{"id":"f9e013cb-a87e-4838-a2e2-dff99cdce846","arxiv_id":"1908.02873","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Universal averaged OPE coefficient asymptotics in large-central-charge 2D CFTs extend to Δ>c/6 only under specific sparseness conditions; the C^2_HLL formula requires the stricter ρ(Δ)≲e^{πΔ}, which permutation orbifolds violate.","lead":"This paper studies when universal formulas for combining operators in 2D conformal field theories stay valid at high energies. It finds one such formula needs a stricter sparseness condition on the spectrum, and that symmetric orbifold theories like the free D1-D5 system fail this condition.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3.1's sparseness condition drops the 16^Δ factor: from (3.4), g_L≈g_vac at β=2π requires C^2ρ ≲ e^{(π−4ln2)Δ}, not C^2ρ ≲ e^{πΔ}.","rationale":"The reader's weakest assumption concerned the external [9] assumptions behind C^2_HHL and C_HHL; those are legitimate but not the most concrete problem. The more direct issue is an internal factor error in Section 3.1: the 16^Δ in (3.4) is dropped when deriving (3.9). This is not a disagreement with the holographic intuition or with [9]; it is an algebraic check on the paper's own sufficient condition. The extended validity of (3.8) is the paper's main new constraint, so this error is load-bearing. It may be repairable by strengthening the condition to C^2ρ≲e^{(π−4ln2)Δ} and re-examining whether permutation orbifolds are still excluded, but as written the derivation does not establish the stated result. I therefore keep the reader's CONDITIONAL verdict (UNCHANGED), with a different primary reason than the one the reader identified.","tokens_in":19734,"tokens_out":21496,"duration_ms":245314,"concrete_test":"Re-derive (3.9) directly from (3.4): impose g_L≈g_vac at β=2π for all Δ≤c/12+ε and verify whether the required inequality is C^2_{OOΔ}ρ(Δ)≲e^{(π−4ln2)Δ} rather than e^{πΔ}. A numerical check with a toy sparse spectrum, e.g. ρ(Δ)=e^{πΔ} and C^2_{OOΔ}=1 for Δ≤c/12 at c=100, would show the light pillow sum exceeds the vacuum term by a factor ∼16^{c/12}, demonstrating failure of the stated condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Eq. (3.4) the pillow sum is g(β)=Σ C^2_{OOi}16^{Δ_i}e^{−β/2(Δ_i−c/12)}, with C the plane OPE coefficients, and the stated sufficient condition in Section 3.1 is C^2_{OOΔ}ρ(Δ)≲e^{πΔ}. But approximating g_L by g_vac at the weakest allowed point β=2π gives, relative to the vacuum term, Σ_{Δ≤c/12+ε} C^2_{OOΔ}ρ(Δ)16^Δ e^{−πΔ}. Because all terms are positive, a sufficient bound on each shell is C^2_{OOΔ}ρ(Δ)≲(e^π/16)^Δ ≈ e^{0.369Δ}. The published condition only implies C^2ρ(16/e^π)^Δ ≲16^Δ, which grows like e^{2.77Δ}; it is not small even for Δ∼c/12 in a CFT with ρ≲e^{πΔ}. Thus the inference from (3.9) to 'g≈g_vac' and the extended validity of (3.8) does not follow as written. The same issue affects the abstract's claim about the e^{πΔ} condition and the use of (3.8) in Section 5. If C^2 in (3.9) was intended to be the pillow OPE coefficient, that contradicts the sentence after (3.4) defining C as the plane OPE coefficient.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies extensions of universal asymptotic formulas for averaged OPE coefficients in holographic 2d CFTs from Δ→∞ down to Δ>c/6, adapting the HKS modular-invariance argument to modular-covariant quantities. It derives sufficient conditions on the light spectrum under which the formulas for C^2_HLL, C^2_HHL, C_HHL, and the density of primary states remain valid, discusses obstacles to extending C^2_HHH, and applies the results to vacuum block dominance. The main results are conditional on the HKS sparseness condition and on additional assumptions imported from reference [9].","tokens_in":20085,"tokens_out":10077,"duration_ms":100100,"significance":"If the statements hold, the paper provides a useful framework for constraining OPE data of holographic CFTs and identifies conditions stronger than extended Cardy behavior, with implications for permutation orbifolds. The paper is transparent about its conjectural parts, explicitly labels unproved input, and gives a clear covariant generalization of the HKS argument. However, the derivation of the central C^2_HLL extension in Section 3.1 