{"id":"bce5a82a-d504-4f15-bfa2-4283218d38fc","arxiv_id":"2502.06388","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The shadows and observed intensities of GUP-inspired regular black holes with a Minkowski core are computed under static and infalling spherical accretion, and are compared with Bardeen and Hayward black holes.","lead":"This paper computes the predicted shadow images and light intensity for a family of regular black holes with a Minkowski core, under two spherical accretion models. It finds that the quantum parameter alpha_0 shrinks the shadow while the deformation parameter n enlarges it, and that these Minkowski-core holes look slightly dimmer and smaller than Bardeen and Hayward holes with de Sitter cores.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Core-type comparison hinges on an unvalidated parameter identification; alternative identifications could erase or reverse the sub-percent shadow differences.","rationale":"Read in good faith: the ray-tracing and intensity formalism follow the standard spherical-accretion framework, and the Schwarzschild limit is correctly recovered. The within-family trends (shadow radius decreases with alpha_0, increases with n; intensity does the opposite) are supported by Tables 1-2 and appear robust. The concern is confined to the cross-core comparison in Sec.4. The reader's weakest assumption identifies the same parameter-correspondence issue, and I agree that it is the critical link. The comparison has no robustness analysis, and the reported differences are at or below the percent level even at the largest allowed alpha_0, so a reidentification test is decisive. I also note minor internal inconsistencies (the Sec.3.1 sentence saying alpha_0 enlarges the shadow, and Fig.3-6 using n=1 despite the stated n>=2 condition), but neither is as load-bearing as the identification issue. The paper does not need rejection, but the conditional verdict is appropriate pending the proposed check.","tokens_in":15964,"tokens_out":12721,"duration_ms":112958,"concrete_test":"Perform a robustness check on Sec.4: for each alpha_0, solve f_dS(r_h^{dS}; g)=0 with g chosen so that r_h^{dS}=r_h^{new}(alpha_0), i.e., match the outer horizon radii instead of the first-order large-r mapping, then recompute b_c and peak I_obs for both static and infalling accretion. If the Minkowski-core hole is no longer smaller and dimmer under this identification, the claimed core-type distinction is an artifact of parameter choice; if the ordering survives, the concern is resolved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The load-bearing step is the identification of alpha_0 in Eq.(3) with the Bardeen/Hayward charge in Eq.(4) via the authors' prior one-to-one correspondence. That correspondence matches only the first correction in a 1/r expansion; the two potentials differ at order alpha_0^2, so the comparison in Sec.4 is a statement about this chosen identification, not about 'Minkowski core vs dS core' in general. The computed differences are small: at alpha_0=0.72, the shadow-radius difference between the new (x=2/3, n=2) BH and Bardeen is ~0.57%; at alpha_0=0.92, the difference between the new (x=1, n=3) BH and Hayward is ~0.11%. A different but equally plausible identification (for example, matching outer horizon radii rather than the first large-r coefficient) could reduce, eliminate, or reverse these differences. Since the final claim is that the two core types can be distinguished observationally, this unvalidated identification is the least secure link in the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the shadows and optical appearances of a family of regular black holes with an asymptotically Minkowski core, defined by the metric in Eqs. (1)-(3), under static and infalling spherical accretion. The authors compute geodesics, photon-sphere radii, critical impact parameters, and observed specific intensities, and then compare these quantities with the traditional Bardeen and Hayward black holes, which have de Sitter cores, using a one-to-one correspondence inherited from earlier work. The central claims are that for the Minkowski-core holes the shadow and photon-sphere radii decrease with increasing quantum parameter α0 while the observed intensity increases, that increasing the deformation parameter n has the opposite effects, and that the Minkowski-core holes have smaller and dimmer shadows than the corresponding dS-core holes, with the difference growing with