{"id":"0efa688d-d118-4c9e-9e8f-4c13d00d8b1a","arxiv_id":"2607.21386","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In near-extremal holographic black holes, the n-th order pole-skipping momentum with mode index q becomes order-independent as T→0: k²_{n,q} → −m²h(r_h) + ½q(q−1)h(r_h)f″(r_h), with q identified as the AdS2 IR conformal dimension.","lead":"This paper derives a simple formula for high-order 'pole-skipping' points of quantum thermal correlators in nearly extremal holographic black holes, showing they all collapse onto a few near-horizon values as temperature goes to zero. The result makes an otherwise complicated tower of special points analytically tractable and ties it to the emergent AdS2/CFT1 structure of the near-horizon region.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Eq. (33) follows from the stated T-graded determinant hierarchy and is corroborated by high-precision numerics; remaining issues are presentation/reproducibility, not central correctness.","rationale":"Read in good faith, the paper's core technical contribution is the T→0 factorization of the pole-skipping determinant. The derivation is transparent and the high-precision numerical agreement in one nontrivial model is strong evidence. The weakest point is indeed the scope: the hierarchy (30) fails if the extremal limit is not a smooth AdS2 throat (h→0, f''→0, or irregular singular point). The paper states this limitation in Sec. II F, so it is not an unacknowledged flaw but it does mean the 'universal' wording in the abstract/conclusion should be read as 'universal within the smooth-AdS2 class.' I therefore do not find a load-bearing error in the central claim, but the reader's CONDITIONAL verdict is appropriate because the sound-channel consistency check is asserted without derivation, the recursion-matrix notation is inconsistent, and no code/data is provided. My read does not change that verdict; it strengthens the view that the remaining issues are presentational rather than mathematical.","tokens_in":24013,"tokens_out":40350,"duration_ms":409724,"concrete_test":"Compute scalar pole-skipping in a different smooth-extremal model, e.g., near-extremal planar RN-AdS, for n=1..5 and q=1..n by solving the full determinant condition, and check that k²_{n,q}(T) converge to Eq. (33) as T→0 and that the linear slope matches Eq. (42). A single high-precision run at one (n,q) with an independent code would already test the claimed universality and the reliability of the numerical method.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central factorization claim (Eq. 31) is a direct consequence of the temperature hierarchy in Eq. (30): at fixed n as T→0, every non-identity permutation in the Leibniz expansion uses at least one O(T) superdiagonal factor, so det M^(n) = ∏ M^0_σσ + O(T). The diagonal elements (Eq. 28) then give the n-independent collapse (Eq. 33). This argument requires h(r_h) and f''(r_h) to be O(1) (smooth extremal limit with a genuine AdS2 throat), and the paper explicitly restricts to this class in Sec. II F, noting that BTZ and planar AdS-Schwarzschild are outside. Within that domain the derivation is internally consistent, and the Dyonic Gubser-Rocha tests (Figs. 4–8, Table II) match both the intercepts and the slopes of Eq. (42) at the 10^-29–10^-15 level. The unshown sound-channel reduction in Sec. II F and the notational mismatch between Eq. (22) and Appendix A are real presentational gaps, and the absence of code/data impedes independent reproduction of the numerics, but I do not find a flaw that would invalidate the central claim within its stated scope.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a near-extremal expansion for high-order pole-skipping of a bulk scalar in holographic black holes. Working at the Matsubara frequencies ω_n = −i2πTn, the authors observe that the near-horizon recursion matrix has a temperature-graded structure: diagonal and lower entries are O(1), superdiagonal entries are O(T), and higher superdiagonals vanish. A Leibniz argument then yields the factorization det M^{(n)} = ∏ M^{(0)}_{σσ} + O(T) (Eq. (31)), so that in the T→0 limit the pole-skipping condition reduces to a product of diagonal factors. This gives the central formula k^2_{n,q} = −m^2 h(r_h) + (1/2)q(q−1)h(r_h)f″(r_h) (Eq. (33)), independent of the order n. Using the AdS2 mass relation (Eq. (13)), the mode index is identified with the IR conformal dimension Δ_IR = q. The authors compute the leading O(T) correction (Eq. (42)), express the result in thermodynamic variables, verify the predictions numerically in the Dyonic Gubser–Rocha model with very high claimed precision, and discuss the consistency with BTZ/AdS-Schwarzschild.","tokens_in":24210,"tokens_out":21922,"duration_ms":218605,"significance":"If the central claim holds, it is a substantial advance: high-order pole-skipping, normally requiring large determinants, becomes analytically tractable in the near-extremal regime and collapses to an n-independent spectrum set by near-horizon data. The identification of the mode index with an AdS2/CFT1 scaling dimension is a clean physical interpretation, and the paper provides a simple, falsifiable