{"id":"e9d7f9ce-35fc-4df0-8afc-48adba48f9f5","arxiv_id":"2601.09397","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In a perturbed Euclidean BTZ black hole, the first-order change in the thermal one-point function is still controlled by the change in the boundary-to-horizon geodesic length, up to m-dependent prefactors.","lead":"This paper tests whether the holographic formula connecting black hole one-point functions to bulk geodesic lengths remains valid when the black hole metric is slightly perturbed. The authors find the relation is robust for radial, horizon-preserving deformations, but several prefactor inconsistencies in the derivation should be fixed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Saddle-point evaluation in Appendix C contradicts the claimed m-scaling: Eq (5.7) implies m^{-1/2} in (5.6), not m^{3/2} as in (5.8); Eq (5.13) also quotes a different power and an unexplained e^{-m r_+}.","rationale":"I read the paper in good faith. The intended result is a first-order-in-\\epsilon statement: under radial, horizon-preserving perturbations, the logarithmic variation of the large-dimension one-point function equals the variation of the renormalized boundary-to-horizon geodesic length. The restriction to \\delta\\chi=0 is explicit and supported by the matter-source construction in Appendix A, so that scope limitation is not itself a defect. The main load-bearing problem is that the final saddle-point computation is internally inconsistent: Eq. (5.7) and the substitution into (5.6) force an m^{-1/2} prefactor, while Eq. (5.8) gives m^{3/2} and Eq. (5.13) gives m^{-3/2} with an unexplained e^{-m r_+}. These are not small numerical prefactors; they change the large-m behavior of the claimed equality. The WKB and exact-matching sections provide plausible evidence for the structure \\delta K_0=-m K_0\\delta\\ell, but the step from that to \\delta\\langle O\\rangle is where the inconsistency appears. My proposed test is a direct recalculation of the Laplace integral in Appendix C, which will settle whether the correct power is m^{-1/2}; if so, the paper can be corrected by adjusting Eqs. (5.8) and (5.13). Until then the central quantitative claim is not established, so a conditional verdict is appropriate.","tokens_in":12110,"tokens_out":13644,"duration_ms":116330,"concrete_test":"Recompute the Laplace integral in Appendix C keeping all factors: start from (5.3), substitute r'=r_+ \\cosh u, use the paper's convention \\ell(r')=\\log r_+ + u so e^{-m\\ell}=r_+^{-m}e^{-m u}, and evaluate \\int_0^\\infty u^{1/2}e^{-m u}\\,du=(\\sqrt{\\pi}/2)m^{-3/2}. Then substitute the resulting I(m) into (5.6). If the final expression is \\sqrt{\\pi}\\,r_+^{-m}e^{-2\\pi r_+}\\delta\\ell(r_+)\\,m^{-1/2}, then Eq. (5.8) must be corrected. Also independently derive \\delta\\langle O\\rangle from the WKB result \\delta K_0=-m K_0\\delta\\ell and check whether the m-power is m^{+1/2}, m^{-1/2}, or m^{-3/2}, and whether e^{-m r_+} in (5.13) is a typo for e^{-m\\log r_+}.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central quantitative verification of (1.4) is internally inconsistent in its m-scaling. Appendix C evaluates I(m) defined in (5.6) by changing variables to r'=r_+ cosh u and using the paper's own convention e^{-m\\ell(r')}=r_+^{-m}e^{-m u}. The near-horizon expansion then gives I(m)=(\\sqrt{\\pi}/2)\\,r_+^{1-m}\\,\\delta\\ell(r_+)\\,m^{-3/2} (Eq. C.2). Substituting this into (5.6), i.e. multiplying by 2m/r_+, yields \\delta\\langle O\\rangle=\\sqrt{\\pi}\\,r_+^{-m}\\,e^{-2\\pi r_+}\\,\\delta\\ell(r_+)\\,m^{-1/2}, not m^{3/2}. Equation (5.8) quotes m^{3/2}; the specific perturbation example in (5.13) quotes m^{-3/2} and replaces e^{-m\\ell_{\\rm hor}} by e^{-m r_+} without any derivation. These three expressions cannot all be correct. Since the claim (1.4) is precisely that \\delta\\langle O\\rangle is governed by \\delta(e^{-m\\ell_{\\rm hor}}), the power of m controls whether that relation holds at leading order. The derivation as written therefore does not establish the central claim; it may be a typographical error, but until the saddle-point computation is corrected the result is quantitatively unverified. The restriction to radial perturbations with \\delta\\chi=0 is explicit and physically motivated in Appendix A, so it is not the main concern; the unresolved algebraic inconsistency is.