{"id":"f3c3a94c-86cf-495e-84ab-06ff3c7437a4","arxiv_id":"2507.19813","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"The paper numerically computes timelike entanglement entropy for quantum BTZ black holes and claims that the effective central charge drops sharply as quantum backreaction crosses a critical value.","lead":"Black holes in a three-dimensional quantum gravity model are studied by measuring how quantum information spreads over a time-like interval. The calculation suggests that when quantum effects grow strong enough, the probe surface breaks apart and the effective number of degrees of freedom falls sharply.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Lorentzian tEE integrals (32)-(33) use sqrt(f+E^2) although eq (12) with lambda=-1 requires sqrt(-f-E^2); the classical limit already shows the discarded imaginary part is O(pi), so the phase transition and c_slab curve are not established.","rationale":"I read the paper as applying the Nunez-Roychowdhury lambda-continuation to define timelike entanglement entropy for qBTZ. The most load-bearing step is the Lorentzian sign in the constraint: eq (12) with lambda=-1 fixes dot r^2 = -f - E^2, while eqs (32)-(33) effectively use sqrt(f+E^2). The reader's weakest assumption identifies exactly this, and I agree that it is the decisive issue. The paper does provide a clear numerical setup and the classical BTZ check is instructive, but that check uses the imaginary-turning-point regime and therefore does not validate the qBTZ numerics, where rc is held real and the classical limit of the integrals is the Euclidean expression. Because the same flawed integrals feed the phase-transition claim and the c_slab curve, the central claim as stated is not supported. The correct response is to keep the reader's REJECT verdict (no adjustment). A revision that fixes the continuation rule, evaluates the full complex area, and validates c_slab against an independent boundary calculation could be reconsidered.","tokens_in":12992,"tokens_out":15490,"duration_ms":184602,"concrete_test":"Set gamma=0 in eqs (32)-(33) with the Sec. IV input r+=0.2, rc=0.1, E=sqrt(r+^2-rc^2). Eq (33) yields the real Euclidean value 2 arccosh(rb/rc), whereas the exact Lorentzian continuation (28)-(30) gives S_tEE=(c/3) log(2rb/r+ sinh(r+ T_L/2)) + i pi c/6 with T_L from (22); already the real parts differ by log(E/r+) and the imaginary part is pi c/6, not 10^-9. Then recompute Figs 1-3 using dot r = sqrt(-f-E^2) with a fixed branch instead of sqrt(f+E^2). If the non-monotonicity of |T_L| or the sharp fall-off of c_slab disappears, the central claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim — a backreaction-driven phase transition of the extremal surface and a sharply falling slab central charge — is computed entirely from the Lorentzian integrals (32) and (33). Those integrals assume the radicand sqrt(f + E^2), but the paper's own Lorentzian constraint (12) with lambda=-1 gives dot r^2 + E^2 = -f. The correct object is the complex extremal area obtained from dot r = sqrt(-f-E^2), not its modulus. This is not a cosmetic sign issue: in the gamma=0 limit the paper's exact classical tEE, eq (30), has Im S_tEE = i pi c/6, so the statement in Sec. IV that the imaginary component is largely suppressed is already false at the one point where the calculation can be checked analytically. Moreover, eq (33) with gamma=0 and real rc (as used in Sec. IV: r+=0.2, rc=0.1, E=sqrt(r+^2-rc^2)) gives a real area 2 arccosh(rb/rc), i.e. the Euclidean continuation, not the Lorentzian result (28) with its log(E/r+) real-part shift and i pi imaginary part. Since the non-monotonicity of |T_L| in Fig. 1(b), the phase transition, and the c_slab curve in Fig. 3 are all outputs of these integrals, a wrong continuation rule would invalidate the central claim. A secondary but related red flag is that the stability criterion (36) is purely imaginary for lambda=-1, so its sign cannot define a real stable/unstable split without a fixed branch convention.