{"id":"d7952c33-a13c-4574-8cc9-840ad0567ac6","arxiv_id":"2607.24909","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.5,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"OPE and QNM representations of the mixed retarded correlator overlap in complex time, giving an explicit map, sum rules, and new QNM asymptotics for large-N thermal CFTs.","lead":"Thermal OPE data and quasinormal-mode data of large-N CFTs are shown to determine each other through an overlapping region of the retarded correlator in mixed time-momentum coordinates. The map yields new QNM asymptotics, sum rules, and a concrete thermal-bootstrap programme.","discovery_kind":"unification","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"The overlap equation (2.44) — the sole basis for the claimed one-to-one OPE↔QNM map — is established only for holographic and Gaussian large-N models; the paper's own App. B.2 shows non-meromorphy at the Wilson-Fisher fixed point, so the advertised \"large-N CFT\" scope exceeds what is demonstrated.","rationale":"Good-faith read: within the regime where the working hypotheses hold, the paper is internally solid. The Mellin–OPE pole relation (5.8) follows rigorously from small-t data alone (the t>t_c tail of the Mellin integral is entire, so OPE poles are determined by the disk expansion); the rigidity argument (5.4) is a correct finite Vandermonde statement; the zeta/Hurwitz-subtracted sum rules of §5.5 are numerically self-consistent and the residuals in (5.63) decrease with cutoff as advertised; the SAdS5 tail formula (4.26) and Padé+Prony extraction are validated against independent bulk numerics (Tables 1–2, Fig. 7) with residuals scaling as the predicted next order n^{-8/3} — genuine independent support, not circularity, since OPE coefficients enter as boundary data. The lightcone/collider section (§6) is presented at scaling level and matches known WKB results. So the load-bearing soft spot is not internal consistency but scope: condition (i)+(ii)+(iii) for (2.44) is, by the paper's own framing in §2.2–2.3 and App. B.2, a \"generic large-N working hypothesis\" that is in fact established only holographically or in Gaussian models, and is demonstrably false at the interacting Wilson-Fisher fixed point. This is exactly the reader's identified weakest assumption, hence agree. Because the paper is honest about the assumption, clearly delimits where it is proven, and the concern is about generality rather than correctness of the examples, the concern does not warrant lowering correctness risk or changing the verdict: CONDITIONAL stands, with the proposed SYK overlap test (feasible with existing tools from [12,19,29]) as the single most informative way to either extend the theorem beyond holography or bound its domain.","tokens_in":50153,"tokens_out":7056,"duration_ms":259248,"concrete_test":"Settle the overlap hypothesis in large-q SYK, the one non-holographic, non-Gaussian model with analytic control of both sides: (a) extract the QNM poles/residues of G_R(t) from the ladder-kernel eigenvalues as in [12,19] and numerically/analytically determine the convergence region of the QNM sum in the complex t plane; (b) independently compute the OPE radius tc from the nearest complex-time singularity of the Euclidean conformal/Schwarzian solution (cf. [29]); (c) check whether the QNM convergence region contains an open segment of the positive real axis with |t|<tc, and whether G_R(t) is genuinely cut-free. If the QNM sum diverges anywhere on Re t>0 inside the OPE disk, or non-bouncing singularities shrink the disk, the claimed generality of (2.44) must be restricted to holographic-like theories; if overlap holds cleanly, the framework's key assumption gains its first non-holographic,","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (nonempty open overlap, Eq. 2.44, making b_Δ̂,k and {r_n,ω_n} interchangeable) requires three conditions simultaneously: (i) G_R(ω,k) meromorphic with simple poles; (ii) the QNM residue sum convergent in a region abutting t=0 on the real axis (right half-plane or wedge); (iii) the only singularities limiting the OPE disk being bouncing-type, so the disk and the QNM region actually intersect. All three are verified only where a holographic product formula [36] or free/mean-field large-N structure exists (BTZ, R-current, SAdS5, large-N O(N); SYK cited from [12]). Meanwhile the paper's own App. B.2 shows that at the Wilson-Fisher fixed point the correlator has branch points at ω=±k already at O(ε), so the discrete QNM representation (1.2) — and hence (2.44) — fails precisely in the generic interacting non-holographic large-N regime (in vector models, damping itself only appears at subleading 1/N, alongside the cuts). Thus the abstract-level scope is broader than the established scope. A secondary soft point: the reverse direction (OPE→QNM, the \"one-to-one\" claim) is constructive only via Padé+Prony numerics or the Carlson completion of §5.6, whose uniqueness needs a type<π growth bound tied to extremal QNM angles; the suggestion that bouncing-singularity locations (large-order OPE data) can supply