{"id":"fc4160f0-5fed-4971-befe-ead651d977df","arxiv_id":"2608.10859","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Combining EHT shadow observations of M87* and Sgr A* with mass and distance priors yields 95% upper limits Q < 0.391 and v < 4.153 on the ModMax black hole charge and nonlinearity parameter.","lead":"Scientists used Event Horizon Telescope images of the black holes M87* and Sgr A* to set upper limits on the charge and a nonlinearity parameter of ModMax black holes: Q < 0.391 and v < 4.153 at 95% credibility. They also simulated thin-disk images and found that larger charge brightens the disk while the nonlinearity parameter slightly dims it.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The v<4.153 headline bound is not identifiable: θtheory depends only on C=Q²e^{-v}, so the reported separate upper limits on Q and v are prior artifacts rather than data-driven constraints.","rationale":"The reader's weakest assumption was the EHT ring-to-shadow mapping, which is a standard model-dependence concern. My stress-test identifies a more direct problem inside the paper's own formalism: the shadow size, and therefore the likelihood, depends only on the combination Q²e^{-v}. Consequently, the abstract's headline separate upper limits on Q and v are not both identifiable from the data. The v<4.153 bound is not a measurement of the ModMax nonlinearity parameter; it reflects the chosen prior range 0≤Q≤1 and the Jacobian of the projection from the degenerate direction. This is load-bearing because the abstract explicitly advertises v<4.153 as the strongest constraint on the nonlinearity parameter. A simple reparameterization test would settle whether the constraint survives an extended Q prior. I do not think this forces outright rejection: the Q<0.391 limit (when marginalized over v) may still carry some information, and the disk-image analysis is illustrative. But the central claim about v must be reframed as a constraint on the product C, or the v bound withdrawn. The reader's verdict of CONDITIONAL remains appropriate, with this specific condition added.","tokens_in":15477,"tokens_out":12284,"duration_ms":127536,"concrete_test":"Reparameterize with C=Q²e^{-v}. (i) Show analytically from Eq. (31) that R_sh is a function of C only. (ii) Re-run the Sgr A* (GRAVITY) MCMC with the same likelihood and priors but extend the Q prior to 0≤Q≤5; if the 95% upper limit on v changes substantially (it should move toward the prior edge), the headline v<4.153 is prior-dominated. (iii) As a cleaner check, sample in the space (M,D,C) with the same shadow data and then map back to (Q,v) under the stated priors; the resulting credible region on v should reduce to the prior transformed by C, matching the reported limits only if the prior on Q is the source of the bound.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The metric (19) enters every observable through the single combination C = Q² e^{-v}. In particular, the photon-sphere and shadow radius (31) depend only on C, so the likelihood (36) is invariant along lines of constant C. The parameters Q and v are therefore not separately identifiable from shadow-size data; only C is. The reported 95% upper limits in Table II are obtained with the arbitrary uniform prior (38), 0≤Q≤1 and 0≤v≤10, and then marginalizing over the unidentifiable direction. The v<4.153 limit is a projection of the prior boundary Q≤1 onto the v axis, not an observational constraint: for any acceptable C, one can keep the fit by choosing Q=sqrt(C e^v), which remains within the prior only for v below roughly -ln(C). Extending the Q prior would move the v bound. The same degeneracy affects the accretion-disk flux statements, which also depend only on C. An internal symptom is that the GRAVITY-only row of Table I has no θ_sh, so Eq. (36) cannot define a likelihood for that panel; the tight v bound reported for 'Sgr A* (GRAVITY)' must have used the EHT shadow size as well, or is otherwise unexplained.