formula that can be tested in other holographic models. The main technical derivation (the Leibniz argument leading to Eq. (31)) is sound and independent of the bulk geometry beyond the assumed AdS2 double-zero structure. The authors are also explicit about the domain of validity (Sec. II F), noting that geometries without a smooth extremal limit are outside. The numerical cross-checks, if reproducible, strongly support the extremal-limit formula. However, as detailed below, the treatment of the leading temperature correction and the consistency check in Sec. II F are not reliable as written, and the numerical verification cannot currently be reproduced.","major_comments":[{"comment":"The linear temperature correction coefficient C_{n,q} is not derived consistently. In Eq. (43), the factors M^{(0)}_{q,q+1} and M^{(0)}_{q-1,q} do not exist: the superdiagonal entries of M at ω_n are pure O(T), so their O(T^0) parts vanish. In Appendix B, Eq. (B6) contains the term (M^0_{q+1,q+1} M^0_{q+1,q}) / M^1_{q,q+1}, which is O(1/T) rather than O(1), and the boundary term uses M^1_{q-1,q} (a superdiagonal quantity) where the transposition argument requires the subdiagonal M^0_{q,q-1}. The explicit expression Eq. (B7) does not follow from the displayed equations. Since Eq. (42) is a central quantitative claim verified numerically in Table II, the authors must provide a correct, self-consistent derivation and state exactly which formula was used to produce the 'analytic' columns.","section":"Sec. II E / Eq. (43); Appendix B"},{"comment":"The consistency check claims that the method 'reproduces the known results λ_L = 2πT and v_B^2 = d/2(d−1)' in BTZ and planar AdS-Schwarzschild. However, this paper analyzes a massive Klein–Gordon scalar; the Lyapunov exponent and butterfly velocity are properties of the gravitational sound/energy-density channel, which is not computed anywhere in the manuscript. If the statement refers to the first scalar pole-skipping point, it should be rephrased; if the sound channel is intended, the derivation must be shown. As written, this assertion is unsupported and should not be presented as a successful consistency check.","section":"Sec. II F"},{"comment":"The alternative derivation in Appendix A is internally inconsistent. Eq. (A10) contains Φ_{n−1} on the right-hand side where the argument requires the coefficient of Φ_0 to vanish; Eqs. (A11) and (A15) are contradictory; the notion of 'two independent parameters (Φ_0, Φ_n)' is not defined; and Eq. (A16) claims an upper bidiagonal form that contradicts Eq. (30), where all below-diagonal entries are O(T^0). The main-text Leibniz proof of Eq. (31) is sufficient and clean; Appendix A should be corrected or removed so that the key factorization is not obscured by an erroneous alternative derivation.","section":"Appendix A"}],"minor_comments":[{"comment":"The numerical verification claims relative errors down to 10^−29 with 100-digit precision, but no algorithm, code, or data are provided. At minimum, the authors should describe how the determinant roots are computed and tracked in T, which precision library is used, and how r_h, h(r_h), and f″(r_h) are evaluated in the extremal limit.","section":"Sec. III B / Table II"},{"comment":"The claim that curves k^2_{n+j,n}(T) for different n 'intersect at a common temperature' is presented as a structural feature, but no analytic formula for the intersection temperature is given. Please either derive it or describe it as an empirical observation of Fig. 7.","section":"Sec. IV"},{"comment":"The notation in Eq. (22) and the later decomposition (27) is confusing: the superdiagonal entries are written as (2jπT − iω) in Eq. (22), then become 2πT(j−n) after substituting ω_n, but the text sometimes refers to them as M^{(0)}_{j,j+1}. Please introduce M^0 and M^1 only for objects that actually have O(1) and O(T) parts, and use separate symbols for the superdiagonal coefficients.","section":"General notation"},{"comment":"The abstract and introduction emphasize universality, but Sec. II F explicitly excludes BTZ and planar AdS-Schwarzschild because the horizon becomes an irregular singular point. A sentence in the abstract or introduction stating the restriction to backgrounds with a smooth extremal limit and O(1) near-horizon data would help set expectations.","section":"Abstract / Sec. II F"},{"comment":"Several references are missing year/volume/page information (e.g., refs. [7,8,10,11,17,18,21]) and some arXiv identifiers are absent. Please bring the bibliography into journal style.