{"id":"43fbc159-ce79-4435-8ab2-03c237378b8f","arxiv_id":"2611.03347","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Finite-c quantum corrections convert sharp complex-time singularity imprints of the black hole interior into smooth Gaussian profiles, while leaving short-distance, OPE-protected singularities intact.","lead":"This paper studies how quantum gravity corrections smooth out the sharp signals that black hole singularities leave in holographic correlation functions. It shows that curved quasinormal-mode trajectories and thermal energy fluctuations both replace those sharp signals with narrow smooth profiles.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"The smoothing mechanism rests on an unproven meromorphic-dominance assumption; the Liouville check analyzes the DOZZ coefficient rather than the full continuum spectral integral, so it does not test the suppression of non-meromorphic remainder.","rationale":"The reader's conditional verdict is appropriate. The paper is explicit that meromorphic dominance and the quadratic QNM deformation are assumptions, and it does not claim to have proven them for a conventional holographic CFT. However, the assumption is load-bearing: without G_mero dominance, the pole-trajectory smoothing is not a statement about the actual correlator, and the O(1/c) loop-level non-meromorphic remainder can be of the same size as the QNM deformation that produces the Gaussian width. My concrete test targets the one place where an exact finite-c calculation exists, and where the presented evidence stops short of the full object: the Liouville analysis analyzes the DOZZ coefficient, not the continuum integral that defines the full HHLL correlator. If the full integral still produces a cut at the order of the claimed smoothing effect, the Liouville example should be interpreted as a toy model of the spectral weight rather than as evidence for the generic mechanism. This reinforces the reader's conditional verdict rather than changing it; no new objection beyond the already-identified weakest assumption is needed.","tokens_in":57127,"tokens_out":12066,"duration_ms":95276,"concrete_test":"Compute in Liouville CFT the full leading-primary contribution to the heavy-state correlator G_H(ω) at finite c by performing the P-integral in (4.9) with the semiclassical block V ≈ z^{h3−hH−hL}, Fourier transforming in cylinder time, and analytically continuing in ω. Then compare, for 1 << β_H ω << c, the full result with the pole-product expression (4.15). If the difference is not parametrically smaller than the O(1/c) Gaussian correction of §3.2, or if a branch cut at the threshold ω = −2κ² survives, the meromorphic-dominance assumption (3.10) is not supported by the paper's exact example, and the generic smoothing claim needs a different check.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.1 decomposes the finite-c correlator as G = G_mero + R and assumes |R| << |G_mero| in the window 1 << β|ω| << c (eqs. 3.8–3.10). This assumption carries the smoothing result: the Gaussian/Stokes argument in §3.2 is a statement about the meromorphic Hadamard product only. The exact finite-c object is not manifestly meromorphic: eq. (3.7) is a sum of δ-functions on the real-frequency axis, so the analytic continuation defining G_mero requires a coarse-graining prescription; the paper gives plausibility arguments but no estimate of R in the window. The two suppression mechanisms cited are not sufficient: O(c) threshold separation removes some cuts, while bulk-loop branch cuts are O(G_N) = O(1/c), which is exactly the order of the quadratic QNM coefficient a2/c that generates the Gaussian width. Thus R can compete with the deformation claimed to smooth the singularity. The one exact finite-c example does not close this gap: in Liouville, the full HHLL correlator is an integral over the continuous primary spectrum (4.9), while the pole analysis in §4.2 replaces the correlator by the DOZZ product (4.15) evaluated on a pole branch. The continuum integral generically produces threshold cuts, e.g., at the physical spectral edge ω = -2κ² (eq. 4.40), which are non-meromorphic contributions not included in the pole-product calculation. Hence the Liouville demonstration establishes a pole trajectory of the spectral weight, not the meromorphy of the full correlator in the intermediate window.