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a static, spherically symmetric ModMax black hole with metric function f(r)=1-2M/r+e^{-v}Q^2/r^2, computes the photon-sphere and shadow radii, and uses EHT shadow-size measurements of M87* and Sgr A* together with GRAVITY mass and distance determinations to fit the parameters (M,D,Q,v) via MCMC with emcee. The reported headline constraints are Q<0.391 and v<4.153 at 95% credibility. The paper also constructs Novikov-Thorne thin-disk flux and temperature profiles and produces backward ray-traced images, finding that the observed disk brightness increases with Q and slightly decreases with v.","tokens_in":15788,"tokens_out":5885,"duration_ms":55118,"significance":"If the reported separate constraints on Q and v were data-driven, the paper would provide a useful observational test of ModMax nonlinear electrodynamics. The geodesic, shadow, and ray-tracing calculations are standard and appear correctly implemented, and the qualitative dependence of the shadow radius on the model parameters is a helpful illustration. However, the central quantitative claim is undermined by an exact parameter degeneracy: every observable considered depends only on the combination C=Q^2 e^{-v}, so the separate 95% upper limits on Q and v are prior artifacts rather than independent measurements. The GRAVITY-only posterior presented in Table II and Fig. 3 is also not defined by the stated likelihood, since Table I lists no shadow diameter for that row. The disk-flux results inherit the same degeneracy. With a reparameterization in terms of C and a clarification of the data entering each posterior, the paper could become a valid constraint study; in its present form the headline claims are not supported.","major_comments":[{"comment":"The metric (19) contains Q and v only through the combination C=Q^2 e^{-v}. Therefore the photon-sphere radius, the shadow radius R_sh in Eq. (31), and the likelihood in Eq. (36) are all invariant along lines of constant C. The MCMC posterior is hence completely flat in the degenerate direction, and the separate 95% upper limits in Table II are projections of the arbitrary uniform prior (38) onto the Q and v axes. For any allowed effective charge C, one can move to Q=sqrt(C e^v) with v up to -ln C without changing the likelihood at all; the reported v<4.153 limit simply reflects the prior boundary Q<=1. The abstract's claim of 'strongest upper limits' on Q and v is therefore not supported by the data. The authors should reparameterize the analysis to constrain C directly, or explicitly identify an observable that separates Q from v; none is presented. The same degeneracy affects the accretion-disk analysis in Section IV, since F(r) also depends only on C.","section":"Section III, Eqs. (19), (31), (36), (38)"},{"comment":"The 'Sgr A* (GRAVITY)' row in Table I has no angular shadow diameter (em dash), so the log-likelihood (36) cannot be evaluated for that dataset. Nevertheless, Table II and Fig. 3 report a posterior labeled 'Sgr A* (GRAVITY)' with best-fit values and v<4.153. Either the GRAVITY-only analysis silently used the EHT shadow diameter together with the GRAVITY mass and distance, in which case the labeling is incorrect, or the likelihood was constructed differently and its definition is missing. The paper must specify exactly which value of theta_sh and sigma_sh enters each of the four posterior rows and justify the label.","section":"Table I and Section III"},{"comment":"The analysis equates the EHT-measured ring angular diameter theta_sh with the photon-sphere shadow radius of the static, spherically symmetric ModMax metric. This mapping is a strong simplification: the EHT ring size depends on the accretion flow morphology, the black hole spin, and the emission region, and the ring is not identical to the photon-sphere shadow. Because the Gaussian uncertainties in Table I are at the few-percent level, a systematic error of that order could shift the reported bounds by more than their statistical uncertainty. The authors should either use a more conservative shadow-size likelihood informed by EHT image modeling, or add and propagate a systematic error term and discuss its effect on the limits.","section":"Section III, Eq. (35) with Table I"}],"minor_comments":[{"comment":"Equation (3) writes F_{\\mu\\nu} as \\partial_\\mu A_\\nu \\partial_\\nu A_\\mu without the antisymmetrization; it should be F_{\\mu\\nu}=\\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu.","section":"Eq. (3)"},{"comment":"The