contains a load-bearing error in the stated sufficient condition; once corrected, the framework may survive, but the current abstract and Section 5 uses of the result are unsupported as written.","major_comments":[{"comment":"The sufficient condition stated in Eq. (3.9) does not imply g ≈ g_vac. From Eq. (3.4), the contribution of a light state of dimension Δ to g at temperature β is C^2_OOΔ 16^Δ e^{-β/2(Δ-c/12)}, where C is the plane OPE coefficient. Comparing this to the vacuum term e^{βc/24} at the weakest point β→2π gives the ratio C^2_OOΔ 16^Δ e^{-πΔ}. Requiring this ratio to be exponentially small for all Δ ≤ c/12 + ε requires C^2_OOΔ ρ(Δ) ≲ e^{-(π - 4 ln 2)Δ} up to subexponential factors, not C^2 ρ ≲ e^{πΔ}. The stated condition only yields C^2 ρ 16^Δ e^{-πΔ} ≲ 16^Δ, which grows like e^{2.77Δ} and is not small relative to the vacuum term. Therefore the inference that (3.8) remains valid for Δ > c/6 does not follow as written. If the author intended C in (3.9) to be the pillow OPE coefficient, that contradicts the sentence after Eq. (3.4) defining C as the plane OPE coefficient. The same issue affects the abstract's claim about the e^{πΔ} condition and the use of (3.8) in Section 5.","section":"Section 3.1, Eq. (3.9)"},{"comment":"The extended validity of (1.2) and (3.31) for Δ > c/6 rests on the 'mild additional assumptions' from [9] listed in footnote 3: factorization of light correlators, subexponential growth of light correlators in medium states, and existence of a large-c expansion of the light contribution to the thermal correlator. These assumptions are not proved in the present manuscript. If any of them fail, the replacement X(β>2π) ≈ X_vac is unjustified and the extended formulas in Sections 3.2 and 3.3 do not follow. The paper should either prove these assumptions or clearly state in the abstract and introduction that the C^2_HHL and C_HHL extensions are conditional on conjectural input from [9] rather than solely on the HKS sparseness condition.","section":"Sections 3.2 and 3.3, footnote 3"}],"minor_comments":[{"comment":"The expression after Eq. (3.39) writes a single summation over k, but the term contains (-1)^{k+k'} and the surrounding text refers to the k,k' plane. This should be a double sum over k and k'.","section":"Section 3.4, Eq. (3.39)"},{"comment":"The sentence 'Since the light OPE coefficients are polynomial in c in large c CFTs' introduces an additional assumption that is load-bearing for the claim that (3.9) is essentially a condition on the density of states; it should be stated explicitly as an assumption.","section":"Section 3.1"},{"comment":"The abstract states that the relevant condition is ρ(Δ) ≲ e^{πΔ}; after repairing the 16^Δ issue, the actual condition is stronger and also involves the OPE coefficients. The abstract should be updated to reflect the corrected condition.","section":"Abstract"},{"comment":"The phrase 'subexponential growth of light correlators in medium states' would benefit from a precise definition of 'medium states' and of the growth rate being bounded.","section":"Footnote 3"},{"comment":"The claim that 'preliminary numerics [28] suggest that this is indeed the case' is supported by a private communication; the authors should either include a plot or a more detailed statement, or soften the claim.","section":"Section 4.2"}],"recommendation":"major_revision","confidential_remarks":"The main issue in Section 3.1 is fixable but it is central: the stated condition for the extended validity of (3.8) is incorrect. The paper's conditional use of reference [9] is acceptable if clearly flagged, but the abstract currently overstates the rigor of the C^2_HHL and C_HHL extensions. I recommend major revision rather than rejection, because the framework is potentially useful and the flaws are local rather than fatal to the overall approach."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nWhat you should know: this paper adapts the HKS modular argument to modular-covariant quantities, which is a genuinely useful move, and it gives clean conditional derivations for C^2_HHL, C_HHL, and the primary density. But its marquee new result—the e^{πΔ} sparseness condition for C^2_HLL—is wrong as written. The condition needs an extra 16^{-Δ} factor.