{"id":"f2f8aae5-6b52-4a8b-bbf8-99fe69f4b4cc","arxiv_id":"2607.24242","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"At large light-like momenta, holographic retarded correlators scale as a horizon-curvature-dependent power of ω, weaker than the naive ω^{2Δ−d}.","lead":"Holographic thermal correlators of scalar operators grow more slowly with frequency on the light cone than dimensional analysis predicts, and the power depends on whether the dual black-hole horizon is flat or curved. The result constrains finite-temperature OPEs and enters rates for on-shell emission processes.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"For spherical horizons (κ=+1) the light-like WKB potential Q(ρ)=f′(1)²(1−f)/(4f²) has an unexamined bulk turning point where f=1 (since f>1 near the boundary), and the spherical scaling law — unlike the planar and hyperbolic cases — has no independent exact or numerical check.","rationale":"Read in good faith, the paper's architecture is strong: the planar claim is triangulated by the exact d=2 correlator (bulk and CFT), d=4 Heun numerics showing power-law convergence of the extracted exponent to (4/3)(2h−1), and the exact large-d hypergeometric solution; the hyperbolic claim is exact in all d (§3.2). These independently verify the reader's flagged assumption (phase-only WKB transport) precisely in the cases where the transport genuinely is phase-only: for planar and hyperbolic horizons f<1 throughout the bulk, Q>0, and the only turning point is the near-boundary layer the paper does analyze (§3.1.1). The least secure point is therefore not the generic WKB assumption the reader identified, but a sharper instance of it: the spherical case, where the geometry itself (f>1 near the boundary for κ=+1) creates a bulk turning point at f=1 that the paper's validity discussion omits, the oscillatory matching of §3.1.2 does not literally apply (imaginary Bessel argument), and no exact or numerical backstop exists. This is not a manufactured objection: the paper's own validity criterion (3.26) diverges at the unexamined turning point, and the exponent — not merely the prefactor — depends on the surviving branch structure after the connection. I agree partially with the reader: same family of concern (WKB transport), but localized to the one channel where it is least secure and least checked. Because standard turning-point analysis very likely restores the claimed exponent (overall real exponential factors cancel in the ratio; the complex in-going wave continues to an O(1) I/K mix), REJECT would be overreach; but ACCEPT/HIGH overstates the current support for the spherical result. CONDITIONAL — conditional on the proposed numerical solve of (3.9) or an explicit connection-formula derivation — is the honest adjustment. A minor separate note: eq. (4.15) prints the sound-channel exponent as 4d/(d+2) while the derivation, eq. (4.16) and eq. (1.12) give 6d/(d+2); this is a typo, not a conceptual issue.","tokens_in":37299,"tokens_out":9787,"duration_ms":512055,"concrete_test":"Numerically solve the radial ODE (3.9) for a spherical Schwarzschild-AdS₅ black hole (κ=+1, d=4) with in-going boundary conditions at ρ=1, on the light-like Casimir locus ω²=l(l+2), for ω up to ~50–100. Extract GR from the near-boundary ratio of normalizable to non-normalizable coefficients and fit the logarithmic slope d log|GR|/d log ω. If the converged slope deviates from (2∆−d)/2=∆−2 by more than a few percent, the spherical scaling law (1.11) fails. As an analytic complement, perform the Airy connection across ρ_t=(a/(1+a))^{1/(d−2)} and verify the near-boundary solution retains both I_{ν′} and K_{ν′} branches with an ω-independent coefficient ratio.