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"This paper analyzes the fate of the bouncing-geodesic complex-time singularities of holographic thermal correlators when the central charge c (equivalently G_N^{-1}) is taken finite. In the strict semiclassical limit, the asymptotically linear QNM spectrum, combined with the thermal product formula and a Gamma-function/Stokes analysis, generates an infinite lattice of complex-time singularities displaced by the bouncing time T_crt (eqs. (2.85)-(2.93)). The central claim, stated as b631 in §3, is that finite-c deformations of the QNM trajectories — parameterized by a polynomial overtone trajectory with a leading quadratic O(1/c) term (eqs. (3.24)-(3.25)) — produce a Gaussian suppression exp(-κω²) in frequency space that smooths the would-be singularities into profiles of width O(β/√c) (eqs. (3.36)-(3.43)), while OPE/KMS-protected seed singularities remain intact (§3.4). The mechanism is illustrated in an exactly solvable quadratic QNM model (§3.3) whose complex-contour effective-field-theory prescription is validated against the exact result (eqs. (3.59)-(3.62)), and in Liouville CFT, where the DOZZ pole trajectory exhibits an exact parabolic deformation of the semiclassical BTZ QNM spectrum (eqs. (4.27)-(4.34)). A complementary ETH-based Gaussian suppression from thermal energy fluctuations is derived in §5 (eqs. (5.19)-(5.21)).","tokens_in":57629,"tokens_out":18742,"duration_ms":119808,"significance":"The paper is a valuable proof-of-mechanism for a concrete, falsifiable proposal: quadratic bending of QNM pole trajectories in frequency space smooths the complex-time singularities associated with bouncing geodesics. The derivation chain is explicit and internally consistent: the Gamma-function and Hadamard manipulations in §2.4 and §3.2 check out, the exact quadratic model in §3.3 is solved in closed form with a verified exact-versus-EFT match (eq. (3.62)), and the Liouville DOZZ calculation in §4 supplies an exact CFT instance of a parabolic pole trajectory with an explicit O(1/c) coefficient (eqs. (4.32)-(4.33)), with the caveat, acknowledged by the authors, that Liouville CFT is not a conventional holographic CFT (§4). The protected-seed-singularity argument (§3.4) correctly separates what can and cannot be smoothed. The paper ships checkable formulas rather than fitting parameters, and its limitations are stated honestly.","major_comments":[{"comment":"The central claim that finite-c QNM deformations smooth the singularities is carried by the meromorphic-dominance hierarchy (3.10), which is stated as an assumption but never quantified. The exact finite-c object (3.7) is a sum of delta functions on the real frequency axis, so the decomposition G = G_mero + R and the analytic continuation defining G_mero require a coarse-graining prescription that is not specified. The two suppression mechanisms offered in §3.1 do not establish the hierarchy in the window 1 << β|ω| << c: thresholds at O(c) are indeed separated, but bulk-loop branch cuts are O(G_N) = O(1/c), i.e., precisely the order of the quadratic coefficient a_2/c that generates the Gaussian width in eq. (3.43); the remainder R can therefore a priori compete with the deformation. No estimate of |R| in the window is given. Since the Gaussian/Stokes argument of §3.2 is a statement about G_mero only, the generic AdS/CFT conclusion (abstract and the sentence b631 in §3.4) exceeds what eqs. (3.8)-(3.10) support unless the hierarchy is either proven or explicitly included in the statement of the result.","section":"§3.1, eqs. (3.8)-(3.10)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The Liouville example does not close the gap identified in §3.1. The full HHLL correlator is the integral over the continuous primary spectrum in eq. (4.9), whereas the pole analysis in §4.2 replaces it by the DOZZ product (4.15) evaluated on a pole branch of the spectral weight. The continuum integral generically produces non-meromorphic structure: the physical spectral edge ω = -2κ² in eq. (4.40) is a threshold of the full correlator, and it coincides with the rightmost zero of the DOZZ coefficient (eq. (4.39)), so analytic continuation must pass around it. In this particular example the threshold lies at O(c), outside the intermediate window, so the conclusions may survive; but the paper neither evaluates the integral (4.9) nor estimates the contribution of the threshold and of the conformal-block factors within