α0.","tokens_in":16082,"tokens_out":6468,"duration_ms":51928,"significance":"The paper applies a standard ray-tracing and spherical-accretion framework to a specific family of regular black holes and recovers the Schwarzschild limit correctly in Tables 1 and 2, which gives confidence in the numerical implementation. If the adopted parameter identification between the Minkowski-core and dS-core families is accepted, the work provides a systematic comparison of shadow and intensity observables between two core types, which is a useful theoretical input for future EHT-type observations. However, the observational distinguishability conclusion is not quantitatively supported, and the manuscript contains at least one internal inconsistency (use of n=1 despite the stated n≥2 horizon condition) that affects part of the parameter scan. The core-type comparison also depends entirely on a single mapping from prior work, so the robustness of the main conclusion is not yet established.","major_comments":[{"comment":"The conditions stated in Sec. 2 after Eq. (4) require n ≥ 2 (together with n ≥ x ≥ n/3) in order to ensure the existence of an event horizon and sub-Planckian curvature. Nevertheless, Sec. 3.1 and Figs. 3 and 4 explicitly include n = 1 configurations (e.g., 'When the spacetime deformation n = 1...' and the four panels of Fig. 4). No proof is given that n = 1 cases satisfy the horizon-existence condition, and they appear to violate the stated constraint. The authors should either restrict the analysis to n ≥ 2 or demonstrate that n = 1 configurations are legitimate black holes under the same criteria; otherwise the reported n-dependence of the intensity and shadow radius includes points that are outside the claimed valid parameter space.","section":"Sec. 2 and Sec. 3.1"},{"comment":"The text reads 'the enhancement of quantum gravity effect can increase the luminosity of photon ring and enlarge the shadow radius for these new regular BHs.' This directly contradicts the immediately preceding sentence and the abstract, which state that larger α0 leads to a smaller shadow radius and photon-sphere radius. Since this sentence reverses the paper's primary trend, it is a load-bearing error and must be corrected, presumably to 'decrease the shadow radius.'","section":"Sec. 3.1, final paragraph"},{"comment":"The comparison between Minkowski-core and dS-core black holes rests entirely on the one-to-one correspondence between Eq. (3) and Eq. (4) taken from Ref. [58]. That correspondence matches only the leading large-r behavior; the differences between the potentials appear at higher order in α0. The paper does not justify that this particular identification is the physically relevant one, and alternative identifications (for example, matching outer horizon radii or photon-sphere radii for each α0) could reduce, eliminate, or reverse the reported differences. The numerical differences are in fact small: in Table 1, for α0=0.72, the critical impact parameter is 4.66393 for the Minkowski-core hole versus 4.69073 for Bardeen (about 0.57%), and in Table 2 for α0=0.92 the difference with Hayward is about 0.11%. The conclusion that the two core types can be observationally distinguished is therefore conditional on the chosen parameter identification. Please state this limitation explicitly and, if possible, test the sensitivity of the conclusions to alternative identifications.","section":"Sec. 4 and Tables 1-2"},{"comment":"The abstract and conclusion state that the two types of regular black holes 'can be distinguished through astronomical observation.' No quantitative observational analysis is provided: the paper does not compare the predicted intensity differences or sub-percent shadow-radius differences with EHT angular resolution, sensitivity, or expected astrophysical contamination. At the sub-percent level, such a claim requires at least an order-of-magnitude estimate of detectability. The authors should either provide such an estimate or soften the claim to a theoretical prediction that would require future observational capabilities.","section":"Sec. 5 and Abstract"}],"minor_comments":[{"comment":"The sentence 'similarly, Hayward BH with a dS core is slightly greater values than the one with a Minkowski core (x = 2/3 and n = 2)' should refer to (x = 1, n = 3) rather than (x = 2/3, n = 2), since the Hayward comparison is made for those parameters in Table 2.","section":"Sec. 2, paragraph after Table 1"},{"comment":"The dimensional discussion in the