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The spherical branch of the central claim (eq. 1.11, GR ∼ ω^{(2∆−d)/2}) rests entirely on the WKB transport argument of §3.1. But for κ=+1 the blackening function in the ρ=r⋆/r coordinate is f(ρ)=1+aρ²−(1+a)ρ^d with a=(R/r⋆)²>0 (horizon condition used). Hence f(ρ)>1 for 0<ρ<ρ_t, ρ_t=(a/(1+a))^{1/(d−2)}∈(0,1): the light-like potential Q(ρ)=f′(1)²(1−f)/(4f²) (eq. 3.20) is negative in an outer evanescent region and positive only between ρ_t and the horizon. The paper's own validity measure, eq. (3.26) ∝ |2f′(1)(f−2)f′/(1−f)^{3/2}|/ω, diverges at ρ_t, so the single-exponential WKB transport from horizon to boundary (eqs. 3.16/3.22) is not justified as written: an Airy-type connection formula across ρ_t is required and is nowhere given. Moreover the near-boundary matching (eqs. 3.42–3.43) assumes oscillatory Bessel J/Y with real argument z; for κ=+1, f⁽²⁾(0)>0 makes z imaginary, so the actual solutions are modified Bessel I/K. This matters for the exponent, not just the prefactor: the scaling law follows from the ratio of the z^{ν′} and z^{−ν′} branches, and if the turning-point connection selected the growing branch I_{ν′} alone, the z^{−ν′} (non-normalizable) coefficient would vanish and the ratio defining GR would be ill-defined or scale differently. Analytic continuation of the complex in-going wave plausibly leaves an O(1) I/K mixture with ω-independent ratio (and any real e^{ω·c} factor cancels in the ratio), so the exponent likely survives — but this must be demonstrated, not assumed. Crucially, the two exact/numerical cross-checks do not exercise this regime: the hyperbolic exact result (§3.2) has κ=−1, where f=1−R²/r²−µR²/r^d<1 everywhere and Q>0 throughout (no bulk turning point), and the planar checks (d=2 exact, d=4 Heun numerics, large-d exact) have f=1−w<1 as well. The spherical case is thus the unique channel where (i) the transport is not phase-only, (ii) the paper's validity criterion fails mid-bulk, and (iii) no independent check exists.<|close|>note<|s","agreement_with_reader":"partial"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper studies holographic retarded two-point functions of scalar primaries (and the stress tensor in its three channels) at finite temperature in the light-like limit |k|=ω with ω→∞. The central claim is that the scaling is anomalous and horizon-curvature dependent: for planar AdS black holes G_R∼ω^{2(2Δ−d)/(d+2)} instead of the dimensional-analysis/OPE expectation ω^{2Δ−d} (eq. 1.9), while for spherical and hyperbolic horizons, with ω² set equal to the horizon Casimir, G_R∼ω^{(2Δ−d)/2} (eqs. 1.10–1.11). The planar result is established by an exact BTZ computation matched to an independent CFT Fourier transform via KLT string-integral identities (§2.1), numerics on the Heun connection problem for AdS5 (§2.2, Fig. 1), a general-d WKB analysis (§2.3), and an exact large-d solution (§5). The hyperbolic exponent is confirmed by an exact hypergeometric result (§3.2). The spherical exponent rests entirely on the WKB transport argument of §3.1. Appendix B reproduces the d=2 anomalous term from a Borel-resummed momentum-space OPE.","tokens_in":37853,"tokens_out":10431,"duration_ms":299260,"significance":"If the results hold, this is a notable and cleanly stated finding: a parameter-free, universal anomalous scaling exponent for light-like thermal correlators at strong coupling, with direct relevance to on-shell emission rates (photon/dilepton, gravitational waves) where light-cone correlators enter. The manuscript's strengths are substantial: the exponents are derived, not fitted; the planar branch is confirmed by four mutually independent methods (exact BTZ plus an independent CFT-side Fourier transform, Heun numerics with error decaying in ω, WKB, exact large-d); the hyperbolic branch has an exact check; and the d=2 OPE/Borel analysis identifies where the anomalous term lives in the momentum-space OPE. The comparisons with generalized free fields, the O(N) model, and the free N=4 glueball correlator (§6) correctly frame the effect as a strong-coupling phenomenon. The stress-tensor channel exponents (1.12) are falsifiable predictions. The spherical branch, however, currently lacks any check independent of the WKB argument, and that argument has a genuine gap (major comments).","major_comments":[{"comment":"For κ=+1 the light-like WKB potential has an unanalyzed bulk turning point. Writing