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper computes holographic timelike entanglement entropy (tEE) for quantum BTZ black holes on a Karch-Randall brane, using a parameter λ to interpolate between Euclidean (λ=+1) and Lorentzian (λ=-1) area functionals. The classical BTZ case is revisited in Section III, and the qBTZ case is treated in Section IV by numerically integrating the extremal-surface area and the timelike interval. The paper claims that increasing the backreaction parameter γ triggers an instability of the extremal surface, producing a phase transition, and that the tEE defines a slab central charge c_slab that falls sharply with γ. The central quantitative results are the non-perturbative curves in Figs. 1-3 and Table I.","tokens_in":13343,"tokens_out":15637,"duration_ms":191518,"significance":"If the computation were correct, it would be a useful step in extending timelike entanglement entropy to a non-perturbative backreaction regime and in connecting tEE to a central-charge-like quantity in a braneworld defect CFT. The paper has some strengths: it works with the exact qBTZ solution, it reproduces the classical BTZ tEE in the Euclidean sector, and it gives explicit analytic expressions in Section III. However, the Lorentzian continuation is not handled in a controlled way: the main integrals are not well defined as written, the classical imaginary part is discarded without justification, and the central-charge claim is a repackaging of the same tEE data. These issues affect the phase-transition and central-charge conclusions directly, so the paper's central claims are not currently supported.","major_comments":[{"comment":"The Lorentzian integrals are not well-defined as written. For λ=-1, Eq. (12) gives \\dot r^2 + E^2 = -f(r), so \\dot r is purely imaginary; substituting this into (13) and (17) requires a branch choice for the square root and for the overall factors of \\sqrt{\\lambda}, and the paper does not state that choice. This is not only a sign issue: for the parameters used in Section IV (r_b=10, r_+=0.2, r_c=0.1, γ≥0), f(r) has a simple zero at r=r_+ inside the integration domain [r_c,r_b], so the integrands in (32) and (33) behave as 1/(r-r_+) near the horizon and both integrals diverge. The finite entries in Table I and the curves in Figs. 1-3 can only be obtained after an additional contour prescription or principal-value regularization, which is neither given nor justified.","section":"Section IV, Eqs. (32)-(33), Table I, Figs. 1-3"},{"comment":"The statement that the imaginary component of tEE is 'largely suppressed' is inconsistent with the classical limit that the paper itself uses as a check. Eq. (30) gives Im S_tEE = iπ c/6 for γ=0, which is parametrically the same size as the real part (of order one in units S0=1/(4G3) for c~O(1)). Table I, by contrast, reports Im S_tEE ~ 10^{-9} S0 in the small-γ regime. Thus the numerical evaluation does not reduce to the exact classical tEE (30); the imaginary part has been discarded rather than computed from the stated integrals.","section":"Section IV, text after Eq. (33), and Section III, Eq. (30)"},{"comment":"The central-charge result is a re-parameterization of the tEE data rather than an independent determination. The quantity c_slab is defined in (38) as T_L ∂_{T_L} S_tEE, and the numerical implementation (40) is a finite difference between the γ≠0 and γ=0 values of the same integrals (32)-(33). The fall-off of c_slab shown in Fig. 3 therefore follows directly from the shape of the tEE curve, so it cannot by itself establish a decrease of the true central charge of the dual CFT. Moreover, the classical check (39) gives c_slab = κ c with an interval-dependent prefactor κ=r_+ E T_L/(12 r_b) ≪ 1; this means c_slab is not equal to the Brown-Henneaux central charge c, and the paper does not show that κ is a constant that can be absorbed or normalized away.","section":"Section IV.B, Eqs. (38)-(40), Fig. 3"},{"comment":"The stability criterion is not well defined for λ=-1. Eq. (36) contains a factor 1/√λ, which is ±i in the Lorentzian signature, so Z(r_c) is purely imaginary; the footnote [65] acknowledges this by saying that one must consider the imaginary component. A sign of a purely imaginary quantity cannot define a stable/unstable split without fixing the branch of √λ and explaining why that sign controls stability under perturbations. Consequently, the predicted transition at γ_c = r_c^3/r_+ is not established.","section":"Section IV.A, Eqs. (35)-(37) and footnote [65]"}],"minor_comments":[{"comment":"The