{"id":"5ad8c0a8-4ba6-4c37-beb5-1b26d5fcc611","arxiv_id":"2507.22965","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A comment claiming that the horizon in a 2024 non-commutative black hole tunneling paper is complex rather than real, and recomputing the tunneling rate under a different twist.","lead":"This comment challenges a 2024 paper's calculation of black hole radiation, arguing that the event horizon used there is wrong because the corrected equation has no real solution. It then redoes the calculation with a different non-commutative model and gets a modified emission rate.","discovery_kind":"replication","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (4) has the wrong sign for the reciprocal of Eq. (2b); with the correct sign the truncated horizon equation has a real root near r≈2.0075, so the all-complex-roots argument collapses.","rationale":"The comment's central claim is that the original horizon formula is wrong because the equation 1/g_rr=0 obtained from the O(Θ²) metric has only complex roots. The entire numerical demonstration depends on Eq. (4), which the text says is obtained by expanding 1/g_rr from Eq. (2b). Direct inversion of Eq. (2b) gives the opposite sign for the O(Θ²) term. The sign is not cosmetic: with the correct sign, the truncated equation has a real positive root near r≈2.0075 for the same values M=1 and Θ=0.01 used in the comment, whereas the printed plus-sign equation has no real root and produces the listed complex pairs. Thus the comment's key evidence is internally inconsistent with the very equation it quotes. The reader's identified weakness, the validity of the small-Θ expansion near the singular point r=2M, is also real: the Θ² correction diverges at r=2M, so no perturbative root calculation near the horizon can be trusted without a matched-asymptotics or all-orders treatment. However, the sign error is more immediate and sufficient to invalidate the comment's 'only complex solutions' assertion. The corrected calculation in Sec. II does not repair this problem because it changes the noncommutative twist from the original Θ23 configuration to ∂r∧∂θ, so it addresses a different model rather than correcting the flawed horizon analysis. A referee should verify the sign by direct inversion of Eq. (2b); if confirmed, the comment's central claim is not supported and the comment should not be accepted in its present form.","tokens_in":7659,"tokens_out":17741,"duration_ms":206684,"concrete_test":"Recompute 1/g_rr directly from Eq. (2b) by inverting g_rr to second order in Θ, without copying Eq. (4). For M=1 and Θ=0.01, evaluate F(r)=(1−2/r)−0.0001 P(r)/(8r^4(2−r)) on r∈(2,2.1) and check whether F crosses zero near r≈2.0075. If it does, the ten complex roots listed in the comment are artifacts of the sign error and the central objection against Eq. (3) is not supported. If the sign in Eq. (4) is claimed to come from a noncommutative inverse-metric definition different from the ordinary reciprocal, the authors should state that definition explicitly and derive Eq. (4) from it; absent that, Eq. (4) is internally inconsistent with Eq. (2b).","verdict_should_be":"REJECT","load_bearing_attack":"Let a=1−2M/r and write Eq. (2b) as g_rr=a^{-1}+Θ²C, with C=M(12M²+Mr(−14+S)−r²(5+S))/(8r²(2M−r)^3) and S=√(1−2M/r). Expanding 1/g_rr to second order in Θ gives a−Θ²Ca², not a+Θ²Ca² as printed in Eq. (4). Since a²=(r−2M)²/r², one obtains Ca²=M P/(8r^4(2M−r)), where P=12M²−14Mr−5r²+r(M−r)S. Thus the correct reciprocal is a − Θ² M P/(8r^4(2M−r)), whereas Eq. (4) has a plus sign. This sign error is decisive: for M=1 and Θ=0.01, the plus-sign equation is positive for all r>2M, and the roots near r=2 are the complex numbers listed in the comment; the minus-sign equation crosses zero at r≈2.0075, a real positive root. Therefore the comment's claim that the mere presence of complex solutions disproves Eq. (3) rests on an internal inconsistency. Additionally, because the Θ² coefficient diverges at r=2M, the truncated equation is a singular perturbation even after the sign is fixed, so the corrected root location should not be treated as a definitive horizon radius; but the specific assertion that all roots are complex is already false.