{"id":"d904c18d-e94c-4c9a-80d5-60a3a004fdd0","arxiv_id":"2507.10874","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":8,"one_line_summary":"A parameter study of geodesics, shadows, scalar-field potentials, GUP-corrected Hawking temperatures, and orbital frequencies for a charged BTZ-like black hole with Bopp-Podolsky electrodynamics and disclinations.","lead":"This paper studies photon orbits, scalar waves, GUP-corrected temperatures, and Keplerian frequencies for a charged three-dimensional black hole with Bopp-Podolsky electrodynamics and a conical defect. The thermodynamic part evaluates temperatures at inner horizons and contains inconsistent formulas, so its central remnant conclusion is not supported.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The GUP-remnant claim fails because f(r_h) is nonzero at every root of g(r)=0 for b≠0: no Killing horizon exists, so the near-horizon expansion behind Eq. (41) and the temperature formula in Eq. (43) are inapplicable.","rationale":"Reading in good faith, the geodesic, shadow, and scalar-field sections appear routine and may contain useful material; the advertised headline is the GUP-corrected Hawking temperature and the prospect of stable remnants. That headline depends entirely on Eqs. (42)-(44) and Table 5. The most load-bearing defect is not merely which root of g(r)=0 is selected, but that for b≠0 the functions f and g in Eq. (2) do not vanish simultaneously. At any root of g, f(r_h)=8 b^2 Q^2 |Λ|>0, so r=r_h is not a Killing horizon for ∂_t; the standard temperature formula used in Eq. (43) presupposes a Killing horizon. The paper's own derivation of Eq. (41) assumes f(r)≈f'(r_h)(r-r_h) near the horizon, which is inconsistent with Eq. (2). The wrong-root choice in Table 5 is a second, independent problem that compounds the failure, but it is not the root cause: even the largest positive root leaves f(r_h) nonzero. The reader identified this as the second load-bearing assumption, hence partial agreement, but it should be elevated above the root-selection issue because it invalidates the temperature formula itself. A single numerical and analytic check, evaluating f(r_h) and redoing the contour integral in Eq. (40) with the exact near-horizon behavior, settles the question. This concern supports the reader's REJECT verdict; it does not change the verdict, so I mark it UNCHANGED.","tokens_in":1029,"tokens_out":2459,"duration_ms":320284,"concrete_test":"Evaluate f(r_h) at the radii used in Table 5 using Eq. (2), e.g., for Λ=-0.1, M=1, Q=1, b=0.1, r0=1 at r_h=0.10767 and at the largest root r_h≈7.0; for b≠0, f(r_h)=8 b^2 Q^2 |Λ|>0. Then recompute ImW+ from Eq. (40) using the exact near-horizon behavior f≈f(r_h)+f'(r_h)(r-r_h) and g≈g'(r_h)(r-r_h). If f(r_h)≠0, the integrand has no first-order pole, so Eq. (41) and hence Eq. (42) do not follow. If an independent derivation nevertheless reproduces Eq. (42) despite nonzero f(r_h), the concern is refuted; otherwise the central remnant claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"From Eq. (2), g(r) = f(r) + 8 b^2 Q^2 Λ. With Λ<0, at any root r_h of g(r_h)=0 we have f(r_h) = 8 b^2 Q^2 |Λ| > 0 for b≠0. Thus the Killing vector ∂_t is not null at r_h, so r=r_h is not a Killing horizon, and the conventional surface-gravity formula T_H = sqrt(f'(r_h)g'(r_h))/(4π) used in Eqs. (43)-(44) does not apply. The derivation of Eq. (41) explicitly invokes the near-horizon expansion f(r) ≈ f'(r_h)(r-r_h), which is valid only if f(r_h)=0; here it is contradicted by Eq. (2). Without that expansion, the integrand of Eq. (40) near r_h behaves as 1/sqrt(r-r_h) rather than having a simple pole, so the tunneling residue πE/sqrt(f'g') and the GUP correction in Eq. (42) are not obtained. This defect is more fundamental than Table 5's root-selection inconsistency: it persists at the largest positive root of g, where f(r_h) = 8 b^2 Q^2 |Λ| remains nonzero. Consequently, the GUP-corrected temperature and the remnant conclusion in the abstract are unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript analyzes geodesics, scalar-field perturbations, GUP-corrected thermodynamics, and Keplerian frequencies for a charged BTZ-like spacetime in Bopp-Podolsky electrodynamics with disclinations, using the metric of Ref. [1] (Eqs. (1)-(2)). It derives effective potentials for null and timelike