this non-circularly is demonstrated only for the R-current. Neither gap touches the examples, which check out (Tables 1–2, Fig. 7, §5.5); the concern is the domain over which (2.44) is a theorem rather than a well-tested ansatz.","agreement_with_reader":"agree"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper studies the retarded thermal two-point function G_R(t,k) of large-N CFTs in mixed time/momentum variables. Its central claim is that the thermal OPE expansion (convergent for 0<|t|<t_c) and the quasinormal-mode sum (convergent for Re t>0 or in a wedge) share a nonempty open overlap in the complex t-plane (Eq. (2.44)), putting OPE data b_Δ̂,k and QNM data {r_n,ω_n} in one-to-one correspondence. Three constructive uses are developed: (1) Padé continuation of the OPE beyond t_c plus Prony fitting to extract low-lying QNMs of the Schwarzschild-AdS5 brane (Tables 1–2); (2) matching at the t=0 and first bouncing singularity to derive analytic large-n QNM asymptotics, including a new n^{−7/3} coefficient ((4.23)–(4.26), Fig. 7); (3) Mellin-space sum rules ((5.9)–(5.10)), with a rigidity result for the spectrum (§5.4), a convergent subtraction scheme tested numerically (§5.5), and a Carlson-theorem uniqueness argument for Mellin completion (§5.6). Supporting examples are worked out in BTZ, R-currents in N=4 SYM, the large-N O(N) model, and an ε-expansion counterexample (App. B.2). Section 6 relates large-k lightcone modes to conformal collider bounds and argues for slower thermalisation near saturation.","tokens_in":50577,"tokens_out":2758,"duration_ms":49674,"significance":"If the claims hold within their demonstrated domain (holographic and mean-field large-N theories), the paper is a significant contribution to the finite-temperature bootstrap: (i) new analytic QNM asymptotics — the n^{−7/3} coefficient in (4.25) and the n′-resummed tail (4.26) are new, confirmed numerically to the expected order in Fig. 7; (ii) a new set of exact constraints on holographic QNM data (the Mellin-space residue formula (5.9) and double-trace sum rules (5.10)), with a rigidity theorem (§5.4) and a convergent asymptotic-subtraction implementation tested quantitatively in (5.62)–(5.63); (iii) an explicit, quantitatively verified extraction of low-lying QNMs of the Schwarzschild-AdS5 brane from OPE data alone (Tables 1–2), with two complementary methods converging from opposite ends of the spectrum; (iv) a falsifiable application of the framework in §5.2, where a toy spectrum satisfying all sum rules is constructed and then ruled out by Ward identities — a genuinely bootstrap-style argument; and (v) a new lightcone connection between conformal collider bounds and thermalisation timescales of stress-tensor correlators (6.7), checked against Gauss-Bonnet WKB. The thermal Mel","major_comments":[{"comment":"The abstract and §1 state the result for 'large-N CFTs' generically, but the overlap equation (2.44) rests on three simultaneous conditions: (i) meromorphy of G_R(ω,k) with simple poles (assumed in §2.2), (ii) convergence of the QNM residue sum in a region abutting t=0 (verified explicitly only for BTZ §2.3, the R-current (2.29)–(2.35), the holographic product formula of [36], and the large-N O(N) model App. B.1), and (iii) bouncing-type singularities as the only obstructions to the OPE disk. The manuscript's own App. B.2 shows that at the Wilson-Fisher fixed point the correlator develops branch points at ω=±k already at O(ε) (Eq. (B.21)), so the discrete QNM representation (1.2) — and hence (2.44) — fails in the generic interacting non-holographic large-N regime (in vector models, damping and cuts both appear at subleading 1/N). The closing sentence of App. B.2 ('the general relation is","section":"§2.2–2.3, App. B.2, Abstract"},{"comment":"The reverse direction of the claimed one-to-one correspondence (OPE→QNM) is constructive in the manuscript only through (a) Padé + Prony numerics on a truncated OPE (§3, Table 2), or (b) the Carlson completion of §5.6. For (b), uniqueness requires the normalised interpolation F(w) to have exponential type < π in the left half-plane, and the paper ties this growth bound to the extremal QNM angles (text after (5.91)). The suggestion that bouncing-singularity locations — i.e., large-order OPE data — can supply this bound non-circularly is demonstrated only for the R-current, where the full correlator is already known in closed form. As written, §5.6 does not establish that the Carlson-class condition can be verified from OPE data alone in a generic case, so the 'one-to-one' claim in §1 (and the statement that 'the defining set of data for the OPE is in one-to-one correspondence with the def","section":"§5.6, §1"}],"minor_comments":[{"comment":"The ansatz (4.7) mentions δ2 ('0 < δ2 − δ1 < 1') but δ2 is never defined or used; presumably only δ1 is intended.","section":"§4, Eq. (4.7)"},{"comment":"The residuals of the raw zeta-regularised sum rules grow with truncation order for q=0 (−5.6×10⁻² → −7.8×10⁻²), which looks counterintuitive before the subtracted scheme (5.57) is introduced. A sentence explaining why