footnote is confusing; in particular, 'express G into a form of M x' is unclear and should be rewritten to show explicitly how α0 is made dimensionless and how the Planck-length powers are absorbed.","section":"Sec. 2, footnote 1"},{"comment":"The integration domain and the treatment of the geodesic path (including the number of times the ray passes through the emitting spherical shell and the handling of turning points) are not specified. For reproducibility, please state the integration limits and the numerical procedure used to evaluate the integral along the full photon path.","section":"Sec. 3.1, Eqs. (16)-(17)"},{"comment":"There is a typo: 'diving into the event horizon' should be 'diving into the event horizon.'","section":"Sec. 1, Introduction"},{"comment":"The sentence 'When the core type is the same, these values increase with the increase of the dimensionless parameters (x and n)' is repeated in the same paragraph; please remove the duplicate.","section":"Sec. 2, paragraph after Fig. 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for the journal and the basic ray-tracing framework is sound. The main risk is the parameter identification used for the core-type comparison; if the authors can show that their qualitative conclusions are robust to alternative reasonable identifications of α0, the paper would be considerably stronger. The n=1 inconsistency should be resolved before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a competent, standard application of the spherical-accretion shadow formalism to the Ling–Wu Minkowski-core regular black holes. The math is standard, the Schwarzschild limit checks out, and the core trends (alpha_0 shrinks the shadow, n enlarges it) are coherent with the tables. The genuinely new piece is the intensity profiles for these specific metrics; that justifies a referee, but not excitement.\n\nWhat it does well: Tables 1–2 give photon-sphere and impact-parameter values for representative parameters and correctly recover Schwarzschild; the intensity-integration formulas are the standard ones from Bambi and the static-vs-infalling comparison is handled correctly. The result that the shadow radius is independent of the accretion model while the intensity depends on it is correct and worth stating.\n\nSoft spots, in order of importance. First, the advertised distinction between Minkowski-core and dS-core BHs rests entirely on the one-to-one correspondence from Ling and Wu (2023), which matches only the leading 1/r potential. A different but equally plausible parameter identification, for example matching outer horizons, can reduce, eliminate, or reverse the sub-percent shadow differences. So the paper's claim that the two core types 'can be distinguished through astronomical observation' is not supported; with shadow-radius differences around 0.5% and even smaller intensity differences under infalling accretion, EHT cannot see this. Second, there is a direct prose contradiction in Sec. 3.1: after Fig. 4 the text says larger alpha_0 gives a smaller shadow radius, then the next paragraph says the quantum effect 'enlarge[s] the shadow radius.' Presumably a typo, but as written it muddies a central result. Third, no code or data are shipped, so the intensity curves are not reproducible from the paper alone. Fourth, the self-citations are heavy but mostly legitimate since the metric and correspondence come from their prior work.\n\nBottom line: this is a standard phenomenology paper, appropriate for a specialist journal after minor revisions. The referee should ask for the contradiction to be fixed, the 'observationally distinguishable' claim to be tempered unless the parameter identification is justified, and ideally for the intensity data or code to be available. I would accept it for peer review; I would not cite it unless I was compiling shadow phenomenology references.","headline":"Competent, incremental shadow phenomenology for a regular BH family, with the core-type comparison resting on an unvalidated parameter identification.","tokens_in":16677,"tokens_out":3146,"would_cite":false,"duration_ms":25557,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.70.