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript is a Comment on Touati and Zaim, Phys. Lett. B 848 (2024) 138335. It claims that the event horizon of the non-commutative Schwarzschild solution used in that paper was incorrectly determined: expanding 1/g_rr to second order in the non-commutativity parameter Θ is said to give only complex roots, so the real horizon radius in Eq. (3) is declared wrong and the subsequent quantum-tunneling calculation is said to be invalid. The Comment then proposes a corrected metric using a ∂r∧∂θ twist following Ref. [10], recomputes the Painlevé–Gullstrand tunneling rate, and argues that the same horizon error was repeated in Refs. [1–4].","tokens_in":7983,"tokens_out":20491,"duration_ms":224898,"significance":"A successful Comment of this sort would be valuable: it would correct a published result, flag a systematic error in a series of papers, and bring the more recent construction of Ref. [10] into the discussion. The manuscript is transparent in presenting the explicit root computation for M=1, Θ=0.01, which is a checkable claim, and it makes a falsifiable prediction for the corrected emission rate. However, the root computation contains a sign error, and the corrected tunneling calculation is not actually performed with the corrected horizon. As it stands, the paper does not establish its central claims; the material is potentially salvageable in revision.","major_comments":[{"comment":"The small-Θ expansion of 1/g_rr in Eq. (4) has the wrong sign. Writing Eq. (2b) as g_rr = a^{-1} + Θ² C with a=1−2M/r and C=M(12M²+Mr(S−14)−r²(5+S))/(8r²(2M−r)³), S=√(1−2M/r), the reciprocal is a − Θ² C a² + O(Θ⁴). Using a²=(r−2M)²/r² and (2M−r)³=−(r−2M)³ gives a + Θ² M P/(8r⁴(r−2M)), where P=12M²−14Mr−5r²+r(M−r)S. Equation (4), with denominator (2M−r), equals a − Θ² M P/(8r⁴(r−2M)). For M=1, Θ=0.01 the correct sign gives a real positive root near r≈2.0075; the ten complex roots listed in §I are an artifact of the sign error. The claim that the 'mere presence of these complex solutions' disproves Eq. (3) is therefore not valid.","section":"§I, Eqs. (2b)–(4)"},{"comment":"Even with the sign corrected, the truncated equation cannot support the conclusion drawn. The Θ² coefficient of 1/g_rr diverges as (r−2M)^{-1} at r=2M (and the coefficient in Eq. (2b) as (r−2M)^{-3}), so the expansion in Θ² is not uniform in a neighbourhood of the horizon. The corrected root at r−2M≈0.75Θ lies in the regime where the Θ² term and the leading term are of the same order, and the O(Θ⁴) terms are not controlled. Thus the root of the truncated polynomial is not a reliable prediction of the exact horizon, and the statements in §I about the absence of a real horizon, or about a 'regular black hole lacking a physical event horizon', are unsupported.","section":"§I, Eq. (4) and following"},{"comment":"The corrected tunneling calculation is not consistent with the stated pole choice. In Eq. (24) both Θ² terms contain a factor 1/(r−2(M−ω′)), and they are multiplied by 1/(1−√(2(M−ω′)/r))², which behaves like 16(M−ω′)²/(r−2(M−ω′))² near r=2(M−ω′). The Θ² integrand therefore has a triple pole at the Schwarzschild pole and zero residue there. The logarithmic correction in Eq. (25) cannot arise from contour integration around r=2(M−ω′); a nonzero Θ² contribution would require locating the Θ-dependent pole of the full integrand, which is not done. Consequently the emission rate (26) and density (27) do not follow from the calculation as presented.","section":"§III, Eqs. (22)–(25)"},{"comment":"The relationship between the horizon critique and the 'corrected solution' is not established. If Eq. (19) is the metric that should replace Eq. (2b), then the event horizon should be determined from the corrected g_rr, and the demonstration that Eq. (3) is wrong should be carried out for that metric; the paper instead analyses a truncated polynomial obtained from the old metric (2b). The comment also states that the same problem was repeated in Refs. [1–4], but no explicit check of those papers' horizon equations is provided. These gaps leave the scope of the claimed correction unclear.","section":"§II, Eq. (19)"}],"minor_comments":[{"comment":"The expansion for r10 contains terms of order Θ, Θ^{3/2}, and Θ²; the fractional power is not explained and appears to be an artifact of expanding a complex root of Eq. (4).","section":"§I, Eq. (5)"},{"comment":"The phrase 'the same issue have been repeated' needs grammatical correction, and the claim that the issue was repeated in Refs. [1–4] is not demonstrated in the body beyond the invalid §I argument.","section":"Abstract"},{"comment":"References [14] and [16] are the same paper (Chaichian, Tureanu, and Zet, Phys. Lett. B 660 (2008) 573) and should not be listed twice.","section":"References"},{"comment":"The caption does not specify the values of M and Θ used for each curve, and the axes are not visibly labelled in the printed figure.","section":"Figure 1"},{"comment":"The notation in the Painlevé–Gullstrand form conflates g(r) with g_rr(r,Θ); the definitions of f, h, and g should be stated explicitly before Eq. (22).","section":"§III, Eq. (22)"}],"recommendation":"major_revision","confidential_remarks":"No issue with the authors' self-citations: Refs [7–9] are used as methodological examples of the tunneling calculation. The main concern is technical correctness, not novelty."