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The core result, the n-independent extremal formula (33), is well supported by the determinant factorization argument and the stated numerical checks, and I judge it as likely correct within the stated class of smooth-extremal-limit backgrounds. However, the treatment of the first-order temperature correction is seriously flawed in both the main text and Appendix B, and the sound-channel consistency claim in Sec. II F is unjustified. These issues are fixable, but they require the authors to redo the perturbation theory and the numerical comparison. I would not recommend acceptance until the corrected derivation is provided and the numerical method is made reproducible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The central result holds up. The temperature-graded determinant factorization (Eq. 31) is a clean Leibniz-expansion argument: any non-identity permutation must use at least one O(T) superdiagonal entry, so det M = product of diagonals + O(T). Eq. (33) follows, and the numerical checks in the Dyonic Gubser–Rocha model are strong enough to make me take the claim seriously. This is the first analytic control of the high-order pole-skipping tower in the near-extremal limit that I know of, and the n-independent collapse is genuinely new relative to the earlier determinant-based treatments.\n\nWhat the paper does well: no fitted parameters, the numerics go to 10^-29 relative error, and the main derivation is self-contained. The AdS2/CFT1 reading — that the mode index q equals the IR conformal dimension Δ_IR=q — is interpretive, not an independent derivation, but it is a natural and consistent way to understand the collapse. The temperature correction formula (42) is also tested against the full determinant and the agreement is convincing.\n\nSoft spots, in proportion: (1) The word “universal” is used too freely. The hierarchy (30) requires h(r_h) and f″(r_h) to be O(1) as T→0; BTZ and planar AdS-Schwarzschild are explicitly outside, so the paper’s own Sec. II F defines the actual scope. That scope is legitimate, but the conclusions should be phrased for it. (2) Sec. II F claims to reproduce sound-channel λ_L and v_B without showing the reduction; as written this is a scalar-field calculation, so either derive the reduction or soften the claim. A referee will want this. (3) There is a real notational/index mismatch between the recursion matrix (22), which has nonzero subdiagonal entries, and the bidiagonal form (A16) used in Appendix A. The iterative derivation seems to assume that the earlier coefficients have already been solved for, but the chain is not spelled out. Fixable, but worth asking for. (4) No code or data. With 100-digit precision claims, independent verification is impossible as submitted.\n\nNone of these touches the central argument, in my reading. The stress-test note is right: within the stated domain, the derivation is internally consistent and the numerics are strong.\n\nThis deserves a serious referee. Send it to peer review. I would cite it.","headline":"The n-independent collapse of high-order pole-skipping in the extremal limit is real; scope and presentation issues, not the core argument, are what need fixing.","tokens_in":24830,"tokens_out":4293,"would_cite":true,"duration_ms":44388,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["11.25.Tq","04.70.-s"],"model":"deepseek-v4-flash","headline":"In near-extremal holographic black holes, all high-order pole-skipping points collapse onto a discrete near-horizon spectrum as temperature vanishes.","keywords":["pole-skipping","near-extremal black holes","AdS2/CFT1","holographic Green's functions","near-horizon geometry","infrared conformal dimension","Dyonic Gubser–Rocha model","Frobenius expansion"],"falsifier":"Compute pole-skipping momenta at several orders n for a near-extremal black hole with a smooth extremal limit and extrapolate the T→0 intercepts; the paper predicts that for fixed q, all orders share exactly the same k²_q. Any leading-order n-dependence in those intercepts—or failure of the determinant to become upper-bidiagonal—would falsify the collapse. A concrete alternative test: take a solution with h(r_h)→0 or f″(r_h)→0 at extremality and verify that the factorization of det M^(n) breaks down, leaving n-dependent momenta.","tokens_in":23775,"feed_emoji":"🕳️","tokens_out":5881,"duration_ms":51518,"temperature":0.7,"pith_summary":"This paper claims that in near-extremal holographic black holes, the entire tower of high-order pole-skipping points collapses as temperature T→0: the n-th order pole-skipping momentum k²_{n,q} becomes independent of the order n, depending only on the mode index q and on near-horizon geometry. This collapse arises because the near-horizon recursion matrix acquires a temperature-graded structure—superdiagonal entries vanish linearly with T—so its determinant factorizes into a product of diagonal elements. The paper further shows that the mode index q equals the IR conformal dimension Δ_IR=q of a scalar in the emergent AdS2 throat, giving a concrete physical meaning to subleading pole-skipping points. If correct, this makes high-order pole-skipping analytically accessible and reorganizes the low-temperature holographic Green's function around a single near-horizon spectrum.","feed_headline":"Pole-skipping towers collapse to one near-horizon spectrum at T→0","feed_subtitle":"Each pole-skipping momentum becomes independent of its order, tying the mode index to an AdS2 conformal dimension.","key_machinery":"The temperature-graded recursion matrix of the near-horizon radial equation. In ingoing Eddington–Finkelstein coordinates, expanding a scalar field around the horizon yields a recursion matrix M(ω,k²). At the Matsubara frequencies ω_n=−i2πTn, the