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper tests whether the holographic relation between thermal one-point functions and the boundary-to-horizon geodesic length, previously established for Euclidean BTZ, survives infinitesimal radial perturbations of the metric. For a perturbation of the form f(r) -> f(r)+epsilon delta f(r), the authors compute the first-order variation of the bulk-boundary propagator using WKB methods, compare it with exact hypergeometric asymptotics, and evaluate the one-point function by saddle-point integration. They claim the central relation delta <O> proportional to delta(e^{-m l_hor}) holds at large mass. The physical setup is clear and the goal is well motivated, but the quantitative verification contains several unresolved algebraic inconsistencies.","tokens_in":12581,"tokens_out":13217,"duration_ms":111028,"significance":"If established, the result would be a useful robustness statement: heavy scalar operators would remain faithful probes of bulk geometry even when the BTZ background is perturbed by a restricted class of matter sources. The paper includes a physically motivated derivation of the allowed perturbation from Einstein equations (Appendix A), a WKB treatment of the Sturm-Liouville problem, and an attempt to justify the WKB approximation from exact large-h hypergeometric expansions. These are useful technical elements. However, the central quantitative verification currently fails: three expressions for the same saddle-point result disagree in the power of m and in the exponential factor, and the exact asymptotic formulas used in the matching are internally inconsistent. The result may be true, but the paper as written does not establish it.","major_comments":[{"comment":"The saddle-point evaluation in Appendix C gives I(m)=sqrt(pi)/2 r_+^{1-m} delta_l(r_+) m^{-3/2} (Eq. C.2). Substituting this into Eq. (5.6) yields delta<O> = sqrt(pi) m^{-1/2} r_+^{-m} e^{-2 pi r_+} delta_l(r_+), not the m^{3/2} quoted in Eq. (5.8). The worked example in Eq. (5.13) quotes m^{-3/2} and replaces e^{-m l_hor} by e^{-m r_+}, again without derivation. These three expressions cannot all be correct. Since the central claim (1.4) is precisely that delta<O> is governed by delta(e^{-m l_hor}), the m-scaling determines whether the geodesic-exponent relation holds at leading order. As written, the result is quantitatively unverified.","section":"Section 5, Eq. (5.8) and Appendix C"},{"comment":"The first-order geodesic variation is defined in (3.4) as an integral of H(r)/sqrt(r_+^2 - r^2) from r_+ to infinity, but Eq. (4.19) and Eq. (5.5) define delta_l(r) = (1/2) integral from infinity to r of H(r')/sqrt(f(r')) dr'. Apart from the missing factor 1/2, the integrand in (3.4) is imaginary for r > r_+ (likely a typo for sqrt(r^2-r_+^2)), and the orientation of the integration limits differs. Since (1.4) equates delta<O> with delta_l_hor, these discrepancies are load-bearing. The sign structure of the preceding perturbation theory also needs checking: from delta K = - G delta_square K, Eq. (3.17) should contain an overall minus sign relative to what is written.","section":"Eq. (3.4) vs Eqs. (4.19)/(5.5)"},{"comment":"The large-h asymptotic formulas used for the exact matching are inconsistent with each other. Eq. (4.30) has a prefactor 1/(2 sqrt(pi) h x^{1/4}), while the derivation in Appendix B, Eq. (B.5), gives 1/(4 sqrt(2 pi h) x^{1/4}); these differ by a factor of order sqrt(h). The paper states that substituting these asymptotic forms into the exact Green function and bulk-boundary propagator recovers the WKB expressions (4.31)-(4.33), but with different h-dependent prefactors this does not follow automatically. The cancellation must be shown explicitly. This matters because Section 4.2 is the justification that the WKB approximation is valid; as written, that justification is not established.","section":"Section 4.2, Eqs. (4.29)-(4.30) and Appendix B"}],"minor_comments":[{"comment":"The line 'H(r)=r^p, p<0 1' appears to contain a stray '1' from the footnote marker; please clean up the typesetting and separate the footnote.","section":"Section 5, near Eq. (5.13)"},{"comment":"The renormalized length l_hor is used before it is properly defined. It would improve readability to state the cutoff prescription, e.g., l(r)=arccosh(r/r_+) minus log(2r), before taking the large-m limit.","section":"Section 2, Eq. (2.16)"},{"comment":"The sentence claiming the result 'should straightforwardly extend to the Lorentzian continuation' is not demonstrated. Either provide a short argument or remove the unsupported claim.","section":"Section 6"},{"comment":"The derivation of Eq. (B.5) should be reconciled with the expression used in Eq. (4.30); the two differ by a non-negligible h-dependent factor (see major comment).","section":"Appendix B"},{"comment":"Reference [13] is given only as an arXiv identifier with no publication status; consider updating if a journal version exists.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is not ready for publication in its present form. The central claim is plausible, but the quantitative verification has severe internal inconsistencies: the m-scaling of the main result disagrees with both the saddle-point appendix and the worked example, and the exact asymptotic matching contains conflicting prefactors. These issues are likely fixable with a careful rederivation, but they are not merely typographical. I would invite a revision that corrects these algebra and asymptotics problems and resubmits."