geodesics, circular photon orbits and shadow estimates, a Klein-Gordon reduction to a Schrödinger-like equation, a GUP-modified Hawking temperature via the tunneling method, and orbital frequencies. The advertised central result is that GUP corrections suppress Hawking radiation and may lead to stable black hole remnants.","tokens_in":25287,"tokens_out":20679,"duration_ms":213530,"significance":"The geodesic and circular photon orbit analysis in Section 3 is standard and the algebraic check of Eq. (24) is sound; the scalar-field reduction in Section 4 follows a familiar route. If the thermodynamic claim were valid, the paper would add a lower-dimensional example of GUP-induced remnants in a nonlinearly charged geometry. However, the thermodynamic section contains a fundamental error: at every zero of g(r) used as the horizon, f(r) is nonzero, so the surface is not a Killing horizon and the near-horizon tunneling calculation is inapplicable. The numerical tables are also inconsistent with the displayed formulas. These problems invalidate the paper's main advertised claim, and they are not local presentation issues.","major_comments":[{"comment":"The tunneling calculation is performed at a surface that is not a Killing horizon. Since g(r) = f(r) + 8 b^2 Q^2 Λ, any root r_h of g(r_h)=0 has f(r_h) = 8 b^2 Q^2 |Λ| > 0 for b, Q ≠ 0 and Λ < 0, so the Killing vector ∂_t is not null at r_h. The near-horizon expansions f(r) ≈ f'(r_h)(r-r_h) and g(r) ≈ g'(r_h)(r-r_h) used before Eq. (41) are therefore not both valid; near r_h the integrand of Eq. (40) behaves as 1/√(r-r_h), not as a simple pole. Equation (41) is not obtained, and Eqs. (42)-(44) and the remnant conclusion are unsupported. The same problem makes Eq. (10) vanish, since that expression contains √g(r_h)=0.","section":"§5, Eqs. (2), (40), and (41)"},{"comment":"Equation (42) does not follow from Eq. (41). With the tunneling probability Γ = exp(-4 Im W+) = exp(-E/T_GUP), Eq. (41) gives T_GUP ≈ (1/4π)√(f'g') [1 - β_GUP E²/f'(r_h)] to first order in β_GUP. The β_GUP-dependent term in Eq. (42) is instead -E²π²β_GUP r_h^6 added to the classical numerator. For the small-horizon rows of Table 5 (b=0.10, r0=1.00, r_h=0.10767) the denominator of Eq. (42) is negative, so that term raises TGUP above TH, exactly opposite to the suppression claimed in the text and shown in Table 5.","section":"§5, Eq. (42)"},{"comment":"Equation (44) is not the explicit form of Eq. (43). Because g'=f', Eq. (43) is |f'(r_h)|/(4π). Equation (44), with the numerator factored as (Λr_h^4 + Q^2 r_h^2 - 4 M b^2 Q^2)^2, reduces to (Λr_h^4 + Q^2 r_h^2 - 4 M b^2 Q^2)/(2π r_h^3), which is -f'(r_h)/(4π), not its absolute value. For the first small-horizon row of Table 5 this gives a negative value (about -3.62), while the table lists 3.62417848. Thus Table 5 is not generated by Eq. (44), and the 'explicitly becomes' claim is incorrect.","section":"§5, Eqs. (43)-(44) and Table 5"},{"comment":"The horizon radii used in Table 5 do not follow the paper's own horizon convention. Section 2 defines the event horizon as the largest positive root of g(r)=0; for b=0.10, r0=1.00, Λ=-0.1, M=Q=1, the largest positive root of Eq. (2) is approximately 7.0, while Table 5 lists r_h=0.10767, the smallest positive root. All temperature values in the r0=1.00 and r0=1.50 rows are therefore evaluated at the wrong root even under the paper's stated convention. Combined with the Killing-horizon problem, this removes the numerical support for the remnant claim.","section":"Table 5 and Sec. 2"},{"comment":"The shadow radii are computed at a fixed rcpo=3.5 that is not a solution of the circular photon orbit condition Eq. (24) for the stated parameters. For example, with M=1, r0=1, Q=0.5, b=0.2, Eq. (24) has no root at r=3.5 (the left-hand side is positive for all r>0 in this parameter set), and with Q=0.1, b=0.1 it likewise is not satisfied at r=3.5. Consequently the Rs entries in Tables 3 and 4 are not shadow radii of the geometry described by Eq. (24), and the monotonic trends reported for Rs in the text are not supported by the displayed equations.","section":"§3.4, Tables 3 and 4"}],"minor_comments":[{"comment":"Equation (34) mixes R and ψ: after the transformation (33), the equation should be R''(r) + (ω² - Veff) R(r) = 0, not R''(r) + (ω² - Veff) ψ(r) = 0.","section":"§4, Eq. (34)"},{"comment":"The