superdiagonal entries M_{j,j+1}=2πT(j−n) vanish linearly with T while diagonal entries retain O(T⁰) near-horizon contributions. A Leibniz-expansion argument then shows the n×n determinant factorizes into the product of its diagonal elements plus O(T), reducing the n-th-order pole-skipping condition to M^{(0)}_{qq}=0 for a single q. This mechanism converts an intractable determinant into a simple algebraic condition.","core_discovery":"The central claim is the n-independence of pole-skipping momenta in the extremal limit: as T→0, k²_{n,q} → k²_q = −m²h(r_h) + ½ q(q−1) h(r_h)f″(r_h), independent of the order n. The paper derives this from a temperature-graded hierarchy in the near-horizon Frobenius expansion: at the Matsubara frequencies ω_n = −i2πTn, the superdiagonal elements of the recursion matrix are O(T), so the determinant factorizes as det M^(n) = ∏_{σ=1}^n M^(0)_{σσ} + O(T). Setting the q-th diagonal factor to zero yields the formula. Substituting into the effective AdS2 mass relation gives m²_eff L²_2 = q(q−1), identifying the mode index with the IR conformal dimension Δ_IR=q in the emergent AdS2/CFT1 duality. The","pith_inferences":["Because the identification Δ_IR=q holds for scalar fields, one might expect a similar near-horizon collapse for fermionic or vector probes; if so, the same temperature-graded factorization could yield a universal classification of pole-skipping spectra across matter content.","The nT regime (large n at fixed T) is left open by the paper; a resummation of corrections in nT could extend the collapse formula to a crossover temperature, possibly connecting to the Schwarzian mode that governs the near-AdS2 dynamics.","The thermodynamic rewriting of the spectrum suggests a practical diagnostic: low-temperature boundary correlators might be used to extract the near-horizon thermodynamic data (entropy density and extremal specific-heat scale) from the position of pole-skipping points alone.","If the collapse persists under higher-order temperature corrections, the pole-skipping spectrum could serve as a clean numerical order parameter for detecting the emergence of AdS2 physics in models that lack an analytic extremal solution."],"forward_implications":["High-order pole-skipping points become analytically computable in near-extremal black holes, well beyond the low orders accessible to standard determinant methods.","The mode index q acquires a physical interpretation as the infrared conformal dimension Δ_IR=q in the emergent AdS2/CFT1 throat, connecting pole-skipping to quantum criticality.","As T→0, the retarded Green's function's pole-skipping skeleton 'forgets' the bulk geometry and is organized by a single discrete near-horizon spectrum, with all Matsubara frequencies sharing the same momenta.","The leading pole-skipping momenta grow linearly with order, k_{n,n} ∝ n, with the slope set by the ratio of the transverse scale to the AdS2 radius.","Leading temperature corrections are linear in T and n-dependent (k²_{n,q} = k²_q + C_{n,q}T), lifting the extremal degeneracy, and can be expressed in terms of thermodynamic quantities such as entropy density and an extremal specific-heat scale."],"fun_headline_variants":["Pole-skipping moments become order-independent as T→0","High-order pole-skipping collapses to one spectrum at T→0","n-dependence vanishes: pole-skipping at extremality","All pole-skipping orders merge in near-extremal limit","Pole-skipping tower degenerates to AdS2 spectrum"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The collapse depends on the near-horizon quantities h(r_h) and f″(r_h) staying finite and nonzero as T→0—i.e., a smooth extremal limit with a genuine AdS2 throat—which the paper itself notes fails for BTZ and planar AdS-Schwarzschild black holes, where the horizon becomes an irregular singular point.","fun_headline_variants_meta":{"raw":{"variants":["Pole-skipping moments become order-independent as T→0","High-order pole-skipping collapses to one spectrum at T→0","n-dependence vanishes: pole-skipping at extremality","All pole-skipping orders merge in near-extremal limit","Pole-skipping tower degenerates to AdS2 spectrum"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000283,"raw_usage":{"total_tokens":1624,"prompt_tokens":974,"completion_tokens":650,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":718,"completion_tokens_details":{"reasoning_tokens":569}},"tokens_in":718,"tokens_out":650,"duration_ms":6479,"temperature":1.0,"reasoning_tokens":569,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T07:36:18.063727+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute pole-skipping momenta at several orders n for a near-extremal black hole with a smooth extremal limit and extrapolate the T→0 intercepts; the paper predicts that for fixed q, all orders share exactly the same k²_q. Any leading-order n-dependence in those intercepts—or failure of the determinant to become upper-bidiagonal—would falsify the collapse. A concrete alternative test: take a solution with h(r_h)→0 or f″(r_h)→0 at extremality and verify that the factorization of det M^(n) breaks down, leaving n-dependent momenta.","supporting_citations":[],"review_version":1}