{"id":"2dbcc58b-0e11-40ea-8b90-5b78a661b4df","arxiv_id":"2608.07412","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The finite-time peak of spread complexity is controlled by the Fourier transform of the nearest-neighbour energy-level spacing distribution.","lead":"This paper derives a simple analytic formula that explains why spread complexity, a measure of how a quantum state spreads in time, peaks for chaotic systems but not for integrable ones. It connects this peak directly to the statistical spacing between energy levels, bridging two standard tools in quantum chaos.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Master formula Eq. (5.6) rests on the unquantified replacement K_unfold≈K_uni and on a factorization of the spacing average in Eq. (5.4); the 6% peak-height error shows the approximation is not exact at the percent level.","rationale":"The paper's central claim would be true if the unfolded Krylov kernel were exactly universal and if spacing statistics were stationary; the master formula then follows from exact identities. What is least secure is the status of those two conditions. The paper itself labels the kernel universality a 'Hypothesis' and leaves the error term δ_err in Eq. (3.3) for future work. The numerical evidence (localization of M, decay of δW_k, agreement in Fig. 9) is genuine and reproducible via the linked code, so the concern is not that the claim is false, but that the derivation contains an unquantified approximation. The 6% peak-height offset shows that the leading-order formula is approximate; without an estimate of the factorization error it is not possible to know whether the residual is due to higher-k terms, to the Wigner-surmise approximation, or to a failure of kernel universality. This is exactly the kind of missing support that should keep the paper conditional rather than accepted as a derivation. No ad hominem or theatrical language is warranted; the authors are transparent about the approximative character of the hypothesis. A concrete finite-size scaling test would settle the issue.","tokens_in":20027,"tokens_out":12928,"duration_ms":125082,"concrete_test":"At D=500, 1000, and 2000 for GUE and GOE, compute the exact ensemble-averaged kernel-weighted phase A_k(τ)=⟨Σ_i K_unfold(i,i+k) cos[(ε_{i+k}−ε_i)τ]⟩ and the factorized uniform-lattice prediction B_k(τ)=(Σ_i K_uni(i,i+k))Φ_k(τ), for k=1,2,3, near the peak. If the normalized difference |A_k−B_k| / [D|Φ_k(τ)|] does not decrease with D, the factorization/kernel-universality assumption behind Eq. (5.4) has a finite error and Eq. (5.6) is not asymptotically controlled. Also recompute the full Eq. (5.6) peak height at D=2000 to see whether the 6% discrepancy shrinks or saturates.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the passage from the exact ensemble average of Eq. (5.2) to the master formula Eq. (5.6). That passage requires two uncontrolled approximations: (i) after unfolding, Kunfold(i,i+k)≃Kuni(i,i+k); and (ii) ⟨Σ_i Kuni(i,i+k) cos[(ε_{i+k}−ε_i)τ]⟩≃(Σ_i Kuni(i,i+k))Φ_k(τ), i.e. the kth-neighbor spacing distribution is independent of i and uncorrelated with the kernel weight. The paper tests (i) through the localization of the overlap matrix M (Figs. 5–6) and the integrated weight difference δW_k (Fig. 8), and tests the combined replacement in Fig. 9. These are numerical checks at D=1000 for GOE/GUE/Poisson; they establish plausibility, not an error bound. Because M has a finite localization width rather than being close to the identity, the substitution Kunfold≃Kuni has no small parameter. Eq. (5.3) defines Φ_k as the unweighted average over the kth-neighbor spacing, while Eq. (5.4) needs the kernel-weighted average; the equality of these two objects is asserted without derivation. The observed 6% peak-height discrepancy between the leading-order formula and the exact unfolded dynamics (Fig. 9 right) shows the combined approximation error is not negligible. The claim that local level statistics control the peak may still be correct, but Eq. (5.6) is presently a conjecture supported by finite-D numerics rather than a derived asymptotic result.