caption repeats 'the scale parameter r0 = 1' twice; one occurrence should be removed.","section":"Fig. 2 caption"},{"comment":"The conversion factor νφ = c³/(2πGM) Ωφ is dimensionally unclear in (2+1) dimensions, where Newton's constant has units of inverse mass; please state the units of G and the intended normalization.","section":"§6, Eq. (52)"},{"comment":"There are typos such as 'chracteristic' in Section 4 and 'Author declare(s)' in the conflict-of-interest statement; these should be corrected.","section":"General"}],"recommendation":"reject","confidential_remarks":"The central advertised result (GUP-suppressed temperature and remnants) is unsupported by a fundamental geometric error, and the discrepancies between the numerical tables and the displayed equations suggest that the tables were not generated from the final formulas. If a revised version is submitted, the authors should re-derive the thermodynamics from a genuine Killing horizon (if one exists) and regenerate all tables from the stated equations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good to have a clear look at this one. The paper is the usual 'take a known metric, run standard methods' exercise: geodesics, circular photon orbits, shadows, scalar field via Klein-Gordon, GUP tunneling, Keplerian frequencies, all for the Bopp-Podolsky BTZ-like solution from Ref [1] with disclinations. The geodesic and scalar-field parts are routine but mostly sound. The equations of motion, the photon-orbit condition (24), the shadow tables, and the Schrödinger-like reduction all hang together algebraically. If you want a catalogue of orbital and wave properties for this specific background, the first half of the paper delivers that.\n\nThe trouble is the GUP-thermodynamics section, which carries the abstract's headline claim about remnants. The problems are load-bearing. From the metric (2), g(r) = f(r) + 8 b^2 Q^2 Λ. For Λ<0 and b≠0, any root of g(r)=0 has f(r) = 8 b^2 Q^2 |Λ| > 0. So ∂_t is not null at r_h; the horizon is not a Killing horizon. The near-horizon expansion f(r) ≈ f'(r_h)(r-r_h) used in the derivation behind Eq. (41) is therefore invalid, and the tunneling integral does not give the simple pole that produces the temperature. The formula T_H = sqrt(f' g')/(4π) in Eq. (43) presupposes a Killing horizon. On top of that, Eq. (42) does not actually follow from Eq. (41), and the numbers in Table 5 do not match between Eqs. (43) and (44) for several rows. The table also evaluates temperatures at the smallest positive root of g in some rows even though Section 2 defines the event horizon as the largest positive root. Any one of these would be a serious issue; together they sink the remnant conclusion.\n\nThis is not a matter of a typo. The thermodynamic section is built on a misidentification of the horizon. The first half of the paper could be salvaged, but the abstract's main physical claim is unsupported. The paper also inherits a solution from Ref [1] that is explicitly perturbative and valid far from the horizon, which raises an additional question about applying it at the horizon at all.\n\nFor peer review: I would desk reject this version. The geodesic and scalar-field material is fine but not novel enough to carry a paper with a broken central claim. The authors could fix the thermodynamic section or, better, drop the GUP/remnant claim and publish a clean 'geodesics and scalar fields' paper. That version would deserve referee time.","headline":"A routine geodesic/scalar-field catalogue for a BP-modified BTZ metric is undermined by a GUP-thermodynamics section whose central derivation assumes a Killing horizon that does not exist.","tokens_in":25883,"tokens_out":4706,"would_cite":false,"duration_ms":53074,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.70.-s","04.60.