the window. The Liouville demonstration therefore establishes the finite-c pole trajectory of the spectral weight, not the meromorphic dominance of the full correlator on which the generic smoothing claim rests; evaluating (4.9) in a suitable limit, or stating explicitly that the demonstration concerns pole data only, would be needed.","section":"§4.2, eqs. (4.9), (4.15), (4.40)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The quantitative prediction — a Gaussian kernel with width O(β/√c) — depends on the assumed deformation shape: a polynomial QNM trajectory with a leading quadratic O(1/c) term (eqs. (3.24)-(3.25)), and, in the exactly solvable model, on the specific coefficients (3.55) chosen to reproduce the square-root Stokes factor (3.56). The paper acknowledges that the deformation coefficients are theory-dependent (sentence b491), but the abstract frames the result as showing generically how finite-G_N corrections modify the holographic imprints of the singularity. Given the first comment, the Gaussian profile in eqs. (3.39)-(3.43) should be presented as the prediction of a specific class of deformations (quadratic bending with Re κ > 0, or with a complex-contour completion for Re κ < 0 as in §3.3 and appendix B), not as the established generic fate. The Liouville trajectory (4.32)-(4.33) is one genuine instance of this class at the DOZZ level, but does not by itself show that all finite-c holographic CFTs realize the quadratic bending.","section":"§3.2-§3.3, eqs. (3.24)-(3.25), (3.55)","authors_acknowledge":"The precise complex coefficients, as well as the detailed form of the finite-c deformation, are of course theory dependent.","authors_acknowledge_p":0.69}],"minor_comments":[{"comment":"The parametric claims κ_α ~ β²/c and width O(β/√c) presuppose a definite scaling of the quadratic coefficient a_2. With the planar AdS-Schwarzschild slope a_1 ~ 1/β in eq. (2.69), eq. (3.36) gives κ_m ~ a_2 β³/c, which reduces to ~ β²/c only if a_2 ~ O(1/β); the intended scaling of a_2 should be stated explicitly.","section":"§1, eq. (1.9); §3.2, eq. (3.36)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The claimed relation between the late-time linear growth rate and the lowest QNM, i ∂ω_{n=0}/∂Δ|_{Δ=0}, requires the normalization conditions G_{Δ=0}(T) = 1, R_0(0) = 1, and vanishing residues R_n(0) for n ≥ 1, which are asserted but not demonstrated for a general residue set; this subsection is peripheral to the main claim, but a one-line derivation of those conditions would make the relation complete.","section":"§6.3, eq. (6.44)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The reduction of the (HL)(HL) conformal block to its leading primary contribution, 1 + O(c^{-1}), is quoted from the published literature [43, 56-58], which is appropriate; stating the parametric form of the O(c^{-1}) correction (for example, whether it is controlled by h_L/c or by h_3/c) would help the reader gauge the reliability of the approximation at the upper end of the intermediate-frequency window β|ω| << c.","section":"§4.1, eqs. (4.9), (4.13)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Minor presentation issues: 'NOVEMBER3, 2026' lacks a space; the abstract's phrase 'produces suppression kernel' is missing an article; §6.6 contains the typo 'equaivalently'; and the sl(2,R) ladder algebra in §6.2 is typeset as 'sl(2,R)' rather than with a fraktur symbol.","section":"Throughout","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern lands on reading the paper: the meromorphic-dominance assumption (3.8)-(3.10) is load-bearing and unquantified, and the Liouville check analyzes the DOZZ spectral weight rather than the full continuum integral (4.9). The authors are honest about these being assumptions (sentences b413, b417, b418, b437, b491), which mitigates but does not remove the gap between the body's conditional statements and the abstract's generic framing. This is fixable within the manuscript's scope by reframing the central claim as a mechanism plus a class of models, or by adding a quantitative estimate of R and a treatment of the continuum integral in the Liouville example. The paper invokes a forthcoming companion work [15] for Picard-Lefschetz details; the main derivations are self-contained, but the editors may wish to confirm that no central proof depends on the companion paper's availability."