abstract uses '95% credible level' while Table II and Fig. 3 use '95% confidence level'; please choose one terminology consistently.","section":"Abstract and Table II"},{"comment":"The combined Sgr A* (EHT+GRAVITY) upper limit on v (4.277) is weaker than the GRAVITY-only limit (4.153), which is unexpected for a combined dataset and is likely a further symptom of the degeneracy; this should be discussed or explained.","section":"Table II"},{"comment":"References [7] and [66] are the same paper (H. M. Siahaan, Phys. Lett. B865, 139479 (2025), arXiv:2409.13967); please merge them or distinguish the specific results being cited.","section":"References"},{"comment":"The redshift factor in Eq. (46) uses \\Omega without explicitly noting that it is the circular-orbit angular velocity from Eq. (43); please define it just before or within the equation.","section":"Section IV, Eq. (46)"},{"comment":"The collaboration name appears as 'GRA VITY' with a space in several places (e.g., Introduction and Table I caption); it should be 'GRAVITY'.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper's headline constraints on Q and v are not identifiable from the data: all observables depend only on C=Q^2 e^{-v}, so the separate upper limits are prior artifacts. The GRAVITY-only posterior is also undefined as described. This is a load-bearing issue, not a cosmetic one, and it requires a reparameterized analysis before publication. I would be willing to reconsider after the authors constrain C directly and clarify which shadow data enter each posterior. The manuscript is within the journal's scope, and the geometric calculations are competently done, but the central quantitative claims need substantial revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The shadow derivation and MCMC implementation are standard and the geometry is correct, but the headline constraints on Q and v do not hold up. The metric (19) depends on Q and v only through the combination C = Q^2 e^{-v}, so the shadow radius and the likelihood are functions of C alone. That means Q and v are not separately identifiable from shadow data. The reported 95% upper limits Q<0.391 and v<4.153 are prior artifacts, not independent observational bounds. The v limit especially is just the Q≤1 prior boundary projected onto the v axis; a wider Q prior would shift it.\n\nThere is real value here. The photon-sphere and shadow-radius derivation is clean, the Novikov–Thorne flux and temperature profiles are computed consistently, and the ray-traced disk images are clearly presented. The specific combination of EHT and GRAVITY data for Sgr A* is new, and the idea of comparing the two data sets is reasonable.\n\nThe soft spots beyond the degeneracy: the GRAVITY-only row in Table I has no shadow diameter, so Eq. (36) cannot define a likelihood for that panel. The paper appears to have used the EHT shadow size with GRAVITY mass and distance, but that is never stated. The disk images are not compared to actual EHT images or spectra, so they remain illustrations. No code or data are provided.\n\nWho is this for? A reader working on EHT shadow constraints for modified gravity will find the methods familiar and the disk images useful as a reference. But as a constraint paper it does not deliver what it claims. The authors should redo the analysis with C as the single parameter and report its posterior, then discuss the mapping to (Q,v) with explicit priors.\n\nMy recommendation: send it to peer review, because the identifiability issue is instructive and fixable, and the disk calculations deserve a careful check. The referee should require the authors to either present the analysis in terms of C or clearly state the prior dependence of their separate limits.","headline":"The separate Q and v constraints are not identifiable from shadow data; the paper's main quantitative claim is a prior artifact, though the geometry and disk calculations are competently done.","tokens_in":16323,"tokens_out":6114,"would_cite":false,"duration_ms":57391,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.70.