{"id":"d11006b7-f400-41bc-893f-4cb0b3f7e9c2","arxiv_id":"2505.09504","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A geodesic Witten diagram built directly in Euclidean BTZ coordinates reproduces the semi-classical Virasoro block and links the half-thermal-period timescale to the probe geodesic reaching the horizon.","lead":"This paper shows that a bulk geodesic calculation in BTZ black hole coordinates reproduces the known semiclassical four-point function in 2D gravity, and identifies the half-thermal-period timescale as the moment the probe geodesic touches the horizon. It matters because it gives a concrete geometric picture for where semiclassical black hole calculations break down, a step toward understanding the information paradox.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The BTZ heavy-geodesic prescription is asserted rather than derived; Eq. (3.7) does not follow from the propagator (3.5), so the horizon interpretation in Section 5 rests on an unproven limit.","rationale":"The reader's weakest assumption correctly identifies the unproven nature of the BTZ propagator prescription (no image sum, global AdS3 propagators in BTZ coordinates, heavy geodesic at r = r_+). I agree that this prescription is the soft spot of the paper. However, I have found a sharper, more concrete defect within that same prescription: Eq. (3.7) is not a straightforward consequence of the stated propagator (3.5). Taken literally, the product of the two heavy bulk-boundary propagators is zero for every finite affine parameter λ, so the entire diagram would vanish; to obtain the claimed constant one needs a limiting argument that is not supplied. This makes the concern more specific than the reader's general reference to 'dramatic steps', because it points to a precise equation whose validity can be checked directly by a finite-L computation. The proposed test would settle the issue: if the regulated L→∞ limit reproduces the constant in (3.7) (up to an unimportant overall prefactor), then the derivation of (3.23) stands and the horizon interpretation is supported; otherwise the central claim is not established. Because the paper itself flags the construction as a new prescription rather than a derivation, and because the verification is a tractable calculation, the appropriate verdict remains CONDITIONAL, agreeing with the reader's assessment. No change to the reader's verdict is needed, but the concrete test should be run before the interpretation in Section 5 is accepted as more than a suggestive analogy.","tokens_in":20345,"tokens_out":29494,"duration_ms":302170,"concrete_test":"Regulate the heavy operators by placing them at the conformal boundary with finite φ separation, x1 = (τ0, φ=−L) and x2 = (τ0, φ=+L), and compute the exact geodesic γ12 connecting them in metric (2.5). Evaluate the λ-integral of G_b∂(x1,y(λ))G_b∂(x2,y(λ)) using the propagator (3.5) along γ12, then take L→∞ and compare the result with the constant used in (3.7). If the limit is not a finite, λ-independent constant (for example, if it scales with L or depends on h_H beyond an overall prefactor), then the placement of the heavy geodesic at r = r_+ is not the L→∞ limit of a bona fide geodesic Witten diagram, and Eq. (3.23) is not derived from (2.1) as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction (Sections 3.2–3.3) places the heavy geodesic at r = r_+ and states that the product of the two heavy bulk-boundary propagators is a constant, G_b∂(x1,y(λ))G_b∂(x2,y(λ)) ∼ 1 (Eq. 3.7). But evaluating the explicit propagator (3.5) at y(λ) = (r_+, τ0, λ/r_+) with x1 = (τ0, φ=−∞) and x2 = (τ0, φ=+∞) gives cosh(r_+Δφ) → ∞ for every finite λ, so each G_b∂ vanishes and the product is identically zero. The paper does not present a regulating procedure (e.g., finite φ-separation L with a subsequent L→∞ limit) that would justify the constant 1. Instead, it appeals by analogy to the conical-defect case [6]. This is not a harmless normalization issue: if the heavy-geodesic product is zero, the entire integral (3.22) vanishes, and if it is not a constant independent of λ, the reduction to the form of [6] and the final expression (3.23) do not follow. Since Eq. (3.23) is the bridge to the horizon-straddling interpretation (Section 5 and Figs. 2–3), the centrality of the horizon is contingent on an unverified limiting prescription.