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Candid take: the comment's central claim does not survive contact with its own Eq. (4). You start from Eq. (2b) and want 1/g_rr up to O(Θ²). Direct inversion gives a − Θ²Ca², with a = 1 − 2M/r, plus a sign: Eq. (4) prints a plus where the algebra gives a minus. That sign is decisive. For M=1, Θ=0.01, the printed equation has no real roots; the corrected one crosses zero near r≈2.0075. So the headline assertion — that the truncated horizon equation has only complex roots and therefore Eq. (3) from Touati-Zaim is wrong — is itself supported by an internal inconsistency. The comparison in Eq. (5) inherits the same problem.\n\nThe comment does a few things well. Flagging that Touati-Zaim never justified their horizon radius is fair game; the Θ² corrections in the metric blow up at r=2M, so a truncated expansion at the horizon is a singular perturbation, and a real horizon at r = 2M[1 + (3/8)(Θ/r_h)²] does not follow automatically from the series. That point stands even after the sign fix. And the ∂r∧∂θ corrected tunneling calculation in Secs. II-III is a real piece of work — it imports the Juric-Kumara-Pozar metric, runs the standard Parikh-Wilczek machinery, and gets a concrete rate with a log correction that suppresses emission. It is new, though not deep, and the statement that surface gravity is ill-defined under that twist is worth taking seriously.\n\nSoft spots besides the sign error: the corrected calculation quietly switches to a different non-commutative twist instead of fixing the original one, so it never resolves the complex-horizon problem it raises; the tunneling result itself treats r=2(M−ω′) as a real pole, ignoring the complexification complaint from Sec. I; and the claim that non-commutativity implies a regular black hole without a physical horizon needs an argument, not just the observation that the roots are complex.\n\nVerdict: the letter is honest in intent and shows the authors know the tunneling machinery, but a signing error sits on the main load-bearing claim. That is exactly what a referee should catch. Send it to review, but the all-complex-roots argument and Eq. (5) need to be redone; the corrected-tunneling section can probably survive on its own.","headline":"The comment's horizon critique is undone by a sign error in its own Eq. (4); with the correct sign the truncated 1/g_rr has a real root near r≈2.0075, so the all-complex-roots argument collapses, though the singular-perturbation worry about the truncated horizon remains worth a referee's attention.","tokens_in":8450,"tokens_out":1311,"would_cite":true,"duration_ms":13172,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.70.Dy","04.60.-m"],"model":"deepseek-v4-flash","headline":"A recent quantum-tunneling calculation uses a wrong black-hole horizon: its horizon equation has only complex roots, so the emission-rate results are invalidated.","keywords":["non-commutative gauge theory of gravity","quantum tunneling","Hawking radiation","event horizon","Schwarzschild black hole","Seiberg-Witten map","Painlevé-Gullstrand coordinates","particle creation"],"falsifier":"Compute the zeros of the untruncated or resummed inverse radial metric for $M=1$, $\\Theta=0.01$. If a real root near $r=2M$ appears when the expansion is not truncated before solving, the claim that the original horizon is not a solution would fail; if no real root appears in the full expression, the comment's central objection is confirmed.","tokens_in":7465,"feed_emoji":"🕳️","tokens_out":12621,"duration_ms":134639,"temperature":0.7,"pith_summary":"This comment takes issue with a recent calculation of particle creation via quantum tunneling from a Schwarzschild black hole in non-commutative gauge theory of gravity. The authors' central point is that the event horizon used in that work is not a solution of the metric the work itself writes down: solving $1/g_{rr}=0$ for the order-$\\Theta^2$ metric gives, for $M=1$ and $\\Theta=0.01$, ten roots and no real positive one. Since the tunneling action is evaluated by integrating around the horizon pole, an incorrect or missing real horizon changes the whole emission spectrum. The comment then supplies a corrected metric under a