-s"],"model":"deepseek-v4-flash","headline":"This paper argues that Event Horizon Telescope shadows and GRAVITY astrometry already bound the charge and nonlinearity parameters of ModMax black holes, with the tightest limits $Q<0.391$ and $v<4.153$ at 95% credibility.","keywords":["ModMax electrodynamics","black hole shadow","Event Horizon Telescope","Sgr A*","M87*","accretion disk","backward ray tracing","parameter estimation"],"falsifier":"Compute a full ray-traced image of a magnetized accretion flow in the ModMax metric and compare the resulting ring diameter with the photon-sphere shadow diameter used in Eq. (35); if they differ by more than the EHT measurement uncertainty for M87* or Sgr A*, the central mapping is biased and the quoted limits on $Q$ and $v$ would not follow.","tokens_in":15307,"feed_emoji":"🕳️","tokens_out":8724,"duration_ms":77913,"temperature":0.7,"pith_summary":"This paper sets out to show that existing shadow observations already speak to the ModMax theory of nonlinear electrodynamics, a single-parameter extension of Maxwell's theory that preserves conformal invariance and electric-magnetic duality. Working in the static, spherically symmetric ModMax black hole spacetime, the authors compute how the total dyonic charge $Q$ and the nonlinearity parameter $v$ change the horizon, photon-sphere, and shadow radii, and then feed the predicted shadow diameter through a Gaussian likelihood into an MCMC sampler using Event Horizon Telescope measurements of M87* and Sgr A* and GRAVITY astrometry of Sgr A*. They report the strongest upper limits $Q<0.391$ and $v<4.153$ at 95% credibility. They also simulate Novikov-Thorne thin-disk images and find that the observed flux rises with $Q$ and falls slightly with $v$, showing that the same parameters that shrink the shadow also brighten the disk.","feed_headline":"Q<0.391: EHT shadows cap ModMax charge","feed_subtitle":"MCMC fits to M87* and Sgr A* also limit the nonlinearity parameter to v<4.153, and disk images show charge brightens the glow.","key_machinery":"The central object is the ModMax black hole metric $f(r)=1-2M/r+e^{-v}Q^2/r^2$, where $v$ acts as an exponential charge-screening factor on the total dyonic charge $Q$. The shadow is carried by the effective potential for null geodesics and the critical impact parameter $R_{\\rm sh}=b_c=r_{\\rm ph}/\\sqrt{f(r_{\\rm ph})}$, converted to an angular diameter by $\\theta_{\\rm sh}=R_{\\rm sh}(2GM/c^2D)$. Parameter estimation is carried by a Gaussian log-likelihood, Gaussian priors on $M$ and $D$, uniform priors on $Q\\in[0,1]$ and $v\\in[0,10]$, and an MCMC sampler. The disk side is carried by the Novikov-Thorne flux integral that converts circular-orbit energies into radiation profiles, plus backward ray tracing that assigns Doppler- and redshift-corrected fluxes to image pixels.","core_discovery":"The paper's central claim is that in the ModMax black hole spacetime $f(r)=1-2M/r+e^{-v}Q^2/r^2$, the parameters $Q$ and $v$ leave opposite imprints on observables: increasing $Q$ decreases the horizon, photon-sphere, and shadow radii, while increasing $v$ pushes all three back toward their Schwarzschild values because $v$ exponentially screens the charge. Using the EHT shadow angular diameters for M87* and Sgr A*, GRAVITY's mass and distance measurements for Sgr A*, and an MCMC fit to the four parameters $(M,D,Q,v)$, the paper finds best-fit values of $Q$ and $v$ consistent with zero and derives 95% upper limits, the tightest being $Q<0.391$ from M87* and $v<4.153$ from Sgr A* GRAVITY data. In the ray-traced thin-disk images, the observed flux increases with $Q$ and slightly decreases with $v$; at $85^\\circ$ inclination the Schwarzschild image reaches about 65% of the $Q=1$, $v=1$ reference brightness, and at fixed $Q=0.5$ the $v=1$ image is about 85% as bright as the $v=0$ image.","pith_inferences":["The paper leaves implicit that, since $v$ enters only through $e^{-v}Q^2$, the constraints really bound the screened charge combination, not $Q$ and $v$ separately; a large charge hidden behind a large $v$ remains compatible with the data and would require a different probe to uncover.","The monotone increase of disk flux with $Q$ suggests that continuum spectra of bright, well-modeled accretion disks could provide an independent test of ModMax-like charge, something the paper