\n\nThe setup is (3.4): g(β)=Σ C^2_{OOΔ}ρ(Δ)16^Δ e^{-β/2(Δ-c/12)}. To have g_L≈g_vac at β=2π, a light shell's contribution relative to vacuum is C^2ρ(16/e^π)^Δ. A sufficient bound is C^2ρ ≲ (e^π/16)^Δ ≈ e^{0.369Δ}. The paper instead claims C^2ρ≲e^{πΔ}. For Δ near c/12 that permits contributions that are not suppressed at all; they grow like 16^Δ relative to the vacuum term. So the inference from (3.9) to (3.8) holding for Δ>c/6 does not go through. The fix is likely to replace e^{πΔ} with e^{(π-4ln2)Δ}, which still excludes permutation orbifolds, but the abstract and Section 3.1 need revision.\n\nThere's also a related slip in (3.8): K/ρ is written as ≈ e^{-π√(c/3)(Δ-c/12)} ≈ 16^{-Δ}e^{-S_BH/2}. Those two right-hand sides are not leading-exponential equivalent in the extended regime; the 16^{-Δ} is an honest factor that should not appear and disappear.\n\nWhat holds up: the covariant HKS framework in Section 2 is a nice generalization, and the treatment of C^2_HHL and C_HHL is appropriately conditional on the unproved assumptions of [9]. The author is transparent that the C^2_HHH extension is conjectural and that the vacuum block section is incomplete. Section 3.4's derivation for primary density is solid.\n\nThe main soft spots beyond the C^2_HLL error: (i) the extended C^2_HHL and C_HHL formulas depend on assumptions (factorization, subexponential light correlators in medium states, large-c light thermal correlator) that are stated but not proven; if any fail, X≈X_vac is unjustified. (ii) Section 5 stacks several unstated assumptions (statistical independence of OPE factors, typicality of OH) before concluding vacuum block dominance; it reads as a roadmap rather than a proof.\n\nShould a serious editor send this to referees? Yes. The framework and the conditional results are worth refereeing, and the sparseness error is concrete and fixable. But I would not cite the paper as it stands. The audience is people working on AdS3/CFT2 bootstrap and OPE asymptotics; they'll be interested, but they should read Section 3.1 with a calculator.","headline":"The covariant HKS framework is useful, but the paper's headline sparseness condition for C^2_HLL loses the 16^Δ factor and is too weak by an exponential amount.","tokens_in":20625,"tokens_out":11823,"would_cite":false,"duration_ms":110062,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["11.25.Hf","04.70.Dy"],"model":"deepseek-v4-flash","headline":"Universal formulas for averaged OPE coefficients of 2d holographic CFTs, normally valid at infinite dimension, survive down to $\\Delta>c/6$ in sparse large-$c$ theories, under conditions that exclude permutation orbifolds.","keywords":["2d conformal field theory","holographic duality","OPE coefficients","modular covariance","black hole entropy","sparseness condition","vacuum block dominance","permutation orbifolds"],"falsifier":"Compute the thermal two-point function $X(\\beta,t)$ for a large-$c$ CFT that satisfies $\\rho(\\Delta)\\lesssim e^{2\\pi\\Delta}$ but contains light multitrace operators, at $\\beta$ just above $2\\pi$; if $X$ deviates from the vacuum sum by more than a subexponential factor, the claim that $C^2_{HHL}$ extends to $\\Delta\\sim c$ collapses. A complementary check is to read off the light density of states of a permutation orbifold such as the free D1-D5 CFT: the paper predicts it exceeds $e^{\\pi\\Delta}$, which would exclude it from the $C^2_{HLL}$ extension.","tokens_in":19483,"feed_emoji":"🕳️","tokens_out":19716,"duration_ms":175470,"temperature":0.7,"pith_summary":"At the core of this paper is a simple question: do the universal asymptotic formulas for averaged operator-product-expansion (OPE) coefficients of two-dimensional conformal field theories, derived in the limit of infinite operator dimension, remain valid when the dimension is only of order the central charge $c$? The author adapts the modular-covariance argument that extends the density-of-states formula, and finds a conditional yes in the extended regime $\\Delta>c/6$. Heavy-light-light coefficients stay universal if the light spectrum obeys the stricter bound $\\rho(\\Delta)\\lesssim e^{\\pi\\Delta}$; heavy-heavy-light and one-point averages require only the standard light-sparseness condition together with a set of stated factorization assumptions about light