80}\\varphi_2\\delta\\varphi_2$, and then reads the $\\delta\\varphi_2$ posteriors from the LVK catalogs. For the shadow part, the photon-sphere radius and impact parameter are computed from the deformed metric, giving an angular diameter $\\Theta = 2b_{\\mathrm{ph}}/D$ that shrinks as $\\ell$ grows.","pith_inferences":["The paper's $a=0$ assumption means $\\ell$ could trade off against spin in a rotating fit; X-ray reflection spectroscopy, which the authors say is in progress, is the natural way to break that degeneracy.","A hierarchical analysis treating $\\ell$ as a population hyper-parameter would convert per-event upper limits into a statement about whether all regular black holes share one mass-distribution parameter; the paper stops short of that.","The proper-time angular-velocity issue in the Kepler-law derivation could rescale the gravitational-wave bounds without affecting the shadow bounds; redoing the derivation in coordinate time would show whether the reported $\\ell$ values shift.","If future shadow measurements shrink the uncertainty below about 2 $\\mu$as, the upper limits on $\\ell$ improve roughly linearly with the angular-diameter error, so the method's power scales with interferometric resolution."],"forward_implications":["The EOS metric can deviate from Schwarzschild by at most about $\\ell = 0.2$ in the region probed by shadows, so any singularity-free core must be hidden deep inside an essentially Schwarzschild exterior.","Gravitational-wave inspiral data from the LVK catalogs are consistent with $\\ell = 0$; the most constraining events place $\\ell$ below 0.05 at the median, so current merger observations do not demand a regular-core modification.","For the allowed range of $\\ell$, the predicted shadow angular diameters stay within the EHT error bars, meaning higher-resolution images or additional sources are needed to see the deformation.","The widening of the lensed- and photon-ring impact-parameter intervals with $\\ell$ implies that, if the deformation is near its upper bound, future very-long-baseline observations of the photon ring could detect it."],"supporting_citations":[{"why":"Defines the Ghosh-Simpson-Visser (eye of the storm) regular black hole metric with parameter $\\ell$ and establishes its astrophysical viability.","marker":"[13, 18, 19]"},{"why":"Supplies the effective-one-body/PN method for mapping a deformed Schwarzschild metric to gravitational-wave phase corrections.","marker":"[67]"},{"why":"Provides the analogous modified Kepler-law calculation used to benchmark the 1PN $\\ell$ term.","marker":"[68]"},{"why":"Defines the parameterized post-Einsteinian phase parametrization $\\Psi = \\Psi_{\\mathrm{GR}} + \\beta u^b$.","marker":"[83]"},{"why":"Gives the PPE convention $\\beta = \\frac{3}{128}\\varphi_2\\delta\\varphi_2\\eta^{-2/5}$ that fixes the sign and normalization of the $\\ell$ mapping.","marker":"[84]"},{"why":"Provides the expression for $\\varphi_2$ in terms of the symmetric mass ratio $\\eta$ used in Eq. (43).","marker":"[85]"},{"why":"Reports the EHT angular diameter of the M87* shadow ($42\\pm3\\,\\mu$as) used to set $\\ell \\leq 0.212$.","marker":"[96]"},{"why":"Reports the EHT angular diameter of the Sgr A* shadow ($51.8\\pm2.3\\,\\mu$as) used to set $\\ell \\leq 0.148$.","marker":"[97]"},{"why":"The GWTC-1, GWTC-2, and GWTC-3 catalogs whose $\\delta\\varphi_2$ posterior samples are fitted to constrain $\\ell$.","marker":"[46-50]"}],"fun_headline_variants":["Eye of storm black hole pinned near Schwarzschild by shadows and waves","M87*, Sgr A* shadows and GW inspirals constrain l to 0.21","Singularity-free black hole l<0.21 from EHT and LVK data","Shadows and inspiral waves tighten regular black hole parameter to 0.21"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The gravitational-wave constraints hinge on identifying the angular frequency in the phase integral with $L = r^2\\Omega$ using the proper-time angular velocity; if that substitution is not the coordinate frequency a distant observer assigns to the orbit, the derived mapping $\\ell = -\\frac{9}{80}\\varphi_2\\delta\\varphi_2$ and the resulting bounds change.","fun_headline_variants_meta":{"raw":{"variants":["Eye of storm black hole pinned near Schwarzschild by shadows and waves","M87*, Sgr A* shadows and GW inspirals constrain l to 0.21","Singularity-free black hole l<0.21 from EHT and LVK data","Shadows and inspiral waves tighten regular black hole parameter to 0.21"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000681,"raw_usage":{"total_tokens":3199,"prompt_tokens":1160,"completion_tokens":2039,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":776,"completion_tokens_details":{"reasoning_tokens":1950}},"tokens_in":776,"tokens_out":2039,"duration_ms":14417,"temperature":1.0,"reasoning_tokens":1950,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:48:23.239753+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute Kepler's law for circular equatorial orbits in the EOS metric from the coordinate angular velocity $\\Omega = d\\varphi/dt$ via the metric components $g^{\\mu\\nu}$; if the $\\ell$ term appears with a different coefficient or at a different order than in Eq. (28), the identification $\\ell = -\\frac{9}{80}\\varphi_2\\delta\\varphi_2$ is wrong and the quoted gravitational-wave constraints do not apply to this model.","supporting_citations":[],"review_version":1}