{"id":"ecfc20ef-c973-43c8-87d6-8a16fc421a31","arxiv_id":"2610.13289","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Codimension-zero holographic complexity saturates at the Heisenberg time due to level repulsion, with the generating function governed by three-point spectral correlators.","lead":"This paper derives the late-time saturation of holographic complexity for codimension-zero bulk regions from spectral level repulsion in quantum chaotic systems. It provides a universal mechanism connecting black hole interiors, random matrix statistics, and Euclidean wormholes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Factorization of ensemble average over matrix elements and spectral correlators is assumed without justification; if matrix elements correlate with spectral fluctuations the derived reduction to codimension-one complexity may break.","rationale":"The reader identified the pole structure as the weakest assumption. That is a valid concern, but the paper motivates it via linear growth and examples. My central concern is a different, more operationally critical assumption: the factorization of the ensemble average. The reader's verdict ACCEPT does not address this, and the paper cites it only as an open question. Therefore a conditional acceptance is appropriate, contingent on testing this factorization in a controlled setting. The paper is otherwise rigorous and well-structured.","tokens_in":61468,"tokens_out":3098,"duration_ms":30084,"concrete_test":"Compute the full ensemble-averaged generating function ⟨e^{-α+C+ - α-C-}⟩ in JT gravity using the exact geodesic-length matrix elements (eq. B.5) and the known two- and three-level spectral correlation functions from the matrix integral (including non-factorized contributions). Compare the result with the factorized approximation (where the matrix elements are averaged independently). If the difference scales as O(e^{-S0}) or smaller, factorization is safe; if the difference is O(1) (i.e., comparable to the leading terms in the generating function), the concern is validated.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The spectral representation (2.40) and the subsequent evaluation of the generating function treat the matrix elements M_α±(E_i,E_j) as statistically independent of the spectral correlations ⟨D(E1)D(E2)D(E3)⟩. The ensemble average is applied only to the density correlators, while the matrix elements are taken as fixed smooth functions. This factorization is not derived; it is assumed. In a fully nonperturbative treatment, the complexity operator and the Hamiltonian share the same random structure, so the matrix elements can fluctuate with the spectrum. The paper acknowledges this possibility in Section 4 (eqs. 4.30–4.31) but does not test it. The clean reduction to ⟨C_Any⟩ = ⟨C_+⟩ + ⟨C_-⟩ (eq. 2.192) and the universal time evolution rely on the factorized decomposition of the three-point function and on the cancellation among quantum sectors (bra-operator, operator-ket, bra-operator-ket). If matrix elements are correlated with spectral fluctuations, these cancellations need not hold exactly, and the central conclusions could be modified at O(1) order.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"This paper develops a spectral formulation of holographic complexity for codimension-zero bulk regions within the complexity=anything proposal. The key result is that the expectation value of codimension-zero holographic complexity reduces to the sum of two codimension-one contributions (eqs. 2.192, 3.72) and exhibits a universal time evolution: linear growth up to the Heisenberg time TH, followed by saturation. The derivation relies on a specific first-order pole structure (2.12) for matrix elements of the complexity generating operator, and on three-point spectral correlation functions of the density of states, which the paper organizes into disconnected, pairwise connected, and fully connected wormhole sectors. An operator interpretation is also given, where finite-entropy corrections render the fixed-complexity basis overcomplete, and a complete orthonormal basis is constructed via Gram–Schmidt orthogonalization.","tokens_in":61859,"tokens_out":3021,"duration_ms":28264,"significance":"If the result holds, it unifies black-hole interior growth, chaotic spectral statistics, and Euclidean wormholes in quantum gravity. The paper extends the spectral approach from codimension-one observables to the more general codimension-zero case, showing that the three-point spectral correlator controls wormhole corrections. The universal prediction of linear growth followed by a plateau at the Heisenberg time is a concrete falsifiable feature of holographic complexity that could be tested in tractable models like JT gravity or DSSYK. The paper also provides a clean operator interpretation of the late-time plateau in terms of finite Hilbert-space dimension.","major_comments":[{"comment":"The spectral representation (2.40) factorizes the ensemble average: matrix elements M_α±(E_i,E_j) are treated as fixed smooth functions while only the density correlators are averaged. The paper acknowledges this factorization as an assumption in Section 4 (eqs. 4.30–4.31) but does not test whether correlated fluctuations between matrix elements and the spectrum could alter the reduction to ⟨C_Any⟩ = ⟨C_+⟩ + ⟨C_-⟩ (2.192) or the cancellation arguments that lead to the plateau. Since the central derivation is load-bearing on this factorized structure, the authors should either justify it (e.g., by arguing that the complexity operator is semiclassically independent of spectral fluctuations within the Thouless regime) or estimate the size of the neglected mixed cumulant.","section":"§2.3.1, §2.4, and §4 (eqs. 4.30–4.31)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The pole structure (2.12) is introduced as a necessary condition for linear growth of ⟨C⟩. The paper demonstrates sufficiency by showing that this pole yields the required exponential decay in the generating function, but the converse — that linear growth forces exactly this pole — is not rigorously proven; other analytic structures (e.g., branch cuts or higher-order poles) might also produce linear growth under the α→0 limit. The claim of necessity should be