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a bulk geodesic Witten diagram prescription for semi-classical Virasoro blocks directly in the Euclidean BTZ coordinate patch, with the heavy operator geodesic placed at r = r_+ and with τ and φ spanning the full real line without quotient identification. The construction yields the known semi-classical block formula, Eq. (3.23), and the authors interpret the half-period τ = π/r_+ as the boundary timescale at which the light geodesic straddles the Euclidean horizon, thereby offering a bulk geometric explanation for the departure of finite-c Virasoro blocks from the semi-classical result seen numerically in Ref. [11]. An appendix gives a position-space to momentum-space Fourier transform identity for bulk BTZ correlators, verified numerically to high precision.","tokens_in":20698,"tokens_out":6288,"duration_ms":68760,"significance":"If the prescription is valid, the paper provides a concrete bulk picture for the Euclidean information-loss timescale in AdS3/CFT2 and explains why finite-c corrections to Virasoro blocks become noticeable near half the thermal period. The main result is not parameter-free, but it is benchmarked against externally known semi-classical blocks from Refs. [6] and [23] and against the finite-c numerics of Ref. [11]. The explicit Fourier-transform identity in Appendix B is a useful technical contribution, with numerical checks to high precision. The paper is candid about the fact that the propagator prescription is a step that is asserted rather than derived, which is also the main source of technical risk.","major_comments":[{"comment":"Equation (3.7) does not follow from the propagator (3.5). For the heavy geodesic y(λ) = (r_+, τ_0, λ/r_+) and boundary points x1 = (τ_0, −∞) and x2 = (τ_0, +∞), one has r_+ Δφ → ±∞ for every finite λ. Each bulk-boundary propagator in (3.5) then behaves as cosh^{-2h_H}(r_+ |Δφ|) → 0, so the product is zero, not ∼1. The displayed reduction to ∼1 appears to interchange the two endpoints or to rely on an unstated regularization; either way, a careful limiting procedure (e.g., boundary points at finite separation L, with the λ-integral performed before L → ∞, or an explicit multiplicative renormalization) is needed. This is load-bearing because the simplification to the λ-integral in Eq. (3.22), and hence the final block (3.23) and the horizon interpretation, all depend on this step.","section":"§3.3, Eq. (3.7)"},{"comment":"The central propagator prescription is asserted rather than derived. The bulk-bulk and bulk-boundary propagators are taken to be the global AdS3 propagators expressed in BTZ coordinates, with no image sum and with τ, φ on the full real line, while the heavy geodesic is placed at r = r_+, the singular tip of the Euclidean cigar. The authors themselves describe this as a dramatic step and justify it by analogy with the conical-defect calculation of Ref. [6], but no path-integral or holographic argument is given. Because the final answer matches independent results, the prescription may be correct, but as written it is an independent axiom. A controlled derivation, or at least a demonstration that a regulated version (r = r_+ + ε with ε → 0, or a finite cutoff on the coordinate patch) gives the same propagators and the same integral, is required for the derivation to be self-contained.","section":"§3.1–§3.2"},{"comment":"The interpretive claim that information loss 'starts becoming substantial when the light operator geodesic starts probing the horizon radius' should be stated with the same precision as the technical result. The calculation shows that as τ approaches π/r_+, the minimal geodesic approaches r = r_+; the additional statement that this is where semi-classical blocks begin to fail compared with finite-c blocks comes from comparing with the numerics of Ref. [11], and the wording in Section 5 moves from a geometric observation to a causal claim about information loss. Please distinguish explicitly what is derived from the geodesic Witten diagram and what is an extrapolation from the numerical comparison.","section":"§5, Figs. 2–3"}],"minor_comments":[{"comment":"There is a parenthesis imbalance in the displayed expression for G_{b∂}(τ_1, y(λ′)) immediately before Eq. (3.18); the argument of the square root and the placement of the closing bracket should be corrected.","section":"§3.4.2, Eq. (3.17)"},{"comment":"The text would benefit from clarifying the orientation of the affine parameter λ and the ordering of the two heavy boundary points φ = ±∞. Since λ = r_+ φ, the statement in Eq. (3.7) that one factor is evaluated at λ → +∞ and the other at λ → −∞ reverses