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Short version: this paper is worth taking seriously. The new content is the quadratic-deformation mechanism: if finite-c QNM trajectories bend as ωn = a0 + a1n + a2 n²/c, the Stokes sectors of the Hadamard product acquire a Gaussian factor and the sharp complex-time singularity becomes a smooth profile of width β/√c. The authors also work out an exact quadratic toy model, extract a finite-c quadratic bending of the BTZ QNM trajectory from the Liouville DOZZ coefficient, and add a separate ETH-based thermal Gaussian smoothing mechanism. The Gamma-function and Hadamard manipulations, the exact Fourier integrals in section 3.3, and the DOZZ pole trajectories in section 4 are explicit and internally consistent. They also state their main assumptions plainly, and the protected-seed argument — OPE singularities and exact KMS images survive, HHLL forbidden images do not — is a genuinely sharp point.\n\nThe soft spots are real but localized. The generic AdS/CFT claim is conditional on the assumed dominance of the meromorphic component in the intermediate window 1 ≪ β|ω| ≪ c, eqs. (3.8)–(3.10). That assumption is doing the work: the exact finite-c object is a sum of δ-functions, so the analytic continuation defining G_mero already needs a coarse-graining prescription, and the paper gives plausibility arguments but no estimate of the remainder R in that window. The two suppression mechanisms they cite do not fully cover the case: bulk-loop branch cuts are O(1/c), the same order as the quadratic QNM coefficient a2/c that generates the Gaussian width, so R can compete with the claimed effect. This is a live concern, not a manufactured one.\n\nThe Liouville example does not close that gap. What the paper analyzes is the DOZZ product evaluated on a pole branch; the full finite-c HHLL correlator is an integral over the continuous primary spectrum, eq. (4.9), and that integral generically produces threshold cuts such as the physical spectral edge at ω = −2κ², eq. (4.40). So the example establishes a pole trajectory of the spectral weight, not meromorphy of the full correlator in the intermediate window. The authors acknowledge that Liouville is not a conventional holographic dual; that caveat is accurate, but the gap remains.\n\nNet: the paper deserves a serious referee. It advances the finite-GN fate of the bouncing-geodesic program with explicit machinery, and the open issues are precisely located: estimate R in the intermediate window, or narrow the generic claim to models where that estimate can be controlled. I would send it to review and ask for that revision.","headline":"A careful, explicit mechanism paper whose main generic claim rests on an unproven meromorphic-dominance assumption; the Liouville example is real but does not test that assumption.","tokens_in":58121,"tokens_out":2326,"would_cite":true,"duration_ms":23884,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["04.70.Dy","11.25.Tq"],"model":"deepseek/deepseek-v4-flash","headline":"At finite central charge, deformed quasinormal-mode trajectories replace the sharp complex-time singularities of bouncing geodesics with smooth profiles of width $\\beta/\\sqrt{c}$, while the OPE- and KMS-protected singularities survive.","keywords":["AdS/CFT correspondence","black hole singularity","quasinormal modes","complex-time singularities","thermal correlators","Hadamard product","Liouville CFT","eigenstate thermalization hypothesis"],"falsifier":"Compute the leading $1/c$ (or $1/N$) correction to the large-overtone QNM frequencies of a genuine holographic CFT, from quantum-corrected bulk wave equations or a $1/c$ expansion of the heavy-light matrix elements, and check whether $\\mathrm{Re}\\,\\omega_n$ bends quadratically as $-(3/c)(\\Delta + 2n)^2$ in the Liouville normalization; then test, in the window $1 \\ll \\beta|\\omega| \\ll c$, whether the two-sided correlator carries the Gaussian factor $\\exp(-\\kappa\\omega^2)$ with $\\kappa\\sim\\beta^2/c$ rather than comparable branch-cut contributions, and whether the would-be singularity at $T = \\pm|T_{\\rm crt}| + i\\beta$ is replaced by a bump of width $\\beta/\\sqrt{c}$.","tokens_in":58589,"feed_emoji":"🕳️","tokens_out":21200,"duration_ms":119100,"temperature":0.7,"pith_summary":"In holographic CFTs the classical black hole singularity leaves a sharp imprint on the analytically continued thermal correlator: an infinite lattice of power-law singularities