including more asymptotic terms without subtraction need not improve the moment would help the reader.","section":"§5.5, table (5.51)"},{"comment":"Table 3 lists higher-order coefficients d_{8/3}, d_{11/3}, d_4, d_5 but gives no independent numerical cross-check (unlike Fig. 7 for the lower orders). If these are extracted by fitting, the fitting procedure and error estimate should be stated; if they are analytic predictions, the agreement criterion should be given.","section":"§4.3, Table 3"},{"comment":"Footnote 6: 'This is is most clearly seen' — duplicated word.","section":"§2.3, footnote 6"},{"comment":"The step from (2.37) to (2.38), discarding 'all perturbative terms at large ω while keeping all the non-perturbative terms', is load-bearing for the wedge-convergence picture but is justified only by citation to [19]. A brief explanation of what is dropped and why the singularity locations are unaffected would make §2.3 more self-contained.","section":"§2.3, Eqs. (2.37)–(2.38)"},{"comment":"In (1.6) the sentence 'Then −7/3 prediction for the QNMs is new' appears garbled; presumably 'The n^{−7/3} prediction'.","section":"§1, Eq. (1.6)"},{"comment":"Notation: ℓ and l are mixed in §2.1 ('taking equally many values for l = 0,1,...'); also the normalisation N is redefined between (3.4) and (4.2) without an explicit flag at the point of redefinition.","section":"§2.1, §4"},{"comment":"Reference [45] is listed with a trailing space in the title and no journal/arxiv identifier; please complete the entry.","section":"References"}],"recommendation":"minor_revision","confidential_remarks":"The results are strong, but the framing in the abstract/introduction claims a generality (\"large-N CFTs\") that the manuscript's own Appendix B.2 partially undercuts. The load-bearing examples are all holographic or mean-field; I recommend the authors be asked to state the precise class of correlators for which (2.44) is established. This is a framing/scope revision, not a correctness problem in the worked examples. Related independent work [45] appears simultaneously; the note added is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The useful core is simple: GR(t,k) has an OPE disk and a QNM wedge that actually overlap, so you can match data both ways. That is not just a slogan. They turn it into Mellin residues, double-trace sum rules, a reorganised large-n tail (including a new n^{-7/3} term), Padé+Prony extraction of low AdS5 modes from OPE coefficients, and a lightcone argument that stress-tensor channels thermalise slower as collider bounds approach saturation.\n\nWhat works: the identities are clean (discontinuity → mixed OPE, Mellin spectral zeta, matching at t=0 and the first bounce). BTZ, R-current, large-N O(N), and Schwarzschild-AdS5 all check out; Tables 1–2 and Fig. 7 are quantitative, not hand-waving. The n' reorganisation of the asymptotic series is a genuine simplification. Circularity is mild: bulk OPE input is an oracle for consistency, not the definition of the map. Simultaneous related work is disclosed.\n\nSoft spot, in proportion: the abstract’s “large-N CFTs” is broader than what is proved. Overlap plus meromorphic simple-pole QNMs is established for holography and free/mean-field large-N. Their own App. B.2 shows Wilson-Fisher already has branch points at O(ε), so the discrete QNM sum (and thus Eq. 2.44 as a theorem) fails in generic interacting non-holographic large-N. That does not sink the examples or the bootstrap programme they sketch; it bounds the domain. Reverse direction (OPE→full QNM spectrum) is constructive mainly via Padé/Prony or Carlson completion, with the growth bound only fully pinned for the R-current. Neither issue spoils the calculations that are actually done.\n\nWho it is for: people doing thermal bootstrap, holographic QNMs, or large-k thermalisation. Worth a reading-group slot. Send to referees; ask them to tighten the stated scope and the non-holographic caveats, not to reject.","headline":"Solid dictionary between thermal OPE and QNMs in mixed (t,k), with real new asymptotics and AdS5 checks; scope is holographic/free large-N, not generic large-N CFTs.","tokens_in":50801,"tokens_out":558,"would_cite":true,"duration_ms":21681,"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":"Thermal OPE data and quasinormal modes are the same data, linked by an overlapping region of the complex-time plane.","keywords":["thermal OPE","quasinormal modes","retarded correlator","Mellin transform","thermal bootstrap","bouncing singularities","conformal collider bounds","AdS black brane"],"falsifier":"Compute a high-overtone quasinormal frequency of the Schwarzschild-AdS5 black brane by independent bulk numerics and check whether it matches the analytic large-n expansion obtained from the OPE singularity matching (Eq. 4.26) to the predicted O(n^{-7/3}) accuracy.","tokens_in":50462,"feed_emoji":"🔗","tokens_out":1017,"duration_ms":18140,"temperature":0.7,"pith_summary":"In a hot large-N conformal field theory the short-distance response of a two-point function is organised by a thermal operator product expansion, while