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this paper tests whether the Kraus-Maloney relation between thermal one-point functions and boundary-to-horizon geodesic length survives a restricted class of static radial BTZ perturbations. The qualitative answer—that at large mass the variation of the one-point function is controlled by the variation of the renormalized geodesic length—is plausible, and the WKB derivation plus the matching to exact hypergeometric asymptotics is genuine work. The perturbation class is narrow (radial, static, δχ=0, sourced by T^t_t = T^r_r), and the authors say so themselves.\n\nThe soft spot is not the narrowness; it is the arithmetic. Eq (3.4) defines δℓ_hor without the factor 1/2 that appears in Eq (4.19). The sign in the intermediate step leading to (4.18) is also not clean. More seriously, the saddle-point evaluation in Appendix C is inconsistent with the quoted results in Section 5. Eq (C.2) gives I(m) ~ m^{-3/2}; since δ⟨O⟩ = (2m/r+) e^{-2πr+} I(m), the leading behavior should be m^{-1/2}. Eq (5.8) quotes m^{3/2} instead, and Eq (5.13) quotes m^{-3/2} plus an unexplained e^{-m r_+} in place of e^{-m ℓ_hor}. These three cannot all be correct. Because the central claim is precisely about the coefficient of e^{-mℓ_hor}, the m power controls whether the relation holds at leading order. As written, the paper does not quantitatively establish (1.4). It may be a batch of typos, but a referee needs to see the corrected computation.\n\nWhat I liked: the WKB-to-exact matching in §4.2 is a legitimate attempt to justify WKB near the horizon; Appendix A gives a clean realization of the metric perturbation from a matter source; and the paper is honest about the restricted class of perturbations. The authors also admit in §4.1 that the result is expected on general grounds, which keeps the novelty modest.\n\nBottom line: this is a paper for people working on holographic one-point functions and geodesic approximations. It deserves a serious referee, but the referee should be asked to verify the saddle-point algebra and the factor of 2 in δℓ. If those are fixed, the result would be a modest but useful robustness check. I would not cite it in its current form.","headline":"A plausible robustness check of Kraus-Maloney that currently fails its own quantitative test: the m-scaling is inconsistent across (5.7), (5.8), and (5.13).","tokens_in":13015,"tokens_out":10229,"would_cite":false,"duration_ms":81287,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Under a horizon-preserving radial perturbation, the variation of a thermal one-point function is controlled by the variation of the boundary-to-horizon geodesic length, in the large-dimension limit.","keywords":["AdS/CFT correspondence","BTZ black hole","thermal one-point function","geodesic length","WKB approximation","black hole perturbations","large conformal dimension","holographic correlators"],"falsifier":"Compute the first-order variation of the one-point function in the large-dimension limit for a perturbation that includes a nonzero δχ(r), such as a metric with e^{2εχ(r)}(f+εδf(r))dt² + ...; if the result differs from e^{-mℓ_hor}δℓ_hor at leading order, the relation is confined to the δχ=0 class.","tokens_in":11979,"feed_emoji":"🕳️","tokens_out":6223,"duration_ms":51658,"temperature":0.7,"pith_summary":"This paper tests whether the exponential relation between thermal one-point functions and boundary-to-horizon geodesic lengths, previously established for the Euclidean BTZ black hole, survives small metric perturbations. It considers a radial deformation f(r)→f(r)+εδf(r) that preserves the horizon and computes both sides to first order in ε. The main result is that, in the large conformal-dimension limit, the variation of the one-point function is proportional to e^{-mℓ_hor} δℓ_hor, with δℓ_hor the first-order change in the renormalized geodesic length. The derivation relies on WKB approximations for the Green function and bulk-boundary propagator, and these approximations are shown to match the exact large-dimension hypergeometric expressions. If correct, the result shows that heavy operators remain faithful probes of bulk geometry even in slightly deformed black hole backgrounds.","feed_headline":"Perturbed black holes keep one-point function tied to geodesic length","feed_subtitle":"Radial horizon-preserving deformations shift both sides alike; heavy operators read the changed geometry.","key_machinery":"The key object is the zero-mode bulk-boundary propagator K0(r) and its first-order variation δK0(r) under the metric perturbation. The computation proceeds via WKB approximations: the Green function takes the form e^{-m|ℓ(r)-ℓ(r')|}/(2m[rr'√(f(r)f(r'))]^{1/2}) and the propagator behaves as e^{-mℓ(r')}/√(r'f(r')^{1/4}), where ℓ(r) is the radial geodesic length. The dominant large-m contribution to δK0 collapses