a=(R/r*)², the spherical blackening factor in the ρ=r*/r coordinate is f(ρ)=1+aρ²−(1+a)ρ^d, so f(ρ)>1 for 0<ρ<ρ_t with ρ_t=(a/(1+a))^{1/(d−2)}∈(0,1). Hence Q(ρ)=f′(1)²(1−f)/(4f²) (eq. 3.20) is negative in an outer evanescent region and vanishes at ρ_t. The paper's own validity measure (3.26) ∝ |2f′(1)(f−2)f′/(1−f)^{3/2}|/ω diverges at ρ_t, contradicting the claimed validity domain (3.30), ω^{−2/(n+2)}≪ρ≪1. The single-exponential WKB transport (3.16)/(3.22) from horizon to boundary is therefore not justified as written for spherical horizons; an Airy-type connection across ρ_t is required and is nowhere given. Unlike the planar exponent (checked exactly in d=2, numerically in d=4, and at large d) and the hyperbolic exponent (checked exactly in §3.2), the spherical exponent in (1.11) has no independent veri","section":"§3.1.1, eqs. (3.20), (3.26), (3.30)"},{"comment":"For κ=+1, f⁽²⁾(0)=2a>0, so the argument z in (3.42) is imaginary and the near-boundary solutions of (3.41) are the modified Bessel functions I_{ν′}(|z|), K_{ν′}(|z|), not the oscillatory J/Y written in (3.42). Correspondingly the large-z asymptotics (3.43) are exponential, not the displayed cos/sin, and cannot match the (evanescent) WKB form (3.22) as written — the displayed matching is internally consistent only for κ=0 and κ=−1. This matters for the exponent, not just the prefactor: the scaling law follows from the ratio of the z^{ν′} and z^{−ν′} branches in (3.44), and if the turning-point connection selected pure I_{ν′} (which has only the z^{ν′} branch), the ρ^{d−Δ} coefficient would vanish or be exponentially suppressed and G_R would be ill-defined. My own estimate is that the in-going wave connects across ρ_t to a K-dominant solution (growing WKB exponential e^{+ω∫κ} maps to e^{−z","section":"§3.1.2, eqs. (3.41)–(3.44)"}],"minor_comments":[{"comment":"The final equality reads GR ≈ w^{2(2∆−d)/(d+2)}; the RHS should be ω, not w.","section":"Eq. (2.73)"},{"comment":"The sound-channel result is printed as ω^{4d/(d+2)}, but eq. (4.14) gives 2ν=6d/(d+2) and eqs. (1.12), (4.16) also give 6d/(d+2). Presumably a typo, but it appears in a numbered result.","section":"Eq. (4.15)"},{"comment":"Both limits are labeled with the subscript 'shear'; they refer to the sound channel.","section":"Eq. (5.30)"},{"comment":"The statement after (1.11) that the scaling 'remains the same' for charged black holes is stronger than §2.3 ('our preliminary calculations indicate') and §3.1.2 (restrictions d>4 planar, d≥3 for κ=±1). Please reconcile and state the charge claims with uniform precision.","section":"§1 vs §2.3/§3.1.2"},{"comment":"Footnote 5 refers to the domain (2.53); presumably (2.55) is meant. Also eq. (5.29) has 'w⁶' where ω⁶ is intended.","section":"Footnote 5; eq. (5.29)"},{"comment":"Please give more detail on the extraction of α (fit range, tolerance of the Heun evaluation, estimated systematic error), and note explicitly in the text that ω≤100 is a numerical-stability cutoff.","section":"Fig. 1, §2.2"},{"comment":"The term 4πTω_n is noted as differing from the usually cited result of [1]; a sentence clarifying its status (contact term vs physical, and scheme dependence) would help, given the emphasis placed on it.","section":"Eq. (2.36)"},{"comment":"It would be useful to state explicitly (e.g., in §3.1.2 or the conclusions) that the spherical exponent currently rests only on the WKB argument, and to compare the exponents quantitatively with the known U(1)-current light-like result, eq. (3.19) of [24].","section":"§3.1.2 / §6"},{"comment":"Assorted typos: 'correaltor' (§1, §6), 'mometa' (§5 title), 'Nordstorm' (§3), 'light-ike' (§5.2), 'anstaz' (§3), 'expresssions' (footnote 11), 'form' for 'from' (several places).","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The planar and hyperbolic branches of the main result are convincingly established by multiple independent methods and I have no concerns there. The spherical branch is one of the two headline formulas in the abstract, yet its sole derivation (§3.1) is internally inconsistent as written: the matching displayed in (3.42)–(3.43) cannot apply for κ=+1, and the turning point at f=1 is unaddressed. My own estimate is that the stated spherical exponent survives the correct connection (K-Bessel dominance), so I expect a revision to confirm rather than overturn the claim — but the fix requires a new calculation, not rewording, hence major rather than minor revision. The authors appear unaware of the spherical turning point; the revision should be straightforward for them to execute."