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an analytic relation between spread (state Krylov) complexity and local spectral statistics. It starts from an exact energy-eigenbasis representation of spread complexity, Eq. (2.19), decomposes the double sum into contributions from k-th diagonals of the Krylov kernel, and observes numerically that the kernel is approximately banded with diagonal weights decaying roughly as k^{-2}. It then introduces an approximate kernel-universality hypothesis: after unfolding, the Krylov kernel of a generic random-matrix spectrum is well approximated by the kernel of a uniform lattice, K_uni. Combining K_uni with the k-th-nearest-neighbour spacing distributions p_k(s) leads to the master formula Eq. (5.6), which expresses the ensemble-averaged unfolded complexity in terms of the Fourier transforms Φ_k of p_k. The paper tests the formula for GOE, GUE, and Poisson spectra at D=1000 and gives an exact resummation of the Poisson limit, showing monotonic approach to the plateau with no finite-time peak.","tokens_in":20376,"tokens_out":6470,"duration_ms":60019,"significance":"If the master formula, Eq. (5.6), is valid, the paper establishes a direct, parameter-free connection between spread complexity and standard local level statistics, identifying the finite-time complexity peak as a consequence of short-range spectral correlations. This would be a useful conceptual and quantitative result for the Krylov-complexity literature, and the paper contains several strong technical components: the exact operator identity (A.7), the clean derivation of the uniform-kernel diagonal sums in Appendix A, the exact Poisson resummation of Section 6, and an openly available numerical repository. The claims are falsifiable and the numerical checks are extensive for GOE/GUE/Poisson at D=1000. However, the central formula rests on two uncontrolled approximations — the kernel-universality replacement K_unfold ≃ K_uni and the factorization of the ensemble average in Eq. (5.3) — so the paper presently delivers a well-motivated conjecture supported by finite-dimensional numerics rather than a fully derived asymptotic result.","major_comments":[{"comment":"The kernel-universality hypothesis is load-bearing and is not quantified. Equation (4.27), K_unfold = M K_uni M^T, is exact, but the replacement K_unfold ≃ K_uni in Eq. (4.30) requires that M be close to the identity in a controlled sense. The numerical evidence, while suggestive, does not provide such control: the localization width of M in Figs. 5–6 is shown only at D=1000, and no scaling with D or estimate of the induced error in the complexity is given. Because M is finite-dimensional and orthogonal rather than close to the identity in an operator norm that shrinks with D, there is no small parameter justifying the substitution. Please state the precise form of the hypothesis and provide either an error bound, a scaling analysis of ‖K_unfold − K_uni‖ or of the induced complexity error as a function of D, or at minimum additional tests for GSE and for intermediate ensembles.","section":"§4, Eqs. (4.27)–(4.30)"},{"comment":"The passage from the exact ensemble average in Eq. (5.2) to the master formula (5.6) uses a second uncontrolled approximation. Equation (5.3) defines Φ_k(t) as the Fourier transform of the global k-th-neighbour spacing distribution p_k(s), but Eq. (5.4) requires the kernel-weighted average of cos[(ε_{i+k}−ε_i)t] over i with weights K_uni(i,i+k). This is valid only if the local spacing distribution is independent of i and uncorrelated with the kernel weight; boundary effects and edge contributions are not estimated. The observed discrepancy between the leading-order peak height, e.g. 0.649 for GUE from Eq. (5.15), and the exact unfolded curve in Fig. 9 (right) shows that the combined approximation error is of order several percent. Please derive or numerically quantify the difference between the factored and full ensemble averages, and report the error separately from the error in the k=1 truncation.","section":"§5, Eqs. (5.3)–(5.4)"},{"comment":"The convergence bound in Eq. (5.7) controls only the tail of the k-sum and does not include the two approximation errors identified above. Therefore it does not, by itself, justify the statement that the leading-order formula reproduces the peak height 'with remarkable accuracy.' The right panel of Fig. 9 shows that the leading-order GUE peak is about 6% higher than the exact unfolded numerical curve; the paper should report this discrepancy explicitly and also compare the analytic peak positions with the exact numerical peak positions. To separate the sources of error, it would be very helpful to evaluate the full master formula using numerically extracted p_k(s) for the GUE and