the natural ordering of x1 and x2; the signs are immaterial for the vanishing issue raised in the major comments, but the notation is confusing.","section":"§3.3"},{"comment":"The dot notation in Eqs. (3.8)–(3.10) is not defined; it should be stated that the dot denotes differentiation with respect to the affine parameter λ′ on the light geodesic.","section":"§3.4.1, Eq. (3.8)"},{"comment":"Appendix B, while technically self-contained, is not used in the main text. A sentence at the start of the appendix explaining how the Fourier identity is expected to connect to the Virasoro-block computation would make its inclusion more transparent.","section":"Appendix B"},{"comment":"The sentences describing 'the half-period' and 'the period' of the Euclidean BTZ cigar would be clearer if β = 2π/r_+ were recalled explicitly at the point where these terms are first used in the interpretive discussion.","section":"Section 5"},{"comment":"The phrase 'trouble at the Euclidean horizon' is evocative, but the abstract already notes that periodic Euclidean singularities are generic in thermal correlators; the title could mislead readers into thinking the paper claims that the horizon is the exclusive cause of those singularities.","section":"Title and abstract"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the main result is likely correct, but the derivation contains a load-bearing step—Eq. (3.7)—that is currently wrong as written and needs a controlled regulator or an alternative justification. This is repairable within the manuscript's scope, so I do not recommend rejection, but the revision must address the missing derivation of the propagator prescription and the heavy-geodesic product before the horizon interpretation can be regarded as established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a serious look. The authors build a geodesic Witten diagram for the semiclassical Virasoro block directly in Euclidean BTZ coordinates, heavy geodesic at r = r+, and reproduce the known block. That is a new route to a known answer, and it gives a clean geometric interpretation: the light geodesic's minimum radius is r+ csc(r+ tau/2), so as the boundary time separation approaches pi/r+, the geodesic reaches the Euclidean horizon. This timescale is half the thermal period, and it matches where the finite-c numerics of Chen et al. show the semiclassical block starting to fail. That observation is the real contribution, and it is solid: it comes straight from the geodesic equations, not from fitting.\n\nThe main soft spot is the heavy-leg product, Eq. (3.7). As written, it says Gb∂(x1,y(λ)) Gb∂(x2,y(λ)) ~ 1, but plugging the explicit propagator (3.5) into y(λ) = (r+, tau0, λ/r+) with x1,x2 at φ = ±∞ gives two vanishing factors, because cosh(r+ Δφ) diverges. The authors wave this away with limits like 'cosh(λ→+∞)' and call the product unity. That is not correct pointwise. The standard fix is a regulated boundary separation L, which makes the product independent of λ but multiplies the whole diagram by e^{-4hH r+ L}; it is a normalization factor, not literally 1. The paper does not supply that regulation, even though it admits the step is dramatic. This is a genuine gap in rigor, but it looks repairable, and it does not touch the Section 5 geodesic interpretation, which is independent of the heavy-leg prefactor.\n\nAppendix B does an explicit Fourier transform of the BTZ two-point function; the match is numerical to 45 decimals, which is honest but not a proof. Fine for a supporting role.\n\nWho is this for? People working on Virasoro blocks, AdS3/CFT2, and bulk mechanisms of information loss. It deserves a serious referee rather than a desk reject: the geometric claim about the half-period is clean, the final block matches known results, and the heavy-leg issue is localized and likely fixable. I would send it out, with the referee asked to focus on the regularization of Eq. (3.7).","headline":"A genuinely new geometric route to the semiclassical Virasoro block with a clean half-period/horizon picture; the heavy-geodesic propagator product needs regularization but the main observation survives.","tokens_in":21196,"tokens_out":5896,"would_cite":true,"duration_ms":61190,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The semi-classical Virasoro block is derived from a geodesic at the Euclidean BTZ horizon, locating Euclidean information loss at half the thermal period.","keywords":["Virasoro blocks","geodesic Witten diagrams","Euclidean BTZ black hole","Euclidean