-s","04.70.Dy"],"model":"deepseek-v4-flash","headline":"For regular black holes with a Minkowski core, raising the quantum parameter α0 shrinks the shadow and brightens the photon ring, while the core itself produces smaller, dimmer images than Bardeen and Hayward black holes.","keywords":["black hole shadow","regular black holes","Minkowski core","spherical accretion","photon sphere","generalized uncertainty principle","Bardeen black hole","Hayward black hole"],"falsifier":"Compute the shadow radius $b_c$ and the static/infalling intensity ratio for a Minkowski-core metric with parameters fitted directly to Sgr A* or M87* data, without imposing the Bardeen/Hayward correspondence. If the best fit places the shadow outside the paper's predicted range for the fitted $\\alpha_0$ and $n$, or if the observed static-versus-infalling intensity ordering is reversed, the claim is falsified. A quantitative target: the predicted $\\sim 0.34\\,M$ decrease in $b_c$ between $\\alpha_0=0.3$ and $0.72$ for the $x=2/3,n=2$ Minkowski-core hole is a difference a high-resolution shadow measurement could in principle confront.","tokens_in":15699,"feed_emoji":"🕳️","tokens_out":12401,"duration_ms":96652,"temperature":0.7,"pith_summary":"The paper tries to establish that the shadow of a regular black hole with a Minkowski core carries an observable fingerprint of quantum gravity. For these singularity-free black holes, built from a generalized-uncertainty-principle (GUP) modified Newtonian potential, a larger quantum parameter $\\alpha_0$ makes the shadow and photon sphere smaller while the observed specific intensity rises; a larger deformation parameter $n$ does the opposite. The paper also claims that Minkowski-core holes are always slightly smaller and dimmer than their Bardeen and Hayward counterparts with de Sitter cores, and that the difference widens as $\\alpha_0$ grows, particularly under static spherical accretion. If correct, black-hole imaging could discriminate which type of regular core is realized in nature.","feed_headline":"Minkowski-core black holes shrink and brighten with α0","feed_subtitle":"The gap from Bardeen and Hayward black holes widens with α0, making the core type potentially visible to telescopes.","key_machinery":"The load-bearing object is the metric function $f(r)=1+2\\psi(r)$ with the exponentially suppressed GUP potential $\\psi(r)=-(M/r)e^{-\\alpha_0 M^x/r^n}$ (Eq. 3), whose core is Minkowski, paired with the one-to-one large-scale correspondence (Eq. 4) to the Bardeen potential at $x=2/3,n=2$ and the Hayward potential at $x=1,n=3$. Photon motion is reduced to the effective potential $V_{\\rm eff}(r)=f(r)/r^2$; the photon sphere is set by $V'_{\\rm eff}(r_c)=0$ and the shadow radius by $V_{\\rm eff}(r_c)=1/b_c^2$. The optical images are produced by ray tracing null geodesics and integrating the redshift-weighted emissivity along the geodesic, with Eq. (17) for static spherical accretion and Eq. (26) for infalling spherical accretion. The same correspondence that fixes which Bardeen/Hayward parameter goes with each $\\alpha_0$ is what allows the core-type comparison to be made at all.","core_discovery":"On the paper's own terms, the central discovery is a monotonic relation: in the Minkowski-core metrics with $f(r)=1+2\\psi(r)$ and $\\psi(r)=-(M/r)\\exp(-\\alpha_0 M^x/r^n)$, the photon sphere radius $r_c$ and the critical impact parameter $b_c$ both decrease as $\\alpha_0$ increases, while the peak of the observed specific intensity increases. Increasing $n$ reverses both trends. For example, at $M=1$, $\\alpha_0=0.72$, $x=2/3$, $n=2$, the paper finds $b_c=4.66393$ for the Minkowski-core hole versus $4.69073$ for the Bardeen hole, and the Minkowski-core photon ring is dimmer under static spherical accretion. The shadow boundary is independent of whether the surrounding accretion is static or infalling; only the intensity changes, with static accretion giving a brighter image. These claims are made for two pairings: $(x,n)=(2/3,2)$ against the Bardeen black hole and $(1,3)$ against the Hayward black hole.","pith_inferences":["Beyond the paper: because the shadow boundary is accretion-model independent while the intensity is not, the ratio of static to infalling intensity is a cleaner probe of the accretion physics; the authors do not make this separation explicit.","Beyond the paper: the same machinery could be applied to a rotating Minkowski-core metric; the spherical-symmetry assumption means the claimed core-type distinction has not yet been tested against the asymmetric images needed for real black-hole observations.","Beyond the paper: the one-to-one Bardeen/Hayward correspondence fixes only large-scale behaviour, so a data-driven parameter fit to the two metrics instead of the assumed mapping could change or even reverse the ordering of