whose complex-time positions are fixed by the bouncing-geodesic time $T_{\\rm crt}$ and by the asymptotically linear quasinormal-mode (QNM) spectrum. This paper argues that those additional singularities are a strict semiclassical artifact and are smoothed out at finite central charge $c$ (finite $G_N$): a quadratic bending of the QNM trajectories, $\\omega_n = a_0 + a_1 n + (a_2/c)n^2$, produces a Gaussian suppression $\\exp(-\\kappa\\omega^2)$ in frequency space that replaces each pole $1/(T-T_*)^{\\nu+1}$ with a smooth bump of width $O(\\beta/\\sqrt{c})$. The smoothing is selective: the seed singularities fixed by the boundary OPE and, for thermal correlators, by the exact KMS condition are preserved. The paper makes the mechanism explicit in Liouville CFT, where the exact DOZZ three-point coefficient places the poles on a parabola whose curvature vanishes as $c\\to\\infty$, recovering the linear BTZ QNM spectrum, and it identifies a second, independent Gaussian smoothing arising from energy fluctuations in the canonical spectral decomposition. If correct, the finite-$c$ correlator cleanly separates boundary-fixed analytic structure from interior-geometry data, turning singularity resolution into a question about $1/c$ corrections to QNM data.","feed_headline":"Quantum corrections smooth the black hole singularity's echo","feed_subtitle":"Bent quasinormal-mode trajectories turn sharp complex-time poles into smooth bumps of width β/√c.","key_machinery":"The load-bearing object is the thermal product formula of ref. [28]: an order-one Hadamard product over the QNM pole quartets that reconstructs the meromorphic two-sided correlator $G_{LR}(\\omega,k)$ from its pole data. Semiclassically, an asymptotically linear family $\\omega_n = a_0 + a_1 n$ turns that product into Gamma functions whose Stirling and reflection asymptotics factor into exponentials that translate the OPE seed singularity through the complex-time lattice $T_{n_+,n_-} = i\\beta/2 + n_+ v_+ + n_- v_-$, with $v_\\pm = \\mp|T_{\\rm crt}| + i\\beta/2$; this translation is what encodes the bouncing geodesic. The finite-$c$ deformation is introduced as a polynomial QNM trajectory with a leading quadratic correction, $\\omega_n = a_0 + a_1 n + (a_2/c)n^2$; inverting to $z_{\\rm sc}(\\omega)$ and feeding the Stokes factor $\\exp(2\\pi i\\, m\\, z_{\\rm sc}(\\omega))$ converts the quadratic term into the Gaussian damping $\\exp(-\\kappa_m\\omega^2)$, $\\kappa_m = 2\\pi i m a_2/(a_1^3 c)$, and the Fourier integral into a parabolic-cylinder (erfc) profile of width $O(\\beta/\\sqrt{c})$. Two further components carry the argument: the exact Liouville DOZZ example, whose Upsilon-function zeros put the poles on a parabola and provide the pole-zero cancellations that remove the $\\omega\\log\\omega$ and $\\sqrt{\\omega}$ terms the OPE forbids; and the ETH-based spectral decomposition, whose microcanonical saddle with $S''(E_\\beta) = -\\beta^2/C_V$ produces the universal Gaussian kernel $\\exp(-\\omega^2/4(\\Delta E)^2)$.","core_discovery":"The central claim is that the complex-time singularities associated with bouncing geodesics exist only in the strict $c\\to\\infty$ limit: they are generated by the asymptotically linear QNM spectrum through the thermal (Hadamard) product formula, and any nonlinear finite-$c$ deformation of the QNM trajectories modifies the Stokes sectors of that product, dressing the large-frequency exponential $e^{i\\omega T_*}$ with a damping factor. For the leading quadratic deformation the dressing is a Gaussian, $\\exp(-\\kappa_m\\omega^2)$ with $\\kappa_m\\sim\\beta^2/c$, and the would-be singularity becomes a smooth, entire profile of width $O(\\beta/\\sqrt{c})$ (eqs. (3.36)-(3.43)); for spectra whose inverse trajectory is sublinear or sufficiently superlinear in frequency, the displaced singularity is likewise absent. The mechanism is selective rather than destructive: the OPE singularity at the fundamental KMS seed and its thermal images are protected by the boundary OPE plus exact KMS, and only the additional bouncing lattice is smoothed (section 3.4). Liouville CFT supplies an explicit finite-$c$ realization: the analytically continued DOZZ coefficient has poles on the parabola $\\mathrm{Re}\\,\\omega = -(\\mathrm{Im}\\,\\omega)^2/(8\\kappa^2)$, a quadratic deformation of the BTZ QNM line, with zeros on the negative real axis whose combined Hadamard data satisfy