GOE ensembles and compare it with the exact unfolded complexity, rather than only the k=1 approximation.","section":"§5, Eqs. (5.7)–(5.8), Fig. 9"}],"minor_comments":[{"comment":"The abstract states that at leading order the finite-time peak is controlled by the Fourier transform of the nearest-neighbour spacing distribution; given the ~6% peak-height discrepancy documented in Fig. 9, it would be more precise to say 'approximately controlled' or to quantify the accuracy in the abstract and in the introductory summary.","section":"Abstract"},{"comment":"The notation p_k(s) should specify precisely which empirical distribution is used: the standard k-th-neighbour spacing distribution for levels in the bulk, a global average over all i, or a distribution with specific boundary conventions (e.g., excluding the first and last k levels). The edge convention matters at finite D because the sums in Eq. (5.4) run over i=0,...,D−1−k.","section":"§5, Eq. (5.3)"},{"comment":"The statement that S^{(D)}_{n,k} → 1 'admit a smooth continuum limit' is heuristic. Since Eq. (A.15) is the only asymptotic input for the diagonal sums, a few lines justifying the limit uniformly enough to control the o(D^2) remainder would make the derivation in Appendix A fully rigorous; at present the reader must supply the convergence argument.","section":"Appendix A, Eq. (A.15)"},{"comment":"The right panel marks the analytic leading-order peak positions with vertical dotted lines but does not mark the exact numerical peak positions. Adding the latter would make the claimed agreement quantitatively transparent.","section":"Fig. 9"}],"recommendation":"major_revision","confidential_remarks":"The paper is clearly written, timely, and likely of interest to the Krylov-complexity and quantum-chaos communities. The main concern is that the central master formula is an approximation whose two key steps — kernel universality and factorization of the spacing average — are not quantitatively controlled. I would support publication after a revision that either provides error bounds or, failing that, gives a scaling analysis in D and a direct numerical test of the full k-sum for GUE/GOE, and that clearly states the conjectural status of Eq. (5.6). The availability of the numerical repository is a positive factor for reproducibility."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper offers a direct analytic formula connecting spread complexity to level-spacing statistics. The master equation, (1.1), writes the ensemble-averaged unfolded complexity as a sum over Fourier transforms of kth-nearest-neighbor spacing distributions, and the numerics for GOE, GUE, and Poisson are in good agreement. The exact diagonal decomposition of the Krylov kernel in Secs. 2-3 is clean, and the observed 1/k^2 decay of integrated off-diagonal weights is a genuinely useful structural insight. The authors are also honest: they label the kernel-universality as a hypothesis and test it from multiple angles.\n\nThe soft spots are in the two steps that take the exact decomposition to the master formula. First, the replacement of the unfolded Krylov kernel by the uniform-lattice kernel has no small parameter; it is supported by numerical checks at D=1000, not by a derivation. Second, Eqs. (5.3)-(5.4) factor the ensemble average of K(i,i+k) cos[(ε_{i+k}-ε_i)t] into separate averages. That requires the kth-neighbor spacing distribution to be independent of the level index i and uncorrelated with the kernel weight. This is asserted, not derived. The 6% overshoot of the leading-order GUE peak height (0.649 vs about 0.61) is a visible reminder that the approximation is approximate. For the chaotic ensembles only the k=1 term is tested against the exact complexity; the full series (5.6) is not separately validated, while for Poisson all k are included in the resummation.\n\nA referee should ask for an explicit statement of these limitations, and ideally for a numerical test that isolates the factorization from the kernel-universality assumption. The treatment of folded-versus-unfolded time also deserves sharpening, since the physical peak location depends on the density-of-states mapping.