shadow sizes; this is the natural next test.","Beyond the paper: the paper's brighter-photon-ring-with-larger-$\\alpha_0$ result implies a possible degeneracy with a slightly smaller Schwarzschild-like object; only joint measurement of radius and brightness, not either alone, would break it."],"forward_implications":["A measured shadow radius at fixed mass would constrain $\\alpha_0$ and $n$: smaller $b_c$ implies larger $\\alpha_0$ or smaller $n$.","Because the shadow and photon-sphere locations do not depend on the accretion model, the shadow radius can be used to compare spacetime models even when the details of the accretion flow are unknown.","At the same parameters, Minkowski-core holes are systematically smaller and dimmer than Bardeen and Hayward holes, so high-resolution images could in principle tell a Minkowski core from a de Sitter core.","The difference between core types grows with $\\alpha_0$, meaning that higher quantum-gravity corrections make the signature easier to detect, not harder.","Under static spherical accretion the difference is larger than under infalling accretion, so the detectability of the core type depends on the astrophysical environment of the hole."],"supporting_citations":[{"why":"Introduces the GUP-motivated Minkowski-core metrics and the one-to-one correspondence (Eq. 4) that links them to Bardeen and Hayward potentials; this is the construction whose shadows are computed.","marker":"[58]"},{"why":"The authors' earlier comparison of photon spheres and marginally stable circular orbits in Minkowski-core versus de Sitter-core holes; motivates the shadow comparison and supplies the parameter pairs.","marker":"[62]"},{"why":"Earlier thin-accretion-disk study of the same two core types showing different optical characteristics; the spherical-accretion images here are the extension of that comparison.","marker":"[15]"},{"why":"One of the two sources for the observed-intensity integral under spherical accretion, including the redshift-factor approach used in Eq. (17).","marker":"[85]"},{"why":"The other source for the spherical-accretion image method, giving the emissivity and proper-length prescription used for both static and infalling models.","marker":"[86]"}],"fun_headline_variants":["Minkowski-core black holes: α0 shrinks shadows, n enlarges them","Static accretion outshines infalling for Minkowski-core black holes","Core type changes black hole shadows: α0 vs n interplay","Shadow size and brightness track α0, n in Minkowski-core holes","Bardeen and Hayward contrasts reveal Minkowski core signatures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the one-to-one correspondence between the quantum-gravity potential used for the new black holes and the potentials of the Bardeen and Hayward black holes; if that mapping is not the physically right one, the claimed differences between the two core types could be an artifact of how the parameters were matched rather than a real observable distinction.","fun_headline_variants_meta":{"raw":{"variants":["Minkowski-core black holes: α0 shrinks shadows, n enlarges them","Static accretion outshines infalling for Minkowski-core black holes","Core type changes black hole shadows: α0 vs n interplay","Shadow size and brightness track α0, n in Minkowski-core holes","Bardeen and Hayward contrasts reveal Minkowski core signatures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000264,"raw_usage":{"total_tokens":1679,"prompt_tokens":1098,"completion_tokens":581,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":714,"completion_tokens_details":{"reasoning_tokens":484}},"tokens_in":714,"tokens_out":581,"duration_ms":6218,"temperature":1.0,"reasoning_tokens":484,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T15:35:48.770817+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the shadow radius $b_c$ and the static/infalling intensity ratio for a Minkowski-core metric with parameters fitted directly to Sgr A* or M87* data, without imposing the Bardeen/Hayward correspondence. If the best fit places the shadow outside the paper's predicted range for the fitted $\\alpha_0$ and $n$, or if the observed static-versus-infalling intensity ordering is reversed, the claim is falsified. A quantitative target: the predicted $\\sim 0.34\\,M$ decrease in $b_c$ between $\\alpha_0=0.3$ and $0.72$ for the $x=2/3,n=2$ Minkowski-core hole is a difference a high-resolution shadow measurement could in principle confront.","supporting_citations":[],"review_version":1}