prefactor in A_ext^{(λ)} appears inconsistent with the on-shell Lagrangian: Eq. (9) with the constraint (11) gives an overall factor 2√λ ∫ dr/\\dot r, whereas Eq. (17) writes 2/√λ ∫ dr/\\dot r. This affects the signs and phases in Eqs. (23)-(24) and (33).","section":"Eq. (17)"},{"comment":"The status of T_L is unclear: (15) gives T_L=-i T_E, which is purely imaginary if T_E is real, while (22) gives a complex T_L for E<r_+ (because tanh^{-1}(r_+/E) has an imaginary part when its argument exceeds unity), and (27) treats T_L as real. The paper should state explicitly whether T_L is real, complex, or a modulus.","section":"Eqs. (15), (22), (27)"},{"comment":"The notation |T_L| in (32) and 'we consider its magnitude only' should be made precise; currently T_L appears both as a complex quantity in (30) and as a real magnitude in the numerical plots, without a definition of the branch or the sign convention.","section":"Eqs. (32)-(33), Fig. 1(b)"},{"comment":"The table lists r_b, r_+, and γ but omits r_c and the corresponding E determined by Eq. (31); this information is needed to reproduce Im S_tEE.","section":"Table I"},{"comment":"The axes labeled '1 γ' and 'c_slab' are unclear; the figure should specify the normalization of c_slab (for example, c_slab/c or c_slab/κc) and the fixed parameters r_+, r_b, r_c.","section":"Fig. 3"},{"comment":"There are several typos and stylistic issues, including 'bacreaction' (Introduction), 'Our staring points' (Section IV), and 'not form the perspective of the bulk AdS4' (Introduction). The PACS field is empty.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The central computation needs to be redone with a careful complex-contour prescription for the Lorentzian integrals. As written, the horizon pole makes Eqs. (32)-(33) divergent for the chosen parameters, and the discarded imaginary part contradicts the paper's own classical limit. I would not ask for a major revision because the phase-transition and central-charge curves are direct outputs of these uncontrolled integrals; the authors would need to reformulate the analytic continuation from scratch, and the qualitative conclusions could change."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This one is straightforward: the central computation does not follow from the paper's own equations. The claimed backreaction-driven phase transition and the sharply falling slab central charge rest on a sign error in the Lorentzian continuation.\n\nWhat the paper does well: the qBTZ setup is clean, the turning-point condition (31) is correctly derived, and the numerical exploration is clearly described. The observation that a critical backreaction gamma_c = r_c^3/r_+ appears is genuinely new, and the figures are readable. I also appreciate that the author checks the gamma=0 limit explicitly, even if that check fails.\n\nThe soft spots are load-bearing. Equation (12) with lambda=-1 gives \\dot r^2 + E^2 = -f, so the radicand in the integrals should be -f - E^2, not f + E^2. The integrals (32) and (33) use sqrt(f + E^2). This is not cosmetic. In the gamma=0 limit, using their integrals gives a Euclidean-looking area 2 arccosh(rb/rc) and a divergent |T_L| from the horizon pole, not the classical result (22) with its log(E/r+) shift and i pi c/6 imaginary part. The paper's statement that the imaginary part is 'largely suppressed' is contradicted by its own classical limit (30), which has Im S = i pi c/6. The tiny imaginary parts in Table I do not match that. Also, the classical check of the central charge definition (39) does not reproduce direct differentiation of (30): from (30), c_slab = (c/6) r_+ T_L coth(r_+ T_L/2), not their kappa c. So the definition is not validated even classically.\n\nThere is a secondary red flag: the stability criterion (36) is purely imaginary for lambda=-1, so its sign cannot define a real stable/unstable split without a fixed branch convention. The footnote [65] acknowledges this but does not resolve it.