{"id":"80ad0a66-5fe0-43c3-8983-5f71574829e2","arxiv_id":"2411.19924","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In a time-symmetric Vaidya-like spacetime, radial null geodesics create boundary bulk-cone singularities only when r_+ < l, and a large-c CFT2 computation reproduces the exact singularity time.","lead":"Boundary observables in a holographic theory can develop extra singularities when light can travel through the extra dimension and return to a different boundary point. The paper shows that in a family of collapsing-shell geometries such singularities exist only below a critical black hole size, and reproduces their timing from a CFT calculation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The truncation to |δ|<π is unjustified: footnote 7 shows singularities outside this region before the light-cone time, so omitted winding channels may alter the claimed bulk-cone divergence.","rationale":"The reader identified the unproven channel-sum approximation and the truncation to |δ|<π as the weakest assumption; my stress test agrees and finds direct textual support for the concern. Footnote 7 explicitly states that before the light-cone time the singularity loci lie outside the integration region, which the paper calls 'certainly unexpected' and defers to future work. Section 5 also concedes that the same truncation misses non-radial bulk cone singularities and even the expected periodic light-cone structure, attributing both to the excluded winding channels. These are not merely aesthetic gaps: they mean the integral (3.35) is not known to be the correct realization of the proposal (3.6), so the exact matching at t_c may be an artifact of the truncation. I gave the paper credit where due: the bulk geodesic computation is straightforward and self-consistent; the monodromy method follows an established technique; there are no fitted parameters; and the equality of (4.4) with (2.14) is a striking quantitative match. But because the CFT calculation is the central evidence for the claimed reproduction of bulk causal structure, an unresolved restriction of the channel sum leaves the strongest claim conditional. The proposed test—extending the integral to the first winding sectors—would settle whether the omitted channels change the singularity structure. Since the reader's verdict already reflects this conditionality, I recommend no change.","tokens_in":18663,"tokens_out":5884,"duration_ms":53243,"concrete_test":"Compute the full channel sum by extending the δ-integration in (3.35) to include the first winding sectors, e.g. |δ| ∈ (π, 3π], using the same vacuum-block integrand but with the iϵ prescription determined separately for each winding topology from the monodromy paths in Fig. 5 (right). Numerically evaluate the resulting integral for a representative subcritical case (e.g., ρ=0.5, t1=−10) and a supercritical case (ρ=2), scanning t2 through t_c and through the light-cone time. If the divergence at t_c appears only for ρ<1 and no divergence appears for t2−t1<π or for ρ>1, the truncation is harmless; if the singularity structure changes, the central claim fails because it rests on an unjustified restriction of the channel sum.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the channel integral (3.35), restricted to |δ|<π, diverges exactly at the bulk cone time t_c for ρ<1 and not for ρ>1, reproducing the geodesic result (2.14). The paper explicitly truncates the sum over channels to the non-winding sector because the winding channels |δ|>π are 'subtle' and it says the trivial sector 'alone is enough.' However, footnote 7 admits that before the light-cone time t2−t1<π, the singularity loci L_± are supported outside |δ|<π, which is 'certainly unexpected' and left unresolved. Since the prescription (4.2) with the iϵ signs is only justified for the |δ|<π sector, the omitted winding channels could introduce additional singularities or shifts that change the analytic structure: they might produce a divergence before t_c, remove the divergence at t_c by interference, or even create one for ρ>1. The Section 5 discussion reinforces this worry by noting that the same truncation fails to capture the usual light-cone periodicity and non-radial bulk cone singularities, which are expected on physical grounds and are attributed to exactly the omitted winding paths. Therefore the reported matching at t_c is only established within an approximation whose validity is not demonstrated; the paper's own text flags this as an unresolved subtlety rather than a harmless technicality.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a time-symmetric null shock-wave spacetime obtained by gluing global AdS to a BTZ/Schwarzschild geometry, parameterized by the horizon radius r_+. The bulk part of the paper argues that radial bulk-cone singularities appear in the boundary two-point function exactly when r_+ < l, with the bulk-cone time given explicitly in eq. (2.14). The CFT part models the state by a heavy operator V on the vacuum and computes the two-point function using a sum of Virasoro identity blocks over channels, obtained by the monodromy method. The resulting integral (3.35) is analyzed both through the singularity loci L_± of the integrand and through a large-Δ saddle-point approximation. The paper claims that the integral diverges exactly at the bulk-cone time (4.4) for ρ = r_+/l < 1 and remains finite for ρ > 1, thereby reproducing the bulk causal-structure