thermal correlators. These conditions exclude permutation orbifolds such as the free D1-D5 CFT, and the results point to new bounds on non-vacuum block contributions in holographic theories.","feed_headline":"Sparse CFTs keep universal OPE asymptotics for Δ>c/6","feed_subtitle":"A modular-invariance argument extends universal OPE asymptotics down to the black-hole threshold.","key_machinery":"The argument runs on a single modular-covariant object: a spectral sum $X(\\beta)=\\sum_i C_i e^{-\\beta(\\Delta_i-c_0)}$ with positive coefficients, transforming as $X(\\beta)=(\\beta/2\\pi)^w X(4\\pi^2/\\beta)$. Splitting $X$ into light and heavy parts and bounding the heavy part by $rX'_H$ with $r=e^{(\\beta'-\\beta)\\epsilon}(\\beta/\\beta')^{w/2}$ shows that at $\\beta>2\\pi$ the quantity is approximated by its light contribution; if that light contribution is in turn close to the vacuum term, an inverse Laplace transform yields the universal asymptotic spectral density. Each OPE coefficient gets its own $X$: the pillow four-point function for $C^2_{HLL}$, the torus two-point function for $C^2_{HHL}$, and the torus one-point function for $C_{HHL}$. The heavy-heavy-heavy case would need the genus-two partition function, whose modular transformation is not known; the paper treats it under two explicit conjectures. The primary-density extension uses the pentagonal number identity to resum eta-function phases, converting the spectral transform into a Bessel-function sum dominated by its zero mode.","core_discovery":"The paper claims that four universal asymptotic quantities of large-$c$ 2d CFTs — the squared heavy-light-light OPE coefficient average $C^2_{HLL}\\approx 16^{-\\Delta}e^{-S_{BH}(\\Delta)/2}$, the heavy-heavy-light average $C^2_{HHL}\\approx e^{-S_{BH}(\\Delta)}$, the heavy-light-heavy one-point average $C_{HHL}\\approx C_{\\chi O\\chi} e^{-2\\pi\\Delta_\\chi\\sqrt{12\\Delta/c-1}}$, and the density of primary states $\\rho_p(\\Delta)\\approx e^{2\\pi\\sqrt{(c-1)/3}(\\Delta-(c-1)/12)}$ — remain valid for every $\\Delta>c/6$ under stated sparseness conditions. For $C^2_{HLL}$ the condition is $\\rho(\\Delta)\\lesssim e^{\\pi\\Delta}$ for $\\Delta<c/12+\\epsilon$, stronger than the standard $e^{2\\pi\\Delta}$ bound and violated by permutation orbifolds including the free D1-D5 CFT. For $C^2_{HHL}$ and $C_{HHL}$ the standard bound suffices provided the light thermal correlator factorizes and grows subexponentially in medium states (footnote 3). The density of primary states extension is proven modulo standard sparseness using the pentagonal number theorem. The motivation for expecting the extension is the thermodynamic stability of large AdS$_3$ black holes, whose entropy must match the universal entropy formula for $\\Delta>c/6$.","pith_inferences":["If the strict light-spectrum condition $\\rho(\\Delta)\\lesssim e^{\\pi\\Delta}$ is genuinely necessary for the heavy-light-light extension, it would provide a sharper holography-versus-orbifold diagnostic than the density-of-states bound alone, one that distinguishes weakly coupled symmetric-product theories from genuine semiclassical bulk duals.","The unproven factorization assumptions of footnote 3 are the main fragility of the $C^2_{HHL}$ and $C_{HHL}$ extensions; a natural numerical test would compute torus two-point functions in symmetric orbifold CFTs at large $N$ and check whether $X(\\beta>2\\pi)$ stays within a subexponential factor of the vacuum sum.","If the block $H$-functions of the conformal block recursion exponentiate when $h\\sim c$, the same extended formulas should hold for averages over primary states with the $c\\to c-1$ shift, unifying the extended density formula, OPE asymptotics, and primary density under one modular-covariant principle.","The vacuum-block-dominance analysis suggests a testable bootstrap constraint: non-vacuum block contributions in heavy-light correlators are exponentially suppressed at generic cross-ratio, with a sharp transition near $z\\to 0,\\infty$; this could serve as a working definition of a holographic CFT."],"forward_implications":["In any large-$c$ 2d CFT whose light spectrum satisfies $\\rho(\\Delta)\\lesssim e^{\\pi\\Delta}$, the averaged heavy-light-light OPE coefficient $C^2_{HLL}\\approx 16^{-\\Delta}e^{-S_{BH}(\\Delta)/2}$ holds for all $\\Delta>c/6$, not only in the $\\Delta\\to\\infty$ limit.","Under the standard light-sparseness bound $\\rho(\\Delta)\\lesssim e^{2\\pi\\Delta}$ plus the footnote-3 factorization assumptions, the heavy-heavy-light average $C^2_{HHL}\\approx e^{-S_{BH}(\\Delta)}$ and the one-point average $C_{HHL}$ remain valid down to $\\Delta>c/6$.","The density of primary states follows the universal density-of-states formula with $c\\to c-1$ for all $\\Delta>c/6$ in sparse large-$c$ theories, via the pentagonal number identity.","Permutation orbifolds, including the free D1-D5 CFT, violate the stricter light-spectrum bound, so the $C^2_{HLL}$ extension does not apply to them even though their density of states may be sparse in the weaker sense.","Assuming the conjectured OPE-density and block properties, heavy-light four-point functions are dominated by the vacuum conformal block except when the light operator approaches the singular points $z\\to 0,\\infty$."],"supporting_citations":[{"why":"Supplies the light-sparseness modular-invariance argument that the paper generalizes from the partition function to modular-covariant correlators.","marker":"[2]"},{"why":"Gives the heavy-light-light OPE coefficient asymptotic from the pillow four-point function, whose extension to $\\Delta>c/6$ the paper studies.","marker":"[3]"},{"why":"Derives the heavy-heavy-light OPE coefficient asymptotic from the torus two-point function and its black-hole emission interpretation.","marker":"[4]"},{"why":"Provides the torus one-point function formula for $C_{HHL}$ and the density of primary states with the $c\\to c-1$ shift.","marker":"[6]"},{"why":"Establishes that at $\\beta>2\\pi$ the thermal correlator is dominated by vacuum data under stated assumptions, the key input for extending $C^2_{HHL}$ and $C_{HHL}$.","marker":"[9]"},{"why":"Gives the thermodynamic stability of large AdS$_3$ black holes that motivates expecting the extended regime in holographic theories.","marker":"[8]"},{"why":"Obtains the genus-two heavy-heavy-heavy OPE asymptotic that the paper discusses in section 4.1 under conjectural extensions.","marker":"[5]"}],"fun_headline_variants":["OPE universality holds down to Δ>c/6 in sparse CFTs","Sparseness extends OPE universality to black-hole threshold","Universal OPE coefficients survive strong sparseness conditions","New constraints from OPE universality in holographic 2d CFTs","Sparse holographic CFTs keep OPE asymptotics near Δ ~ c"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole extension for heavy-heavy-light and one-point averages rests on the unproven assumption that at low temperature ($\\beta>2\\pi$) the light degrees of freedom are all that matter for the thermal correlator, with heavy states contributing only tiny corrections; if that fails, the extended formulas (3.18) and (3.31) do not follow.","fun_headline_variants_meta":{"raw":{"variants":["OPE universality holds down to Δ>c/6 in sparse CFTs","Sparseness extends OPE universality to black-hole threshold","Universal OPE coefficients survive strong sparseness conditions","New constraints from OPE universality in holographic 2d CFTs","Sparse holographic CFTs keep OPE asymptotics near Δ ~ c"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000481,"raw_usage":{"total_tokens":2409,"prompt_tokens":1005,"completion_tokens":1404,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":621,"completion_tokens_details":{"reasoning_tokens":1307}},"tokens_in":621,"tokens_out":1404,"duration_ms":10751,"temperature":1.0,"reasoning_tokens":1307,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:32:36.356763+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the thermal two-point function $X(\\beta,t)$ for a large-$c$ CFT that satisfies $\\rho(\\Delta)\\lesssim e^{2\\pi\\Delta}$ but contains light multitrace operators, at $\\beta$ just above $2\\pi$; if $X$ deviates from the vacuum sum by more than a subexponential factor, the claim that $C^2_{HHL}$ extends to $\\Delta\\sim c$ collapses. A complementary check is to read off the light density of states of a permutation orbifold such as the free D1-D5 CFT: the paper predicts it exceeds $e^{\\pi\\Delta}$, which would exclude it from the $C^2_{HLL}$ extension.","supporting_citations":[],"review_version":1}