\n\nThe author is clearly capable, and the paper is readable. A corrected version that fixes the continuation rule, computes the full complex tEE, and validates c_slab against an independent boundary calculation could be worth another look. But as submitted, the central claims are not established, and I would not send this version to a referee.","headline":"The claimed phase transition and falling central charge rest on a sign error in the Lorentzian continuation, so the paper's central results are not established, though the setup and numerics are clearly presented.","tokens_in":13938,"tokens_out":4978,"would_cite":false,"duration_ms":58308,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Timelike entanglement entropy of quantum BTZ black holes falls sharply with increasing backreaction, tracing a phase transition in the extremal surface.","keywords":["timelike entanglement entropy","quantum BTZ black hole","Karch-Randall braneworld","central charge","backreaction","phase transition","extremal surface","defect CFT"],"falsifier":"Recompute the quantum BTZ integrals for the extremal area keeping the full complex result instead of dropping Im S_tEE, and check whether the real part still shows the same critical backreaction γc = $rc^{3}$/r+; if the imaginary part is not numerically small at the values in Table I, or if it changes sign near the transition, the reported tEE and central-charge curve would need revision.","tokens_in":12721,"feed_emoji":"🕳️","tokens_out":6505,"duration_ms":67335,"temperature":0.7,"pith_summary":"This paper tries to establish that timelike entanglement entropy, computed holographically from an extremal surface inside a Karch-Randall braneworld, can probe quantum backreaction effects on quantum BTZ black holes. It claims that beyond a critical backreaction the extremal surface becomes unstable and disintegrates, producing a phase transition, and that the same computation yields an effective central charge for the dual 2d defect CFT which drops sharply as backreaction grows. If true, timelike entanglement entropy is a non-perturbative probe of quantum backreaction and a way to define the central charge of the defect CFT. The analysis uses a Lorentzian continuation of the area functional and reproduces the classical BTZ results in the zero-backreaction limit.","feed_headline":"Backreaction breaks extremal surfaces and drops central charge","feed_subtitle":"Holographic timelike entanglement entropy tracks a quantum BTZ phase transition and yields an effective central charge for the 2d defect…","key_machinery":"The load-bearing object is the extremal-area functional A = 2∫ dζ sqrt(λf(r)t˙² + r˙²/f(r)) for a constant-ϕ geodesic, regulated by the parameter λ = ±1, where λ = −1 selects the Lorentzian timelike entanglement entropy. The argument then rests on three pieces: the turning-point constraint r˙² + E² = λf(r), the stability function Z(rc) = (d/dr)(πG(r)/F'(r))|_{r=rc} whose sign marks whether the extremal surface is stable or unstable, and the slab central charge cslab = TL ∂_TL S_tEE evaluated as a difference ratio relative to the classical BTZ solution.","core_discovery":"The paper claims that for quantum BTZ black holes in a Karch-Randall braneworld, the holographic timelike entanglement entropy decreases as the quantum backreaction parameter γ grows, and that this decrease is caused by an instability of the extremal surface: at a critical backreaction γc = $rc^{3}$/r+ the stability function Z(rc) changes sign, the connected extremal surface disintegrates into a pair of disconnected surfaces, and a phase transition occurs. Using the slab definition cslab = TL ∂_TL S_tEE, the paper further claims that the central charge of the dual 2d defect CFT falls rapidly from its classical value (satisfying c/6 = 1/(4G3)) and saturates for large backreaction. The conclusion is that timelike entanglement entropy provides a non-perturbative measure of how quantum backreaction reduces the effective degrees of freedom of the dual CFT.","pith_inferences":["If the imaginary part of the timelike entanglement entropy is not genuinely negligible at larger backreaction, the real-part-only central charge curve could miss an additional branch or develop a discontinuity, so a complexified version of cslab might be a sharper diagnostic.","The same λ-parametrised area functional could be applied directly to charged and rotating quantum BTZ solutions, where the critical backreaction and the central-charge fall-off would likely shift in a calculable way.","The sign-change criterion Z(rc) suggests a general pattern: any brane backreaction that