\n\nWho is this for? Researchers in Krylov complexity, quantum chaos, and holography who want a spectral mechanism for the complexity peak. It is a solid, honest paper with a plausible new result, and it deserves peer review. With revisions that clearly delimit the approximations, it could become a standard reference for this connection.","headline":"A plausible and useful master formula connecting spread complexity to level-spacing statistics, built on an uncontrolled but numerically well-supported kernel-universality and a factorization that deserves more scrutiny; worth refereeing.","tokens_in":20871,"tokens_out":4585,"would_cite":true,"duration_ms":39002,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81Q50"],"pacs":[],"model":"deepseek-v4-flash","headline":"Spread complexity is controlled by local spectral statistics through a master formula that ties the finite-time peak to the nearest-neighbour spacing distribution.","keywords":["spread complexity","Krylov complexity","level spacing statistics","quantum chaos","random matrix theory","Poisson statistics","kernel universality","Lanczos algorithm"],"falsifier":"Construct a spectrum engineered to have Wigner nearest-neighbour spacings but non-random higher-order correlations, compute the exact unfolded Krylov kernel and ensemble-averaged spread complexity by Lanczos iteration, and compare with Eq. (1.1); if the $k>1$ terms contribute beyond the predicted $1/(4k^2-1)$ tail, or the peak height deviates by more than the $O(1/\\sqrt{D})$ fluctuations noted in Section 3, the universality and factorization assumptions fail.","tokens_in":19828,"feed_emoji":"📈","tokens_out":7409,"duration_ms":61061,"temperature":0.7,"pith_summary":"Spread complexity tracks how quickly a quantum state spreads across the Krylov basis built from repeated Hamiltonian action; for chaotic systems it shows a growth–peak–plateau shape, and it has been unclear which spectral correlations set the peak. This paper claims the answer is strictly local: after unfolding the spectrum, the ensemble-averaged complexity is fixed by the Fourier transforms of the $k$th-nearest-neighbour spacing distributions, with the nearest-neighbour term dominant. The central formula expresses $\\langle C_{\\rm unfold}(\\tau)\\rangle/D$ as $\\frac12\\left(1-2\\sum_{k=1}^\\infty \\Phi_k(\\tau)/(4k^2-1)\\right)$, so all ensemble dependence enters through level-spacing statistics. If correct, it unifies the chaotic (Wigner–Dyson) and integrable (Poisson) limits and turns the complexity peak into a quantitative probe of level repulsion.","feed_headline":"Level-spacing statistics set spread complexity's peak","feed_subtitle":"A master formula ties the chaos peak to nearest-neighbor spacings, unifying chaotic and integrable spectra.","key_machinery":"The central object is the Krylov kernel $K(i,j)$ in the ordered energy basis, together with its diagonal decomposition $C(t)=C_0+\\sum_{k=1}^{D-1} C_k(t)$, where $C_k(t)=2\\sum_i K(i,i+k)\\cos[(E_{i+k}-E_i)t]$. The load-bearing approximation is the uniform-lattice kernel $K_{\\rm uni}$ constructed from discrete Chebyshev polynomials, whose large-$D$ diagonal sums give the weights $W_k\\sim -D/[2(4k^2-1)]$; these weights make the diagonal series rapidly convergent and turn level-spacing distributions, through their cosine transforms $\\Phi_k(\\tau)$, into the sole ensemble-dependent input.","core_discovery":"The paper derives an exact energy-space representation of the Krylov kernel, $K(i,j)=\\frac{1}{D^2}\\sum_{n=0}^{D-1} n\\,p_n(E_i)p_n(E_j)$ with $p_n$ the Lanczos polynomials, and shows that this kernel is approximately banded: its integrated off-diagonal weights decay roughly as $1/k^2$. It then proposes an approximate kernel-universality hypothesis: after unfolding, the Krylov kernel of a generic spectrum is well approximated by the kernel of a uniform lattice, built from discrete Chebyshev (Gram) polynomials. Substituting this universal kernel into the diagonal decomposition and averaging over the $k$th-neighbour spacing distributions gives the master formula Eq. (1.1), with the nearest-neighbour term alone reproducing the GOE and GUE peak heights. For Poisson spectra the same series resums exactly and produces a monotone rise to the plateau with no peak.","pith_inferences":["Editorial extension: the same master formula should predict peak positions and heights for intermediate ensembles once the appropriate nearest-neighbour spacing distribution (for example a Brody interpolation) is inserted; the paper does not compute this.","Editorial extension: if kernel universality holds beyond the Gaussian and Poisson cases, the spread-complexity peak height becomes a directly measurable proxy for the level-repulsion parameter