different twist and derives a corrected tunneling rate, while noting that the usual surface-gravity definition of temperature is not well defined in this setting. The issue matters because the horizon location is the load-bearing input for black-hole radiation predictions, and a wrong horizon invalidates the derived spectrum.","feed_headline":"Tunneling paper's black-hole horizon has no real root","feed_subtitle":"A new comment finds only complex roots for the horizon equation and says the emission-rate result must be redone.","key_machinery":"The load-bearing object is the radial metric component $g_{rr}$ of the non-commutative Schwarzschild solution and the horizon condition $1/g_{rr}=0$. The argument's mechanism is simple: if the only roots of that condition are complex, the supposed real horizon at $r_h^{\\mathrm{NC}}$ does not exist in the metric being used, and the contour integral for the tunneling amplitude, which is taken around the pole at the modified horizon, has no physical pole to encircle. The replacement calculation uses the $\\partial_r\\wedge\\partial_\\theta$ twist and the Seiberg-Witten map to build the corrected metric, then repeats the Painlev\\'e-Gullstrand tunneling computation for the corrected radial function.","core_discovery":"The paper's central claim is that the event horizon in the commented work was incorrectly determined: for the order-$\\Theta^2$ metric used there, the equation $1/g_{rr}=0$ has no real positive root when $M=1$ and $\\Theta=0.01$; all ten solutions are complex. The real horizon radius $r_h^{\\mathrm{NC}} = r_h\\left(1+\\frac{3}{8}(\\Theta/r_h)^2\\right)$ that the original paper quotes therefore does not follow from its own truncated metric. An expansion of the outermost root has leading imaginary terms $2M + \\frac{3i\\Theta}{4} + O(\\Theta^{3/2})$, so the original truncation omitted the leading imaginary contribution and used the wrong factor for the real correction. Because the tunneling calculation integrates around the pole at the horizon, all subsequent results must be revised. The comment goes on to recompute the metric and the tunneling probability using the $\\partial_r\\wedge\\partial_\\theta$ twist, obtaining $\\mathrm{Im}S = 2\\pi\\omega(2M-\\omega) + \\frac{25}{32}\\pi\\Theta^2[\\ln M - \\ln(M-\\omega)]$, and observes that the same horizon error appears in the companion papers [1--4].","pith_inferences":["I infer that the all-complex answer should be checked across a range of $M$ and $\\Theta$; the single example ($M=1$, $\\Theta=0.01$) is suggestive but does not by itself show the original horizon formula fails for all parameter values.","I infer that the discrepancy between the original factor $3/8$ and the corrected factor $5/32$ is a sign of twist-dependence: non-commutative corrections are not unique until the deformation prescription is fixed, so comparing predictions across papers requires comparing the chosen twist rather than just the value of $\\Theta$.","I infer that if the deformed spacetime is horizonless, the quantum-tunneling method loses its defining boundary condition, and particle creation should instead be modeled as radiation from a regular compact object; testing this requires the full non-perturbative metric."],"forward_implications":["The original tunneling rate, particle number density, and any derived temperature in the commented paper should be recomputed, because every one of those quantities is evaluated at the horizon that the comment finds to be missing or incorrect.","If the order-$\\Theta^2$ configuration genuinely has no real horizon, the object may be a regular, horizonless spacetime, and the tunneling picture of Hawking radiation needs to be replaced or reinterpreted rather than merely corrected.","The corrected first-order result is $\\Gamma(\\omega,\\Theta)\\sim \\exp\\left[-4\\pi\\omega(2M-\\omega)-\\frac{25}{16}\\pi\\Theta^2(\\ln M - \\ln(M-\\omega))\\right]$, which shows the non-commutative correction suppressing particle creation as $\\Theta$ grows.","Under the $\\partial_r\\wedge\\partial_\\theta$ twist, the surface gravity is not well defined, so standard geometric black-hole thermodynamics does not apply to this non-commutative background.","The same horizon error is reported in the cited companion papers [1--4], which therefore also need scrutiny."],"supporting_citations":[{"why":"Supplies the metric components (2), the disputed horizon formula (3), and the tunneling calculation being corrected.","marker":"[5]"},{"why":"Supplies the refined non-commutative framework, the corrected metric (19), and the statement that surface gravity is not well