the long-time response is organised by a sum of damped collective modes called quasinormal modes. This paper shows that the two expansions of the mixed time-and-momentum retarded correlator are valid in overlapping regions of the complex-time plane, so they must agree there. That agreement supplies an explicit dictionary: OPE coefficients become residues of a Mellin transform, while the quasinormal frequencies and residues appear as the spectral data of the same transform. Matching the two sides at singularities yields new asymptotic formulae for high-overtone modes; analytic continuation of a finite OPE sum recovers the lowest modes; and Mellin-space sum rules force every double-trace moment of the quasinormal data to vanish. The same dictionary applied at large spatial momentum implies that stress-tensor correlators ring down more slowly as the conformal collider bounds approach saturation. The result is a concrete thermal bootstrap in which ultraviolet OPE data and infrared quasinormal data constrain each other.","feed_headline":"Thermal OPE and quasinormal modes are the same data","feed_subtitle":"An overlap in complex time turns short-distance coefficients into long-time ringdown frequencies","key_machinery":"The mixed retarded correlator GR(t,k) written once as a thermal OPE series and once as a quasinormal residue sum (Eq. 2.44). Their common domain of convergence, together with the Mellin transform that turns the QNM sum into a spectral zeta function whose poles are the OPE data, carries the entire dictionary.","core_discovery":"The OPE and QNM representations of the mixed retarded correlator GR(t,k) share a nonempty open set of the complex-t plane on which they are identical. Consequently the defining OPE coefficients are in one-to-one correspondence with the quasinormal frequencies and residues, realised concretely by Mellin residues and by an infinite family of double-trace sum rules.","pith_inferences":["Once the overlapping-region map is accepted, any independent bound on OPE coefficients (unitarity, collider bounds, etc.) immediately becomes a bound on allowed quasinormal spectra, and vice versa.","The same complex-time overlap should exist for charged or rotating black branes; the extra purely imaginary modes would only affect the OPE singularity, leaving the bouncing-singularity matching intact.","Non-holographic large-N models with finitely many normal modes (such as the critical O(N) vector model) furnish the simplest solvable points of the bootstrap, where the Mellin transform reduces to a finite exponential polynomial."],"forward_implications":["High-overtone QNM asymptotics, including new fractional powers, are completely fixed by the local expansion of the thermal OPE near its first singularity.","Low-lying QNMs can be extracted from a finite number of OPE coefficients by Padé continuation plus Prony fitting, without solving any bulk wave equation.","An infinite set of Mellin-space sum rules forces every double-trace moment of the QNM data to vanish, rigidly constraining any finite deformation of the spectrum.","At large spatial momentum the light-cone OPE implies that stress-tensor correlators thermalise more slowly as the conformal collider bounds approach saturation.","The same dictionary supplies a practical thermal bootstrap in which ultraviolet OPE data and infrared QNM data mutually constrain each other."],"fun_headline_variants":["OPE and QNM data map via complex-time overlap","Thermal OPE coefficients equal quasinormal residues","Short-distance OPE fixes long-time QNM ringdown","Mellin sum rules tie OPE spectrum to QNM frequencies","Retarded correlator equates OPE with quasinormal modes"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"That the retarded correlator is meromorphic with only simple poles whose residue sum converges throughout a right half-plane or wedge, so the only singularities that limit the OPE disk are bouncing-type singularities.","fun_headline_variants_meta":{"raw":{"variants":["OPE and QNM data map via complex-time overlap","Thermal OPE coefficients equal quasinormal residues","Short-distance OPE fixes long-time QNM ringdown","Mellin sum rules tie OPE spectrum to QNM frequencies","Retarded correlator equates OPE with quasinormal modes"]},"model":"grok-4.5","effort":"low","cost_usd":0.004152,"raw_usage":{"total_tokens":1239,"prompt_tokens":766,"num_sources_used":0,"completion_tokens":87,"cost_in_usd_ticks":41524000,"prompt_tokens_details":{"text_tokens":766,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":386,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":766,"tokens_out":87,"duration_ms":6336,"temperature":1.0,"reasoning_tokens":386,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T06:06:34.773753+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute a high-overtone quasinormal frequency of the Schwarzschild-AdS5 black brane by independent bulk numerics and check whether it matches the analytic large-n expansion obtained from the OPE singularity matching (Eq. 4.26) to the predicted O(n^{-7/3}) accuracy.","supporting_citations":[],"review_version":1}