-m"],"model":"deepseek-v4-flash","headline":"Adding a minimum-length correction to the Hawking temperature suppresses thermal radiation and can leave a stable remnant of this charged BTZ-like black hole.","keywords":["BTZ black hole","Bopp-Podolsky electrodynamics","generalized uncertainty principle","Hawking temperature","black hole remnant","null geodesics","disclination","Keplerian frequencies"],"falsifier":"Take the Table 5 parameters $M=1$, $Q=1$, $\\Lambda=-0.1$, $b=0.1$, $r_0=1.0$: the event horizon defined as the largest positive root of $g(r)=0$ is a larger root, not the $0.10767$ used in the table, so computing the corrected temperature at both roots settles whether the suppression pattern survives. A second check is to evaluate $f(r_h)$ at the tabulated roots; if $f(r_h)$ is not numerically zero, $\\sqrt{f'(r_h)g'(r_h)}/(4\\pi)$ is not the standard surface gravity, and a calculation using the proper Killing-horizon relation gives a different temperature.","tokens_in":24734,"feed_emoji":"🕳️","tokens_out":15140,"duration_ms":144950,"temperature":0.7,"pith_summary":"This paper studies a circularly symmetric charged BTZ-like black hole pierced by disclinations, a conical-defect parameter $\\beta=1-4\\lambda$, in Bopp-Podolsky electrodynamics, a nonlinear electrodynamics whose nonminimal coupling makes the two metric functions $f(r)$ and $g(r)$ unequal. It derives the effective potential for null geodesics, circular photon orbits, the critical impact parameter and shadow radius, and the Keplerian orbital frequency, showing how the Bopp-Podolsky coupling $b^2$, electric charge $Q$, cosmological constant $\\Lambda$, and disclination parameter shift these observables. Its central new claim is a GUP-corrected Hawking temperature: tunneling computed with a minimum-length uncertainty relation gives a negative correction proportional to $\\beta_{\\rm GUP} E^2$, so quantum gravity suppresses Hawking radiation most strongly for small horizons, which the authors interpret as evidence for stable black hole remnants. If the paper is right, this charged BTZ-like black hole would not evaporate completely, and the predicted shadow sizes and orbital frequencies give concrete signatures to search for in observations.","feed_headline":"Quantum-gravity corrections cool black holes, hinting at remnants","feed_subtitle":"A minimum-length uncertainty lowers the Hawking temperature most for small black holes, suggesting evaporation ends in a remnant.","key_machinery":"The argument runs on three pieces. The spacetime is fixed by the metric functions $f(r)=-M-\\Lambda r^2-4b^2Q^2M/r^2-2Q^2\\ln(r/r_0)$ and $g(r)=-M-\\Lambda r^2+8b^2Q^2\\Lambda-4b^2Q^2M/r^2-2Q^2\\ln(r/r_0)$, whose inequality $f\\neq g$ encodes the Bopp-Podolsky nonminimal coupling. The quantum step is the GUP-modified commutation relation, which deforms the Klein-Gordon equation; the WKB ansatz then gives, after a near-horizon expansion and residue evaluation, an imaginary action $\\mathrm{Im}\\,W_+=\\pi E/\\sqrt{f'(r_h)g'(r_h)}\\,\\bigl(1+\\beta_{\\rm GUP}E^2/f'(r_h)\\bigr)$. Feeding this into the tunneling probability $\\Gamma\\sim\\exp(-2\\,\\mathrm{Im}\\,S)=\\exp(-E/T_{\\rm GUP})$ produces the closed-form GUP-corrected temperature that carries the remnant claim.","core_discovery":"The paper's central claim is that GUP corrections change the thermal life of a charged BTZ-like black hole in Bopp-Podolsky electrodynamics. Starting from the modified commutation relation $[x_i,p_j]=i\\hbar\\delta_{ij}(1+\\beta_{\\rm GUP}p^2)$, the authors deform the Klein-Gordon equation, apply the WKB tunneling ansatz $S=-Et+W(r)+j\\phi$, and evaluate the imaginary part of the action at the horizon to obtain a corrected temperature whose $\\beta_{\\rm GUP}\\to 0$ limit is $T_H=\\sqrt{f'(r_h)g'(r_h)}/(4\\pi)$. The correction is negative and grows with the emitted particle energy $E^2$, and the paper's Table 5 and Figure 7 show the corrected temperature decreasing as $\\beta_{\\rm GUP}$ increases, most strongly for small horizon radii. The authors interpret this as suppression of Hawking radiation that can stabilize the black hole into a remnant, and they connect the same parameter set to observable quantities: the shadow radius grows with charge, and the Keplerian frequency $\\Omega_\\phi=(1/\\beta)\\sqrt{-\\Lambda+Q^2/r^2-4b^2Q^2M/r^4}$ steepens with the Bopp-Podolsky coupling $b^2$.","pith_inferences":["The temperature table's remnant pattern is computed at the smallest positive root of $g(r)$ (for example $r_h\\approx 0.10767$ for $r_0=1$), whereas the paper defines the event horizon as the largest positive root; re-evaluating the corrected temperature at that larger root could erase the claimed suppression, so the remnant conclusion should be checked against the paper's own horizon definition.","The closed-form corrected temperature contains no $\\beta$ (disclination