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new result is clean: holographic retarded correlators of scalar primaries (and all three stress-tensor channels) at large light-like frequency pick up an anomalous power set by horizon curvature—ω^{2(2Δ−d)/(d+2)} for planar, (2Δ−d)/2 for spherical/hyperbolic when ω^{2} matches the horizon Casimir—rather than the vacuum ω^{2Δ−d}. That is not in the prior literature beyond one fragmentary U(1) current remark in AdS5.\n\nWhat they do well is the cross-check stack. Exact BTZ (bulk + CFT Fourier via KLT) and exact hyperbolic hypergeometric both give the claimed powers. AdS5 planar is checked numerically with Heun connection; large-d reduces to hypergeometrics that recover the d→∞ limit; WKB reproduces everything and extends to stress-tensor channels. Free-field, large-N O(N), and free Yang-Mills comparisons correctly show the effect is Einstein-holography specific. No free parameters, no circular fitting. Math and citations look standard and honest.\n\nSoft spot, in proportion: the spherical (κ=+1) branch rests only on the §3.1 WKB argument. For the usual blackening function there is a bulk turning point where f=1, the paper’s own validity measure diverges, and the near-boundary solutions become modified Bessel rather than ordinary J/Y. No Airy connection is supplied, and neither the hyperbolic exact result nor the planar checks exercise this regime. Analytic continuation probably still yields the same exponent (overall factors cancel in the ratio), but that is an assumption, not a demonstration. Planar and hyperbolic claims do not share this hole.\n\nThis is for people working holographic thermal correlators, light-cone emission rates, or finite-T OPEs. Worth a serious referee. I would engage; the planar/hyperbolic core is already usable.","headline":"Solid multi-method result on horizon-dependent light-like scaling; planar and hyperbolic are locked down, spherical WKB needs a connection formula the paper never writes.","tokens_in":38152,"tokens_out":495,"would_cite":true,"duration_ms":17870,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Retarded correlators of holographic CFTs scale anomalously on the light cone, with the power fixed by the curvature of the dual black-hole horizon.","keywords":["retarded correlators","light-like momenta","AdS/CFT","black-hole horizon","anomalous scaling","WKB approximation","thermal CFT","stress tensor"],"falsifier":"Evaluate the exact retarded correlator for a holographic CFT whose bulk equation is solvable (BTZ, hyperbolic black hole, or large-d planar black hole) at large light-like frequency and check whether the measured power equals 2(2Δ−d)/(d+2) or (2Δ−d)/2 according to horizon topology.","tokens_in":37695,"feed_emoji":"⚫","tokens_out":946,"duration_ms":32015,"temperature":0.7,"pith_summary":"At finite temperature the retarded two-point function of a scalar primary normally grows at large frequency like the zero-temperature power fixed by dimensional analysis. On the light cone that leading contribution vanishes, and the paper shows that the surviving growth is weaker and depends on the horizon topology of the dual black hole. For planar horizons the correlator scales as ω to the power 2(2Δ−d)/(d+2); for spherical or hyperbolic horizons (when frequency squared equals the Casimir on the horizon) it scales as ω to the power (2Δ−d)/2. The claim is established by exact BTZ and hyperbolic-black-hole solutions, numerical Heun analysis in AdS5, WKB matching in general dimension, and large-d checks, and is extended to all three channels of the