the OPE's ultraviolet constraints. A complementary mechanism follows from the thermal spectral decomposition: the concavity of the black hole entropy, $S''(E_\\beta) = -\\beta^2/C_V$, yields a Gaussian factor $\\exp(-\\omega^2/4(\\Delta E)^2)$ with $\\Delta E\\sim 1/\\sqrt{G_N}$, which convolves every singularity sector into a smooth profile of time width $O(\\sqrt{G_N})$ (eqs. (5.20)-(5.21)).","pith_inferences":["The specific Gaussian shape is an artifact of the quadratic truncation: any nonlinearity in the inverse QNM trajectory eliminates the displaced singularity, so the paper's message is best read as 'the bouncing-geodesic singularity is unstable to finite-$c$ effects', with the Gaussian profile being the leading, not the universal, form of the smoothing.","The protected-seed principle suggests a general diagnostic: any feature of the analytically continued correlator not pinned by the OPE plus (for thermal states) KMS is semiclassically unstable, so the bouncing lattice is best understood as Stokes data of a QNM product rather than exact spectral data of the CFT.","The same bending mechanism should shift other QNM-built observables, notably the late-time linear growth of the black hole interior and complexity-like quantities, since the lowest QNM controls that growth rate; $1/c$ corrections to the QNM trajectory would then produce a subleading shift, a testable effect in exactly solvable cases like JT gravity.","A natural extension the paper leaves implicit is to include the $\\omega$-dependence of the ETH spectral function $F_O(E,\\omega)$ instead of freezing it at the saddle, which would couple the two smoothing mechanisms and could produce a regime where the total suppression is the product of two Gaussian kernels with widths $\\beta/\\sqrt{c}$ and $\\sqrt{G_N}$."],"forward_implications":["The semiclassical singularity lattice $T = \\pm m|T_{\\rm crt}| + i\\beta(2p+m+1)/2$ is resolved at finite $c$ into smooth bumps of width $O(\\beta/\\sqrt{c})$, so the black hole interior leaves a finite, calculable signature in the correlator instead of a divergence.","The genuine OPE singularity and its exact KMS images survive the smoothing, so finite-$c$ effects erase only the bouncing-geodesic lattice, separating boundary-fixed analytic structure from interior-geometry data.","The Liouville DOZZ example shows the mechanism concretely: exact poles lie on a parabola at finite $c$ and reproduce the linear BTZ QNM spectrum only as $c\\to\\infty$, demonstrating that quadratic QNM bending is realized, not merely assumed, in a solvable CFT.","At finite $c$ zero families (absent in the semiclassical product formula) are required to cancel $\\omega\\log\\omega$ and $\\sqrt{\\omega}$ contributions forbidden by the OPE, opening the possibility of bootstrapping the finite-$c$ pole-zero data from UV constraints plus thermal analyticity.","The spectral-decomposition mechanism adds a sector-independent Gaussian damping $\\exp(-\\omega^2/4(\\Delta E)^2)$ with $\\Delta E\\sim 1/\\sqrt{G_N}$, so together the two mechanisms supply complementary quantum resolutions of the singularity's holographic imprint."],"supporting_citations":[{"why":"Establishes the bouncing-geodesic singularity: the semiclassical two-sided correlator is dominated by complex-conjugate geodesics on the physical branch, while the real bouncing branch lives on a secondary sheet with a singularity at $T = T_{\\rm crt}$.","marker":"[7]"},{"why":"Supplies the thermal product formula that reconstructs the meromorphic two-sided correlator from its QNM pole data, the bridge through which the whole smoothing argument passes.","marker":"[28]"},{"why":"Shows that asymptotically linear QNM families generate the complex-time singularity lattice and provides the AdS black-brane asymptotic spectrum used to fix the lattice directions.","marker":"[37]"},{"why":"Provides the finite-$c$ resolution of forbidden singularities underlying the paper's distinction between protected OPE/KMS seeds and smoothable semiclassical images.","marker":"[44, 46, 47]"},{"why":"Identifies the poles of the Liouville DOZZ coefficient with the semiclassical BTZ QNM spectrum, the starting point for the paper's exact finite-$c$ example.","marker":"[45]"},{"why":"Gives the DOZZ formula for the exact Liouville three-point coefficient, the input that makes the finite-$c$ pole-zero structure explicitly computable.","marker":"[50, 51]"},{"why":"Supplies