softened or supported by an additional argument (e.g., by requiring that the generating function’s Fourier transform be exponential in t to leading order).","section":"§2.2, eq. (2.12)","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The regularization δ(E−E')² ⇒ ρ(E) δ(E−E') is presented as a prescription. While it is justified by the discrete‑to‑continuum map, the paper should note that this step is a choice and that alternatives (e.g., point-splitting) could modify subleading additive constants. The current text is clear about the ambiguity; this is more a remark than a correction.","section":"§2.5.1, eq. (2.138)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"In the expression for R_3 (eq. 2.60), the sign convention for the cyclic determinant should be checked: the last line contains terms −ρ_i (K_jk)², etc. Please confirm that the signs are consistent with the standard determinantal formula det K = ρ_i ρ_j ρ_k + 2K_ij K_jk K_ki − ρ_i (K_jk)² − ρ_j (K_ik)² − ρ_k (K_ij)².","section":"§2.3.1, eq. (2.60)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Figure 2 is schematic; it would benefit from labelling the axes with explicit time and complexity scales (e.g., marking t ∼ e^{S_BH}, T_H). The current caption is clear, but the figure itself could be more informative.","section":"§1 & Fig. 2","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The definition of I^(1)(s) includes a term −½ sgn(s) (eq. C.42). Please clarify whether this term is required to ensure the antisymmetry I^(1)(s) = −I^(1)(−s) and whether it affects the short‑distance behaviour in eq. (C.55).","section":"§C.1.3, eqs. (C.42), (C.45)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"In the footnote giving the plateau value for a Krylov basis of dimension K, the average is (K−1)/2, not K/2. Please correct K/2 to (K−1)/2.","section":"§3.3.1, eq. (3.68)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The notation (q^{4Δ}; q²)_∞ is a q‑Pochhammer symbol. For readers less familiar with q‑series, a brief definition or reference would be helpful.","section":"§B.3, eq. (B.29)","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The paper is technically thorough and presents a coherent framework for codimension-zero holographic complexity. The main nontrivial assumption — factorization of the ensemble average — is acknowledged but not tested. I suspect that within the narrow microcanonical window and Thouless regime the factorization holds to leading order, but because the entire reduction (2.192) depends on it, the authors should provide a more concrete justification or at least a scaling argument. The pole-structure claim (2.12) is used as an input; this is legitimate but the text should be more precise about its necessity. If these points are addressed, the paper is publishable in JHEP."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"This paper is a careful extension of the codimension-one spectral complexity framework (Miyaji et al., 2507.23667) to codimension-zero observables. The new piece is the involvement of three-point spectral correlators. The authors show, in some detail, that the codimension-zero generating function reduces to the sum of two codimension-one ones and exhibits the same universal time evolution – linear growth for an exponentially long time followed by saturation at the Heisenberg time. They also provide a Gram–Schmidt operator construction that reproduces the same result. The calculations look solid within the stated assumptions (narrow microcanonical window, Thouless regime, GUE statistics), and the authors are transparent about where divergences arise and how they regularise them. Credit is due for working through the wormhole-sector cancellations explicitly.\n\nThe main unresolved issue – flagged in the paper itself but not resolved – is the factorization assumption used in the spectral representation (eq. 2.40 and following). The matrix elements of the complexity generator are treated as smooth, universal functions that are statistically independent of the spectral fluctuations. The ensemble average is applied only to the density correlators. This is a clean starting point, but it is an input, not a result. If the matrix elements fluctuate jointly with the spectrum, which is plausible in a fully nonperturbative treatment, the clean reduction to ⟨C_Any⟩ = ⟨C_+⟩ + ⟨C_-⟩ and the associated cancellation of extra wormhole sectors could break at O(1) order. The authors acknowledge this possibility in Section 4 (eqs. 4.30–4.31) but do not test it. The central result rests on this assumption.\n\nIs this a fatal flaw? Not necessarily. The paper shows that within the factorized approximation the story hangs together beautifully. And many readers will be willing to buy the input as a physically motivated ansatz. But a strict referee will want a quantitative discussion of what happens when the assumption is relaxed – perhaps a minimal model where the matrix elements pick up a small correlated fluctuation, or an argument that the effect is suppressed by an additional 1/S_0 factor. As it stands, the paper has a load-bearing input that is acknowledged but not defended.\n\nThe paper is well written and treats its limitations fairly. It is a genuine contribution for the holographic complexity community, especially those working on the spectral approach. I would support peer review, but I would ask the authors to address the factorization assumption more concretely – either by providing a test, a bound, or a clear statement of why it is the natural leading-order approximation.  \n\nWho should read this? Anyone interested in black hole interiors, quantum complexity, or the interplay between wormholes and spectral statistics. The formal parts are heavy but the central logic is clear.\n\nVerdict: send to a serious referee, but flag the factorization issue.","headline":"Solid extension of the spectral mechanism to codimension-zero, but a load-bearing factorization assumption about matrix elements and spectral fluctuations is acknowledged but untested.","tokens_in":62287,"tokens_out":1445,"would_cite":true,"duration_ms":15562,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":null,"created_at":"2026-10-12T16:08:18.970302+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}