transition. It also sketches a von Neumann algebra interpretation and discusses, in Section 5, the absence of the expected periodic light-cone structure and of non-radial bulk-cone singularities.","tokens_in":18912,"tokens_out":7836,"duration_ms":74583,"significance":"If the CFT calculation is valid, the paper provides a nontrivial new example where large-c, multi-channel Virasoro identity blocks reproduce Lorentzian bulk causal structure, including a sharp presence/absence transition in a one-parameter family of geometries. The bulk geodesic computation is explicit and dimension-independent, the monodromy calculation is detailed, and no free parameters are fitted: ρ is fixed by the heavy-operator data through (3.17), and the claimed divergence time is a falsifiable prediction that matches the bulk answer. The paper is also commendable for transparently flagging the unresolved winding-channel issue in footnote 7 and in Section 5. The main limitation is that the central CFT result depends on an assumed channel-sum prescription and on a truncation to |δ|<π whose validity is not established; this makes the result conditional rather than conclusive.","major_comments":[{"comment":"The central divergence at t_c is derived from the integral (3.35) restricted to |δ|<π, and the iϵ prescription (4.2) is justified only in that sector. Footnote 7 states that before the light-cone time the singularity loci L_± are supported outside |δ|<π, which the author calls 'certainly unexpected,' and Section 5 attributes the absence of the usual periodic light-cone structure and of non-radial bulk-cone singularities precisely to the excluded winding paths. Because those winding channels are not controlled, the matching at t_c could in principle be shifted, cancelled, or supplemented by additional singularities from the omitted sectors. This is a load-bearing assumption, not a harmless technical restriction. The manuscript should either provide an argument that the omitted sectors cannot change the analytic structure near t_c, or explicitly state the result as conditional on the non-winding truncation.","section":"§4.1, footnote 7; §5"},{"comment":"The identification of the exact large-c correlator with a sum of Virasoro identity blocks over all channels is imported from [12] under 'mild assumptions' that are not stated or verified here. This is load-bearing because the claimed divergence is extracted from the singularities of this approximate sum. The agreement with the bulk geodesic time (2.14) is supporting evidence for the proposal of [12], but it does not by itself validate the channel-sum replacement; an independent check, such as a solvable limit or a demonstration that the omitted primary blocks are subleading in the relevant Lorentzian regime, would be needed to make the CFT derivation self-contained.","section":"§3.2, eqs. (3.6) and (3.35)"}],"minor_comments":[{"comment":"The horizon radius is written as r_+ in the text but as r_h in eqs. (2.5)-(2.7); please use one notation throughout.","section":"§2, eqs. (2.5)-(2.7)"},{"comment":"The displayed boundary value (4.10) would be clearer with an explicit bracket separating the prefactor ρ^2/4 from the product of hyperbolic sines; as printed it is easy to misread.","section":"§4.1, eq. (4.10)"},{"comment":"The introduction refers to 'the correlator (5.1)', but the two-point function is introduced in (1.1); the cross-reference should be corrected.","section":"§1, around eq. (1.1)"},{"comment":"The saddle-point analysis for ¯t>0 and for general ρ is described only qualitatively, and Fig. 10 is computed for a single parameter set (Δ=3, ρ=0.7, ¯t=0). The text should state explicitly that the general-case saddle selection is a heuristic extrapolation, since the direct integral argument of §4.1 is the primary support for the bulk-cone divergence.","section":"§4.2, Fig. 10"},{"comment":"The definition of the regularized state V_{-τ}|0⟩ as a product of n operators would benefit from a few words on the normalization N_n and on how the continuum limit is taken, since these details are imported from [11].","section":"§3.2, eq. (3.3)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a good fit for a hep-th journal and the bulk calculation is clean. I would not reject on the basis of the unproven channel-sum assumption alone, since the author is explicit about it and the agreement with the bulk result is suggestive. However, the published version should not state the CFT reproduction as unconditional; either the winding-sector issue should be addressed or the claims should be carefully qualified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper reports a new class of bulk cone singularities in a one-parameter family of Vaidya-like spacetimes, with a sharp threshold at r_+=l. The bulk side is clean and the CFT calculation is a nontrivial application of the monodromy method and the channel-sum proposal from Anous-Hartman-Rovai-Sonner. I think the paper is worth serious referee time.\n\nThe genuinely new thing is the radial bulk cone singularity, absent in earlier works which considered angular-momentum-carrying geodesics. The geodesic computation in Section 2 is elementary but correct, and the identification of t_c(t1) is explicit. The CFT calculation is the real meat: the integral over identity blocks with the tangency condition at delta=sigma=0 producing the divergence at exactly the bulk cone time is a neat mechanism, and the appendix supporting the pinch is careful. Credit where due: no fitted parameters, no invented entities, and the paper is admirably honest about its own unresolved points.