the ETH ansatz for energy-basis matrix elements used to set up the thermal spectral decomposition behind the Gaussian smoothing mechanism.","marker":"[60–62]"},{"why":"Provides the ETH saddle-point treatment of thermal two-point functions from which the entropy-induced Gaussian factor $\\exp(-\\omega^2/4(\\Delta E)^2)$ is taken.","marker":"[63]"}],"fun_headline_variants":["Finite-c corrections smooth the black hole singularity","Bouncing-geodesic singularities become smooth at finite c","Finite c bends quasinormal modes and softens the singularity","Gaussian smearing smooths the black hole singularity's echo","Finite-c physics rounds the bouncing-geodesic singularity"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The central claim collapses if, in the intermediate frequency window $1 \\ll \\beta|\\omega| \\ll c$, the exact finite-$c$ correlator is not dominated by a meromorphic component whose QNM poles follow a polynomial trajectory with a leading quadratic deformation, because then the Gaussian suppression kernel and the predicted $\\beta/\\sqrt{c}$ smoothing profile would not be the generic mechanism operating in real holographic CFTs.","fun_headline_variants_meta":{"raw":{"variants":["Finite-c corrections smooth the black hole singularity","Bouncing-geodesic singularities become smooth at finite c","Finite c bends quasinormal modes and softens the singularity","Gaussian smearing smooths the black hole singularity's echo","Finite-c physics rounds the bouncing-geodesic singularity"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000416,"raw_usage":{"cost":0.0004163978,"is_byok":false,"cost_details":{"upstream_inference_cost":0.0004163978,"upstream_inference_prompt_cost":3.38338e-05,"upstream_inference_completions_cost":0.000382564},"total_tokens":3325,"prompt_tokens":1223,"completion_tokens":2102,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":1064,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":2032}},"tokens_in":1223,"tokens_out":2102,"cost_source":"openrouter_usage","duration_ms":16464,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":2032,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-11-03T14:57:32.907295+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Compute the leading $1/c$ (or $1/N$) correction to the large-overtone QNM frequencies of a genuine holographic CFT, from quantum-corrected bulk wave equations or a $1/c$ expansion of the heavy-light matrix elements, and check whether $\\mathrm{Re}\\,\\omega_n$ bends quadratically as $-(3/c)(\\Delta + 2n)^2$ in the Liouville normalization; then test, in the window $1 \\ll \\beta|\\omega| \\ll c$, whether the two-sided correlator carries the Gaussian factor $\\exp(-\\kappa\\omega^2)$ with $\\kappa\\sim\\beta^2/c$ rather than comparable branch-cut contributions, and whether the would-be singularity at $T = \\pm|T_{\\rm crt}| + i\\beta$ is replaced by a bump of width $\\beta/\\sqrt{c}$.","supporting_citations":[{"cited_title":"The Black Hole Singularity in AdS/CFT","cited_arxiv_id":"hep-th/0306170","evidence_quote":"Establishes the bouncing-geodesic singularity: the semiclassical two-sided correlator is dominated by complex-conjugate geodesics on the physical branch, while the real bouncing branch lives on a secondary sheet with a singularity at $T = T_{\\rm crt}$."},{"cited_title":"A thermal product formula","cited_arxiv_id":"2304.12339","evidence_quote":"Supplies the thermal product formula that reconstructs the meromorphic two-sided correlator from its QNM pole data, the bridge through which the whole smoothing argument passes."},{"cited_title":"Bouncing off a stringy singularity,","cited_arxiv_id":null,"evidence_quote":"Shows that asymptotically linear QNM families generate the complex-time singularity lattice and provides the AdS black-brane asymptotic spectrum used to fix the lattice directions."},{"cited_title":"On CFT and Quantum Chaos","cited_arxiv_id":"1603.03020","evidence_quote":"Identifies the poles of the Liouville DOZZ coefficient with the semiclassical BTZ QNM spectrum, the starting point for the paper's exact finite-$c$ example."},{"cited_title":"Bounds on chaos from the eigenstate thermalization hypothesis","cited_arxiv_id":"1906.10808","evidence_quote":"Provides the ETH saddle-point treatment of thermal two-point functions from which the entropy-induced Gaussian factor $\\exp(-\\omega^2/4(\\Delta E)^2)$ is taken."}],"review_version":1}