stress-tensor correlator. A reader cares because the anomalous exponents are universal short-distance data that remain after the identity OPE term drops out and that directly encode the infrared geometry of the dual black hole.","feed_headline":"Horizon curvature rewrites light-cone correlator powers","feed_subtitle":"Holographic CFTs grow more slowly once frequency equals momentum, with the exponent set by planar versus curved black holes.","key_machinery":"WKB transport of the ingoing horizon solution across the bulk, which carries frequency only as an overall phase; the retarded Green’s function is then fixed solely by the order of the near-boundary Bessel functions whose argument is set by the first non-constant term in the blackening factor.","core_discovery":"In AdS/CFT the large-frequency light-like retarded correlator of a scalar primary of dimension Δ grows as ω^{2(2Δ−d)/(d+2)} for planar black holes and as ω^{(2Δ−d)/2} for spherical or hyperbolic black holes (when ω^{2} equals the horizon Casimir), rather than the naive dimensional-analysis power ω^{2Δ−d} that holds at generic momenta.","pith_inferences":["The light-cone thermal OPE must contain a specific tower of operators whose coefficients reproduce the reduced powers once the identity contribution cancels; extracting those coefficients would give a pure CFT derivation of the horizon dependence.","The same WKB matching applies to charged or higher-derivative black holes whenever the near-boundary blackening expansion is unchanged, predicting charge-independent exponents for d>4.","Position-space light-cone singularities previously linked to bulk null geodesics are the natural Fourier dual of these anomalous momentum-space powers."],"forward_implications":["On-shell photon and dilepton emission rates at strong coupling acquire anomalous frequency dependence set by horizon curvature.","Stress-tensor correlators inherit light-cone exponents 2d/(d+2), 4d/(d+2) and 6d/(d+2) in the scalar, shear and sound channels.","Large-d planar correlators reduce exactly to zero-momentum BTZ correlators, giving a dimensional-reduction check of the same exponents.","Any CFT dual to an asymptotically AdS black hole of the stated topology must exhibit these reduced powers once momenta equal frequency."],"fun_headline_variants":["Horizon curvature sets anomalous light-like correlator scaling","Light-like retarded correlators track black-hole horizon shape","Planar horizons yield slower light-cone correlator growth","Curved horizons alter light-like CFT correlator exponents","AdS horizon geometry rewrites scalar correlator powers on light cone"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"The WKB solution carries the horizon boundary condition to the boundary only as a phase, with no amplitude mixing from turning points or sub-dominant potential terms.","fun_headline_variants_meta":{"raw":{"variants":["Horizon curvature sets anomalous light-like correlator scaling","Light-like retarded correlators track black-hole horizon shape","Planar horizons yield slower light-cone correlator growth","Curved horizons alter light-like CFT correlator exponents","AdS horizon geometry rewrites scalar correlator powers on light cone"]},"model":"grok-4.5","effort":"low","cost_usd":0.005053,"raw_usage":{"total_tokens":1452,"prompt_tokens":806,"num_sources_used":0,"completion_tokens":67,"cost_in_usd_ticks":50528000,"prompt_tokens_details":{"text_tokens":806,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":579,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":806,"tokens_out":67,"duration_ms":9478,"temperature":1.0,"reasoning_tokens":579,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T20:08:52.172250+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Evaluate the exact retarded correlator for a holographic CFT whose bulk equation is solvable (BTZ, hyperbolic black hole, or large-d planar black hole) at large light-like frequency and check whether the measured power equals 2(2Δ−d)/(d+2) or (2Δ−d)/2 according to horizon topology.","supporting_citations":[],"review_version":1}