\n\nThe soft spot is the channel-sum truncation. The paper sums only the non-winding sector |delta|<pi and explicitly leaves the winding channels |delta|>pi aside, with footnote 7 admitting that singularities outside the integration region appear before the light-cone time, which is 'certainly unexpected'. That is a genuine gap. The claimed CFT reproduction of the bulk cone divergence is established only within that truncated prescription, and the missing channels could in principle change the analytic structure. I don't think the gap is fatal: the divergence at t_c comes from a local pinch inside the integration region, and the omitted channels are positive contributions, so adding them would not cancel a divergence. But the paper does not prove that the |delta|<pi sector is the right sector, and the authors themselves don't claim to resolve it. So a referee should push on this point, but it shouldn't be a desk-reject reason.\n\nThe von Neumann algebra discussion in Section 5 is speculative and brief, but clearly labeled as such. The saddle-point analysis is a useful complement, and the plots seem consistent.\n\nOverall: this is a solid subfield contribution, likely to be cited by people working on bulk-cone singularities and large-c blocks. It deserves a serious referee. My recommendation: send it to review, with a request to address the truncation issue or at least sharpen the discussion of why the non-winding sector is sufficient.","headline":"A clean bulk-side result with a genuinely new radial bulk cone singularity, and a CFT derivation that is impressive but conditional on an unresolved truncation of the channel sum.","tokens_in":19431,"tokens_out":2576,"would_cite":true,"duration_ms":23627,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.70.-s","11.25.Hf"],"model":"deepseek-v4-flash","headline":"Radial bulk cone singularities in a time-symmetric Vaidya-like spacetime turn on sharply at $r_+=l$, and a large-$c$ CFT$_2$ correlator built from Virasoro identity blocks diverges at exactly the bulk cone time.","keywords":["bulk cone singularities","Vaidya spacetimes","Virasoro identity blocks","monodromy method","AdS3/CFT2","BTZ black holes","causal structure","conformal blocks"],"falsifier":"Evaluate the channel integral (3.35) numerically for $\\rho < 1$ and confirm the divergence as $t_2$ approaches $t_c$; then include the winding channels $|\\delta| > \\pi$ with a consistent $i\\epsilon$ prescription and check whether the divergence survives.","tokens_in":18427,"feed_emoji":"🕳️","tokens_out":11279,"duration_ms":85101,"temperature":0.7,"pith_summary":"The paper studies boundary two-point functions in a family of time-symmetric, spherical null shock-wave geometries (Vaidya-like spacetimes) labeled by the horizon radius $r_+$ of the black hole that forms after the shock bounces off the boundary. It claims a sharp transition in bulk causal structure at $r_+=l$: a radial null geodesic can leave the boundary, cross the origin, and return to an antipodal boundary point only when $r_+<l$, so the associated radial bulk cone singularities exist only below this threshold. Above it, every such geodesic falls into the black hole and the singularities are absent. The central result is a CFT$_2$ calculation: the correlator for the state created by a heavy operator, approximated by a sum of large-$c$ Virasoro identity blocks over channels, diverges at exactly the bulk cone time obtained from the geodesic. The paper's significance is that a purely boundary conformal-block sum reproduces the bulk causal structure, including the presence/absence transition at $r_+=l$.","feed_headline":"Black hole smaller than AdS length turns on bulk cone singularities","feed_subtitle":"A boundary conformal-block sum diverges at the exact time a radial null geodesic returns to the antipode.","key_machinery":"The machinery is the sum over Virasoro identity blocks in all channels, evaluated with the monodromy method, a technique that obtains large-$c$ conformal blocks from the monodromy of solutions to a second-order differential equation (3.8) around operator-contraction paths. The resulting channel integral (3.35) runs over the crossing-point parameters $(\\sigma, \\delta)$, and its singular loci $L_\\pm(t_1, t_2)$, where $B_\\pm = 0$, control the correlator's singularities. The bulk cone singularity appears when $L_+$ and $L_-$ first touch tangentially at $\\delta = \\sigma = 0$, which yields $t_c(t_1)$; the light cone singularity appears when the loci reach the boundary $|\\delta| = \\pi$ at $t_2 - t_1 = \\pi$. The parameter $\\rho = r_+/l$ enters through the state's energy via $\\rho = \\sqrt{4K - 1}$.","core_discovery":"The central claim is that the family of time-symmetric Vaidya-like spacetimes (2.1) has a transition at $\\rho = r_+/l = 1$. Below the threshold, a radial null geodesic sent from the boundary before the shock can reach an antipodal boundary point at a later time, producing a bulk cone singularity in the boundary two-point function; above the threshold the black hole and white hole horizons overlap and no such radial geodesic exists. In AdS$_3$/CFT$_2$ the paper derives the CFT correlator (3.35) as a sum over Virasoro identity blocks computed by the monodromy method, and shows that it diverges precisely at $t_2 = t_c(t_1) = (1/\\rho) \\log[(\\tanh(\\rho t_1/2) - \\rho^2)/(\\tanh(\\rho t_1/2) + \\rho^2)]$, matching the bulk geodesic time (2.14). The divergence arises because the two singularity loci $L_\\pm$ of the channel integrand meet tangentially at $\\delta = \\sigma = 0$; for $\\rho > 1$ the bulk cone time is complex and the loci never intersect.","pith_inferences":["I infer that the omitted winding channels $|\\delta| > \\pi$ are what would produce bulk cone singularities from null geodesics that wind around the black hole, along with the periodic light-cone structure on the compact spatial circle; the paper flags this as an open question.","The tangency mechanism should hold for generic angular separations $\\theta$, with the meeting point of $L_\\pm$ moving away from $(0,0)$; one could derive a $\\theta$-dependent bulk cone time from the same analysis of (4.1).","Because the divergence is a pinch of the channel integral rather than a saddle-point artifact, it should persist to all orders in the probe dimension $\\Delta$; the saddle-point calculation only sets the strength of the divergence.","The proposed algebra-type transition could be tested through the modular spectral analysis of the two-point function (5.1), where the bulk-cone divergence should leave a visible signature in the spectral function determining the type."],"forward_implications":["For $\\rho < 1$, the boundary two-point function in the Vaidya-like state diverges at the bulk cone time $t_c(t_1)$, so this class of bulk cone singularities is visible in a purely boundary quantity.","For $\\rho > 1$ the would-be bulk cone time is complex and the channel integral remains finite, matching the absence of radial bulk cone singularities.","The light cone singularity is reproduced by the same integral: it occurs when the singularity loci $L_\\pm$ reach the boundary of the integration region at $t_2 - t_1 = \\pi$.","These radial bulk cone singularities are a new class, distinct from the angular-momentum bulk cone singularities of static black holes, which are present both below and above the transition.","The same $r_+ = l$ transition shows up in the type of the single-trace von Neumann algebra of the state: type I below the threshold and type III above it."],"supporting_citations":[{"why":"Supplies the proposal that the Lorentzian correlator is approximated by a sum of Virasoro identity blocks over all channels, eq. (3.6), which the CFT calculation adopts.","marker":"[12]"},{"why":"Supplies the operator construction of the shock-wave state and the monodromy setup for computing the correlator.","marker":"[11]"},{"why":"Provides the monodromy method used to compute the large-c identity blocks.","marker":"[13]"},{"why":"Establishes that Euclidean correlators are dominated by Virasoro identity blocks at large central charge.","marker":"[14]"},{"why":"Defines bulk cone singularities and their signatures in AdS/CFT, the phenomenon this paper extends.","marker":"[2]"},{"why":"Provides the static black hole bulk-cone singularities from angular-momentum geodesics that are contrasted with the radial class studied here.","marker":"[10]"},{"why":"Supplies the gravitational time-delay theorem used to place bulk cone singularities after the light cone singularity.","marker":"[1]"},{"why":"Supplies the subregion-subalgebra duality used to relate the r_+ = l transition to the type of the single-trace algebra.","marker":"[15]"}],"fun_headline_variants":["New bulk cone singularities emerge below the AdS length threshold","Antipodal bulk cone singularities switch on for small black holes","Large c conformal blocks reproduce bulk cone singularities in small black holes","Bulk cone singularities appear when black hole radius drops below AdS length","Sub-AdS black holes host new bulk cone singularities"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that the exact large-c correlator is faithfully represented by the sum over Virasoro identity blocks over all channels, eq. (3.6), and that the non-winding sector $|\\delta| < \\pi$ alone captures the bulk cone divergence.","fun_headline_variants_meta":{"raw":{"variants":["New bulk cone singularities emerge below the AdS length threshold","Antipodal bulk cone singularities switch on for small black holes","Large c conformal blocks reproduce bulk cone singularities in small black holes","Bulk cone singularities appear when black hole radius drops below AdS length","Sub-AdS black holes host new bulk cone singularities"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001034,"raw_usage":{"total_tokens":4362,"prompt_tokens":962,"completion_tokens":3400,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":578,"completion_tokens_details":{"reasoning_tokens":3309}},"tokens_in":578,"tokens_out":3400,"duration_ms":21198,"temperature":1.0,"reasoning_tokens":3309,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:41:31.710975+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the channel integral (3.35) numerically for $\\rho < 1$ and confirm the divergence as $t_2$ approaches $t_c$; then include the winding channels $|\\delta| > \\pi$ with a consistent $i\\epsilon$ prescription and check whether the divergence survives.","supporting_citations":[{"cited_title":"Conformal symmetry in two-dimensional space: Recursion representation of conformal block,","cited_arxiv_id":null,"evidence_quote":"Provides the monodromy method used to compute the large-c identity blocks."}],"review_version":1}