does not fit.","A next-generation interferometric image resolving the photon ring rather than the shadow edge could directly test the assumed equality between observed ring diameter and the photon-sphere shadow diameter, potentially tightening or overturning the quoted limits.","The same MCMC pipeline could be applied to other nonlinear electrodynamics metrics that reduce to the same screened-charge form, giving a uniform comparison of observational bounds."],"forward_implications":["The EHT and GRAVITY data, under the paper's shadow-diameter mapping, exclude ModMax charges $Q\\gtrsim0.4$ and nonlinearity parameters $v\\gtrsim4.2$ at 95% credibility for the two supermassive black holes studied.","M87* gives the tighter charge bound, while Sgr A* GRAVITY data give the tighter $v$ bound, so combining shadow and astrometric data sets does more than repeating the same measurement.","A larger total charge makes the shadow smaller and the disk brighter, so the same ModMax parameters that would hide the shadow would make the accretion flow easier to see.","The best-fit mass and distance recovered from the MCMC agree with the independently measured values, indicating the constraints are not driven by forcing the masses away from their observed values.","Because increasing $v$ shifts observables back toward Schwarzschild values, large $v$ values are harder to exclude than large $Q$ values; the data bound $v$ from above at $4.153$."],"supporting_citations":[{"why":"supplies the EHT angular shadow diameter and uncertainty for M87* used in the MCMC likelihood.","marker":"[69]"},{"why":"supplies the EHT angular shadow diameter for Sgr A*.","marker":"[70]"},{"why":"supplies the GRAVITY mass and distance measurements for Sgr A* used as priors and as an independent data set.","marker":"[71]"},{"why":"derives the static spherically symmetric Einstein-ModMax black hole solution with the metric function carrying $e^{-v}Q^2$.","marker":"[5]"},{"why":"provides the emcee MCMC sampler used to obtain posterior distributions.","marker":"[72]"},{"why":"provides the Novikov-Thorne thin-disk flux formula on which the radiation profiles and images are based.","marker":"[73]"},{"why":"supplies the backward ray-tracing scheme used to generate the simulated disk images.","marker":"[80]"}],"fun_headline_variants":["EHT caps ModMax charge at 0.391","Shadow data bound ModMax: Q<0.391, v<4.153","Charge shrinks shadows, nonlinearity restores them","Q<0.391 and v<4.153: ModMax constrained","EHT and GRAVITY fit tightens ModMax limits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole constraint rests on assuming the observed bright ring has the same angular size as the theoretical shadow of a static, spherically symmetric ModMax black hole; if the glowing accretion flow shifts the apparent ring size, the quoted limits shift with it.","fun_headline_variants_meta":{"raw":{"variants":["EHT caps ModMax charge at 0.391","Shadow data bound ModMax: Q<0.391, v<4.153","Charge shrinks shadows, nonlinearity restores them","Q<0.391 and v<4.153: ModMax constrained","EHT and GRAVITY fit tightens ModMax limits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001223,"raw_usage":{"total_tokens":5103,"prompt_tokens":1096,"completion_tokens":4007,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":712,"completion_tokens_details":{"reasoning_tokens":3915}},"tokens_in":712,"tokens_out":4007,"duration_ms":27412,"temperature":1.0,"reasoning_tokens":3915,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:36:22.588357+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute a full ray-traced image of a magnetized accretion flow in the ModMax metric and compare the resulting ring diameter with the photon-sphere shadow diameter used in Eq. (35); if they differ by more than the EHT measurement uncertainty for M87* or Sgr A*, the central mapping is biased and the quoted limits on $Q$ and $v$ would not follow.","supporting_citations":[{"cited_title":"Akiyama, Astrophys","cited_arxiv_id":null,"evidence_quote":"supplies the EHT angular shadow diameter for Sgr A*."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the Novikov-Thorne thin-disk flux formula on which the radiation profiles and images are based."}],"review_version":1}