makes the effective potential non-monotonic will trigger an extremal-surface phase transition, so timelike entanglement entropy may serve as a universal backreaction detector.","The slab central charge could be interpreted as a running c-function along the backreaction direction, and checking its monotonicity would connect this computation to a c-theorem for defect CFTs."],"forward_implications":["Timelike entanglement entropy can act as a probe of quantum-backreaction-induced phase transitions in braneworld holography, because the extremal surface disintegrates beyond a critical backreaction.","The central charge of the 2d defect CFT can be read off from timelike entanglement entropy, approaching the classical relation c/6 = 1/(4G3) for small backreaction and saturating for γ ≥ 1.","In the zero-backreaction limit, the algorithm reproduces the known classical BTZ timelike entanglement entropy, including the imaginary piece iπc/6.","The stability function Z(rc) gives a geometric criterion for backreaction-induced destabilization: a sign change from negative to positive signals the phase transition.","The transition resembles known entanglement phase transitions in confining gauge theories, but here it is driven by quantum backreaction on the brane rather than by the geometry of the confining background."],"supporting_citations":[{"why":"Supplies the quantum BTZ solution and the backreaction parameter that the entire computation starts from.","marker":"[4]"},{"why":"Introduces timelike entanglement entropy through analytic continuation of the Ryu-Takayanagi prescription.","marker":"[34]"},{"why":"Provides the geodesic parameterisation and area functional for BTZ timelike entanglement entropy that this paper extends to quantum BTZ.","marker":"[37]"},{"why":"Introduces the λ = ±1 parametrisation used to select the Lorentzian timelike entanglement entropy without explicit analytic continuation.","marker":"[47]"},{"why":"Supplies the stability function Z(rc) whose sign change signals the phase transition.","marker":"[52]"},{"why":"Defines the slab central charge from the derivative of the entanglement entropy.","marker":"[57]"},{"why":"Motivates the central-charge definition through refinement of entanglement entropy as a count of degrees of freedom.","marker":"[58]"}],"fun_headline_variants":["Quantum backreaction cuts central charge in BTZ black holes","Timelike entanglement traces phase transition in quantum BTZ","Backreaction drives defect CFT central charge down","Holographic tEE sees central charge drop with backreaction","Quantum BTZ: backreaction lowers central charge via tEE"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the analytic-continuation rule that Lorentzian timelike entanglement entropy follows from setting λ = −1 in the area functional and keeping only the real part of the resulting extremal area, even though the classical limit carries an imaginary term iπc/6; if that rule or the neglect of the imaginary part is wrong, the entropy, the phase transition, and the central-charge curve all change.","fun_headline_variants_meta":{"raw":{"variants":["Quantum backreaction cuts central charge in BTZ black holes","Timelike entanglement traces phase transition in quantum BTZ","Backreaction drives defect CFT central charge down","Holographic tEE sees central charge drop with backreaction","Quantum BTZ: backreaction lowers central charge via tEE"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000168,"raw_usage":{"total_tokens":1210,"prompt_tokens":845,"completion_tokens":365,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":461,"completion_tokens_details":{"reasoning_tokens":281}},"tokens_in":461,"tokens_out":365,"duration_ms":4519,"temperature":1.0,"reasoning_tokens":281,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:00:40.493215+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the quantum BTZ integrals for the extremal area keeping the full complex result instead of dropping Im S_tEE, and check whether the real part still shows the same critical backreaction γc = $rc^{3}$/r+; if the imaginary part is not numerically small at the values in Table I, or if it changes sign near the transition, the reported tEE and central-charge curve would need revision.","supporting_citations":[],"review_version":1}