and could serve as a chaos diagnostic in many-body spectra where level statistics are already accessible.","Editorial extension: the $1/(4k^2-1)$ hierarchy suggests a quantitative spectral explanation for the maximum wormhole size and its subsequent relaxation in holographic duals, a connection the paper raises but leaves open.","Editorial extension: the diagonal decomposition could extend to temperature-dependent or weighted complexity measures by filtering the degree matrix in the kernel definition; this is a numerically testable generalization."],"forward_implications":["The finite-time peak height and position of spread complexity become computable directly from the nearest-neighbour spacing distribution, so stronger level repulsion (GUE versus GOE) yields a higher, slightly later peak.","Keeping only the $k=1$ diagonal already captures the peak, because higher diagonals are suppressed by the factor $1/(4k^2-1)$ and the omitted tail is bounded by $1/[2(2m+1)]$.","For Poisson spectra the full series resums to $C_P(\\tau)/D=\\frac12\\int_0^1 dx\\, \\tau^2/[(1-x^2)^2+\\tau^2]$, which rises monotonically to the plateau $D/2$ without developing a finite-time maximum.","The unfolded-time peak positions convert to physical time through the local density of states, giving $t_{\\rm peak}/D\\simeq 0.88$ for GUE and $0.85$ for GOE under the bulk-density approximation.","The late-time plateau value $D/2$ is a universal outcome of the master formula for both chaotic and integrable spectra."],"supporting_citations":[{"why":"Defines spread complexity and the Krylov basis for states, the quantity the entire paper studies.","marker":"[10]"},{"why":"Observed that the complexity peak height varies with spectral statistics and symmetry class, the phenomenon the paper explains.","marker":"[44]"},{"why":"Provides the late-time diagonal ensemble value and the suppression of the peak in integrable limits, the baseline the Poisson analysis reproduces.","marker":"[48]"},{"why":"Supplies an earlier kernel-based spectral representation and late-time plateau discussion that the paper refines and extends.","marker":"[49]"},{"why":"Established a connection between spread complexity and level-spacing distribution in a two-level system, a precursor of the general formula.","marker":"[59]"},{"why":"Introduced a diagonal decomposition of the spectral form factor that motivates the analogous kernel-diagonal decomposition used here.","marker":"[60]"}],"fun_headline_variants":["Peak of spread complexity set by level spacings","Level statistics dictate spread complexity's peak","Analytic formula ties complexity peak to spacing","Spacing Fourier transform shapes complexity peak","From chaotic to integrable: one spacing formula"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the approximate kernel-universality hypothesis of Section 4: after unfolding, a generic spectrum's Krylov kernel is close enough to the uniform-lattice kernel that replacing one by the other leaves the ensemble-averaged spread complexity unchanged, together with the factorization in Eq. (5.3) that replaces correlations between different $k$-separations by the single $k$th-neighbour spacing distribution.","fun_headline_variants_meta":{"raw":{"variants":["Peak of spread complexity set by level spacings","Level statistics dictate spread complexity's peak","Analytic formula ties complexity peak to spacing","Spacing Fourier transform shapes complexity peak","From chaotic to integrable: one spacing formula"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00023,"raw_usage":{"total_tokens":1470,"prompt_tokens":922,"completion_tokens":548,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":538,"completion_tokens_details":{"reasoning_tokens":481}},"tokens_in":538,"tokens_out":548,"duration_ms":5036,"temperature":1.0,"reasoning_tokens":481,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:27:07.406440+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a spectrum engineered to have Wigner nearest-neighbour spacings but non-random higher-order correlations, compute the exact unfolded Krylov kernel and ensemble-averaged spread complexity by Lanczos iteration, and compare with Eq. (1.1); if the $k>1$ terms contribute beyond the predicted $1/(4k^2-1)$ tail, or the peak height deviates by more than the $O(1/\\sqrt{D})$ fluctuations noted in Section 3, the universality and factorization assumptions fail.","supporting_citations":[],"review_version":1}