to -m K0 δℓ(r), which directly links the propagator correction to the geodesic length variation. The paper also derives the large-h asymptotics of the hypergeometric functions appearing in the exact zero-mode Green function and propagator, showing they reproduce the WKB expressions, th","core_discovery":"The central claim is that the one-point function of a heavy primary operator, dual to a bulk scalar of mass m, obeys δ⟨O⟩ = δ(e^{-mℓ_hor}) ∝ e^{-mℓ_hor} δℓ_hor at first order in a radial, horizon-preserving perturbation of Euclidean BTZ. The paper proves this by computing the first-order change in the zero-mode bulk-boundary propagator, δK0(r) = -m K0(r) δℓ(r), using WKB forms for the propagator and Green function, and then verifying these WKB forms are nothing but the large conformal-dimension limits of the exact hypergeometric solutions. The result is confirmed for a concrete power-law perturbation H(r)=r^p, where the saddle-point evaluation of the one-point function reproduces the geodesi","pith_inferences":["The restriction to δχ=0 means the result does not cover perturbations that alter the redshift function; if such a deformation were included, the one-point function variation might acquire an additional contribution not captured by δℓ_hor.","One could test the universality of δK0 = -m K0 δℓ(r) by applying the same WKB procedure to Schwarzschild-AdS or higher-dimensional black holes, where exact hypergeometric verification is harder but the leading-order relation should still hold.","The saddle-point structure hints that time-dependent perturbations could be probed by inserting the perturbation's Fourier modes; the zero-mode dominance might be lost, potentially yielding corrections beyond δℓ_hor that encode the time dependence.","Because the relation is parameter-free and background-independent within the WKB regime, it may be a general property of holographic one-point functions for heavy operators, possibly extending to non-black-hole geometries with a natural 'horizon' cutoff."],"forward_implications":["The exponential one-point function/geodesic-length relation is stable under the class of horizon-preserving radial perturbations, so heavy operators can be used to measure small metric deformations from the boundary.","For any static perturbation of this type, the first-order correction to a thermal one-point function can be computed directly from δℓ_hor without solving the full bulk wave equation.","The structure of the derivation suggests the same relation holds for any static black hole geometry where the WKB approximation for the propagator is valid, not just BTZ.","The Euclidean result carries over to Lorentzian static time-slices, so similar geodesic-imprint statements apply to real-time thermal correlators (as the paper notes).","The dominant contribution to δ⟨O⟩ comes solely from δK, while variations of the measure and of ⟨χ²⟩ are subleading at large m, simplifying future computations."],"fun_headline_variants":["Geodesic length still ties one-point function to horizon in perturbed black holes","Perturbed black hole still ties heavy operators to horizon geodesic","One-point function shift equals geodesic length shift in perturbed BTZ","WKB verifies geodesic rule for one-point functions in deformed black holes","Perturbations don't break the geodesic connection for heavy one-point functions"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The perturbation is required to be purely radial with no change to the redshift function (δχ=0), which is sourced only by matter with T^t_t = T^r_r; if the perturbation has time dependence or off-diagonal components, the simple proportionality between δ⟨O⟩ and δℓ_hor could fail.","fun_headline_variants_meta":{"raw":{"variants":["Geodesic length still ties one-point function to horizon in perturbed black holes","Perturbed black hole still ties heavy operators to horizon geodesic","One-point function shift equals geodesic length shift in perturbed BTZ","WKB verifies geodesic rule for one-point functions in deformed black holes","Perturbations don't break the geodesic connection for heavy one-point functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000708,"raw_usage":{"total_tokens":3006,"prompt_tokens":707,"completion_tokens":2299,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":451,"completion_tokens_details":{"reasoning_tokens":2199}},"tokens_in":451,"tokens_out":2299,"duration_ms":14528,"temperature":1.0,"reasoning_tokens":2199,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T10:36:41.948465+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the first-order variation of the one-point function in the large-dimension limit for a perturbation that includes a nonzero δχ(r), such as a metric with e^{2εχ(r)}(f+εδf(r))dt² + ...; if the result differs from e^{-mℓ_hor}δℓ_hor at leading order, the relation is confined to the δχ=0 class.","supporting_citations":[],"review_version":1}