parameter), even though the text says the topological defect modifies the thermodynamic quantities; the remnant mechanism as written does not depend on the defect, so any claim about its interplay with quantum gravity needs a separate derivation.","Because the suppression scales with $E^2$, different particle species emitted at different energies would experience different effective temperatures, changing the shape of the Hawking spectrum rather than merely lowering the luminosity; this is a testable consequence distinct from a uniform temperature drop."],"forward_implications":["As $\\beta_{\\rm GUP}$ grows, the corrected temperature falls below the classical Hawking temperature for every horizon radius, with the suppression strongest for small $r_h$, so micro black holes would evaporate more slowly and could end as stable remnants.","The $\\beta_{\\rm GUP}\\to 0$ limit reproduces the semiclassical temperature, so the GUP result is presented as a consistent deformation of the standard tunneling temperature.","The heat capacity develops divergences and sign changes for nonzero Bopp-Podolsky coupling $b$, signaling phase transitions in the thermodynamic stability of these black holes.","The shadow radius and the Keplerian frequency $\\Omega_\\phi$ both grow with charge and with $b^2$, steepening the frequency gradient near the hole and giving parameter-dependent signatures for timing and imaging searches."],"supporting_citations":[{"why":"supplies the charged BTZ-like metric with Bopp-Podolsky corrections, the $f(r)$ and $g(r)$ functions used throughout","marker":"[1]"},{"why":"provides the surface-gravity/Hawking-temperature relation that is the semiclassical starting point","marker":"[88]"},{"why":"gives the GUP-modified Klein-Gordon equation and WKB tunneling scheme that yield the corrected temperature","marker":"[65]"},{"why":"supplies the tunneling-probability framework $\\Gamma\\sim\\exp(-2\\,\\mathrm{Im}\\,S)$ used to identify the temperature from the imaginary action","marker":"[62]"},{"why":"the earlier GUP result that quantum corrections suppress Hawking temperature and suggest remnants, which the paper's interpretation builds on","marker":"[82]"},{"why":"motivates the disclination parameter $\\beta=1-4\\lambda$ appearing in the angular metric and orbital frequencies","marker":"[83–85]"}],"fun_headline_variants":["GUP lowers Hawking temperature, hints at black hole remnants","Quantum gravity cools charged black holes, suggests remnants","GUP corrections suppress Hawking radiation in BTZ-like black holes","Charged black holes get cooler with GUP, possible remnant"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The remnant conclusion assumes the Hawking temperature is correctly given by $\\sqrt{f'(r_h)g'(r_h)}/(4\\pi)$ evaluated at the chosen root $r_h$ of $g(r)=0$, even though $f(r_h)$ is not zero there, so the horizon is not a Killing horizon for the time translation and the standard surface-gravity formula may not apply.","fun_headline_variants_meta":{"raw":{"variants":["GUP lowers Hawking temperature, hints at black hole remnants","Quantum gravity cools charged black holes, suggests remnants","GUP corrections suppress Hawking radiation in BTZ-like black holes","Charged black holes get cooler with GUP, possible remnant"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000201,"raw_usage":{"total_tokens":1467,"prompt_tokens":1125,"completion_tokens":342,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":741,"completion_tokens_details":{"reasoning_tokens":272}},"tokens_in":741,"tokens_out":342,"duration_ms":4125,"temperature":1.0,"reasoning_tokens":272,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:25:50.049192+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the Table 5 parameters $M=1$, $Q=1$, $\\Lambda=-0.1$, $b=0.1$, $r_0=1.0$: the event horizon defined as the largest positive root of $g(r)=0$ is a larger root, not the $0.10767$ used in the table, so computing the corrected temperature at both roots settles whether the suppression pattern survives. A second check is to evaluate $f(r_h)$ at the tabulated roots; if $f(r_h)$ is not numerically zero, $\\sqrt{f'(r_h)g'(r_h)}/(4\\pi)$ is not the standard surface gravity, and a calculation using the proper Killing-horizon relation gives a different temperature.","supporting_citations":[],"review_version":1}