defined under the chosen twist.","marker":"[10]"},{"why":"Provides the quantum-gravitational correction methodology for particle creation that the comment uses to redo the tunneling action.","marker":"[6]"},{"why":"Foundational tunneling calculation: defines the imaginary-action contour and the emission-rate formula that the original and corrected calculations both follow.","marker":"[18]"},{"why":"Foundational construction of Schwarzschild corrections in non-commutative gauge theory of gravity; the deformed tetrad expansions in Eqs. (7)--(17) build on it.","marker":"[16]"},{"why":"Introduces the Seiberg-Witten map used to construct the deformed gauge connection and tetrads in the corrected metric.","marker":"[11]"},{"why":"Supplies the deformation of Einstein's gravity in the non-commutative gauge approach, underpinning the framework the comment adopts.","marker":"[12]"}],"fun_headline_variants":["No real horizon root ruins noncommutative black hole tunneling rate","Comment says black hole tunneling paper misidentified horizon","Imaginary horizon roots invalidate noncommutative black hole tunneling","No real horizon for noncommutative black hole tunneling paper","Black hole tunneling result flawed: horizon equation has no real root"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that truncating $1/g_{rr}$ at order $\\Theta^2$ and then solving for zeros is legitimate at the would-be horizon $r\\approx 2M$; if the expansion is not uniform there, the absence of real roots in the truncated polynomial may be an artifact rather than a disproof of the original real horizon.","fun_headline_variants_meta":{"raw":{"variants":["No real horizon root ruins noncommutative black hole tunneling rate","Comment says black hole tunneling paper misidentified horizon","Imaginary horizon roots invalidate noncommutative black hole tunneling","No real horizon for noncommutative black hole tunneling paper","Black hole tunneling result flawed: horizon equation has no real root"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001331,"raw_usage":{"total_tokens":5391,"prompt_tokens":902,"completion_tokens":4489,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":518,"completion_tokens_details":{"reasoning_tokens":4403}},"tokens_in":518,"tokens_out":4489,"duration_ms":34644,"temperature":1.0,"reasoning_tokens":4403,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:29:28.922783+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the zeros of the untruncated or resummed inverse radial metric for $M=1$, $\\Theta=0.01$. If a real root near $r=2M$ appears when the expansion is not truncated before solving, the claim that the original horizon is not a solution would fail; if no real root appears in the full expression, the comment's central objection is confirmed.","supporting_citations":[{"cited_title":"Quantum tunneling from Schwarzschild black hole in non- commutative gauge theory of gravity,","cited_arxiv_id":null,"evidence_quote":"Supplies the metric components (2), the disputed horizon formula (3), and the tunneling calculation being corrected."},{"cited_title":"Constructing noncommutative black holes,","cited_arxiv_id":null,"evidence_quote":"Supplies the refined non-commutative framework, the corrected metric (19), and the statement that surface gravity is not well defined under the chosen twist."},{"cited_title":"Quantum gravitational corrections to particle creation by black holes,","cited_arxiv_id":null,"evidence_quote":"Provides the quantum-gravitational correction methodology for particle creation that the comment uses to redo the tunneling action."},{"cited_title":"Hawking radiation as tunneling,","cited_arxiv_id":null,"evidence_quote":"Foundational tunneling calculation: defines the imaginary-action contour and the emission-rate formula that the original and corrected calculations both follow."},{"cited_title":"Corrections to Schwarzschild solution in noncommu- tative gauge theory of gravity,","cited_arxiv_id":null,"evidence_quote":"Foundational construction of Schwarzschild corrections in non-commutative gauge theory of gravity; the deformed tetrad expansions in Eqs. (7)--(17) build on it."},{"cited_title":"String theory and noncommutative geometry,","cited_arxiv_id":null,"evidence_quote":"Introduces the Seiberg-Witten map used to construct the deformed gauge connection and tetrads in the corrected metric."},{"cited_title":"Deforming Einstein’s gravity,","cited_arxiv_id":null,"evidence_quote":"Supplies the deformation of Einstein's gravity in the non-commutative gauge approach, underpinning the framework the comment adopts."}],"review_version":1}