horizon","information loss","semi-classical limit","AdS3/CFT2","thermal correlators"],"falsifier":"Redo the double integral (3.22) with the BTZ image-summed bulk-bulk propagator (B.14) instead of the unquotiented one; if the resulting $W_{2h,0}(\\tau)$ differs from (3.23) before $\\tau=\\pi/r_+$, or if finite-$c$ Virasoro block numerics already depart from the semi-classical block well before half the period, the horizon-straddling claim is falsified.","tokens_in":20157,"feed_emoji":"🕳️","tokens_out":14505,"duration_ms":118882,"temperature":0.7,"pith_summary":"This paper tries to give the Euclidean information-loss puzzle in two-dimensional holographic CFTs a concrete bulk location. In the semi-classical limit $c\\to\\infty$, heavy-light-light-heavy correlators $\\langle O_H O_L O_L O_H\\rangle$ reduce to Virasoro blocks whose Euclidean-time singularities repeat periodically, while a unitary finite-$c$ CFT allows only the coincidence singularity; the open question is what makes the semi-classical approximation fail. The answer proposed here is geometric: evaluate the block as a geodesic Witten diagram directly in Euclidean BTZ coordinates, with the heavy geodesic pinned at the horizon $r=r_+$. The light geodesic carrying boundary time separation exists only for $0\\le \\tau \\le \\pi/r_+$, and at the upper endpoint it straddles the Euclidean horizon. The horizon is thereby identified as the place where semi-classical blocks begin to break down, so horizon-scale physics is where Euclidean information must be restored.","feed_headline":"Half the thermal period is where Euclidean information loss begins","feed_subtitle":"A geodesic pinned to the black hole horizon reproduces the semiclassical Virasoro block and sets the information-loss scale.","key_machinery":"The machinery is the geodesic Witten diagram of Eq. (2.1), evaluated in Euclidean BTZ coordinates instead of the conical-defect background used previously. The heavy operators sit at $r=r_+$ and $\\phi=\\pm\\infty$, so the heavy geodesic runs along $\\phi$ with affine parameter $\\lambda=r_+\\phi$. The light operators lie on the boundary at fixed $\\phi$ and times $\\tau_1,\\tau_2$; solving (3.8)-(3.9) gives $r=r_+\\csc\\psi\\cosh\\lambda'$ and $\\tan\\theta=\\tan\\psi\\tanh\\lambda'$, with $\\psi=r_+(\\tau_2-\\tau_1)/2$. The distinctive input is that the bulk-bulk and bulk-boundary propagators are the global AdS$_3$ propagators rewritten in BTZ coordinates through the chordal variable (3.4), with no BTZ image sum. That combination produces Eq. (3.23) and the existence bound $0\\le\\tau\\le\\pi/r_+$ for real geodesics.","core_discovery":"The paper's central claim is that the semi-classical Virasoro block for HLLH correlators, $W_{2h,0}(\\tau)$, is reproduced by a geodesic Witten diagram calculated on the unquotiented Euclidean BTZ patch, provided the heavy-operator geodesic sits at $r=r_+$ and the propagators are the global AdS$_3$ propagators written in BTZ coordinates with no image sum. The computation gives $$W_{2h,0}(\\tau)\\sim \\bigl[\\sin(r_+\\tau/2)\\bigr]^{2h-4h_L}\\;{}_2F_1(h,h;2h;1-$e^{{ir_+\\tau}}$)\\;{}_2F_1(h,h;2h;1-$e^{{-ir_+\\tau}}$),$$ and, just as importantly, the light geodesic with time-separated boundary endpoints exists only for $0\\le \\tau\\le \\pi/r_+$, reaching the horizon at the upper limit. The authors read this as the geometric origin of Euclidean information loss: half the thermal period is the boundary timescale at which the probe geodesic straddles the horizon, and it is precisely the scale at which numerical finite-$c$ Virasoro blocks begin to depart from their semi-classical form. They stress that no periodic thermal circle is used on either the bulk or the boundary, so the periodicity of the singularities emerges from the $r_+$ scale itself.","pith_inferences":["Beyond the paper: if the horizon is the onset locus, finite-$c$ blocks should deviate most strongly from semi-classical behavior near half the thermal period, and the appendix's momentum-space transform (B.20) gives a concrete way to look for that deviation.","Beyond the paper: modifying the prescription near $r=r_+$ — smearing the heavy geodesic or adding a regulating tip — is the natural place to seek corrections that match finite-$c$ numerics.","Beyond the paper: the costless horizon-straddling geodesics parallel winding strings in cigar resolutions, suggesting a worldsheet version of the diagram could expose the unitarity-restoring mechanism.","Beyond the paper: because the interior $r