{"id":"65893e1b-723e-4cba-9d5a-929f4aa00374","arxiv_id":"2412.18737","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"SYK spectral form factor moments match random matrix statistics at low order, with a k^2/N^{q-2} correction from spectral edge fluctuations that is amplified by sparsification.","lead":"The paper computes the moments of the spectral form factor in the SYK model of quantum chaos, showing random-matrix behavior at large N but deviations from it at high moment order due to spectral edge fluctuations. It predicts the leading correction grows with the moment order and shrinks with the number of random couplings, and tests this in sparse SYK numerically.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (3.37) hinges on an unevaluated edge quantity ΔE whose claimed 1/N^{q-2} scaling rests on an extensivity assumption that the paper's own saddle-point construction does not establish.","rationale":"I read the paper's central argument as: (i) in the large-N limit the ramp moments equal the RMT values k! or (2k-1)!!, with the combinatorial factors emerging from replica-symmetry breaking; and (ii) the leading 1/N correction is of the form (1 + k(k-1) q!/(4 N^q) T^2 |ΔE|^2 + ...), which by extensivity of ΔE gives 1/N^{q-2} and therefore a breakdown of RMT-like behavior at k ~ N^{q/2-1}. The warm-up model in §2 is exactly solvable and demonstrates the analogous structure (2.18) with q!/q^2 / N^{q-2}, and the derivation of Eq. (3.37) from Feynman diagrams in §3.4/Appendix B is internally consistent: the leading diagram is identified by power counting, and the vanishing of dangerous subleading terms is argued carefully. The q = 2 section is a separate, self-contained matrix-integral analysis that appears coherent and agrees with numerics at the level of plateau and jumps. The sparse-SYK numerics genuinely support the qualitative claims: k(k-1) growth of B and a deviation that increases with sparsification. In particular, the microcanonical filtering in Fig. 8 is an independent and meaningful check that the unwanted correction is edge-dominated, corroborating the structural picture. However, the single formula that converts this structural picture into a quantitative prediction, Eq. (3.37), depends on ΔE, which the paper does not compute, and the paper explicitly acknowledges that the saddle-point solution used to describe the edge regime is not controlled: §3.2 states the image-sum construction (3.11) is only accurate for β_aux ≪ T, while the ΔE integral is dominated by β_aux ≳ T (the edge). This is precisely where the reader's weakest assumption lands: the edge contribution and the extensivity of ΔE are assumptions. If they fail, the central scaling k ~ N^{q/2-1} is unsupported, even though the qualitative departure from RMT for high moments might survive. The numerics in §5 do not resolve this: they never vary N while holding q and the fitting procedure fixed to extract α(N); the p-dependence fit is 1/p^{1.2}, not 1/p, so the quantitative proportionality to the number of random parameters is already imperfect in the regime probed. For these reasons I agree with the reader's conditional verdict: the framework is credible and the structural conclusions likely correct, but the headline quantitative formula needs a controlled evaluation of ΔE or an independent derivation of its N-dependence before the acceptance threshold is met.","tokens_in":45993,"tokens_out":3194,"duration_ms":26296,"concrete_test":"Compute ΔE(β_aux) of Eqs. (3.34)-(3.35) by numerically solving the full saddle-point equations for the ramp saddle points of [24] (e.g. by iteration as in SSS) for N = 16, 20, 24, 26 with q = 4, and evaluate the integral (3.38) for moderate T. Then compare T^2|ΔE|^2 against the assumed extensive scaling ~ c N^2; if the ratio T^2|ΔE|^2/N^2 decreases with N, or if the integrand is not dominated by β_aux ≳ T (i.e. the naive extrapolation of the Schwarzian edge behavior is not self-consistent), Eq. (3.37)'s claimed 1/N^{q-2} scaling fails. As a complementary check, perform the same extraction from full SYK exact diagonalization at N = 16-26, q = 4, fitting the k(k-1) correction amplitude to see whether it indeed scales as N^{-2}.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central quantitative claim is Eq. (3.37): the leading correction to the ramp-region moments is k(k-1) q!/(4 N^q) T^2 |ΔE|^2, which is then promoted to a 1/N^{q-2} statement by declaring that ΔE is extensive in N. The paper itself flags (§3.4, surrounding Eqs. (3.34)-(3.38)) that ΔE(β_aux) is not computed, that the saddle-point correlators of [24] used to evaluate it are constructed by image-summing thermofield-double correlators and are explicitly accurate only for β_aux ≪ T, while the moment defining ΔE is dominated by β_aux ≳ T, near the spectral edge. The claim that this edge regime still controls and that ΔE is extensive is an assumption, not a derivation. All of the paper's headline scaling — k ∼ N^{q/2-1} for the breakdown of RMT-like moments — depends on T^2|ΔE|^2 scaling as N^{q-2}, i.e. |ΔE| ~ N (since T is kept fixed in the large-N ramp analysis). If ΔE instead scales as a lower power of N, or acquires a different T-dependence, the deviation threshold shifts. The paper's own numerics do not settle this: the sparse-SYK fits (Figs. 6-7) confirm the k(k-1) dependence and approximate 1/p behavior in p, but the fit gives p^{-1.2}, not p^{-1}, and no N-scaling check of the coefficient α is provided. Thus the most load-bearing step is the assumed extensivity/edge-control of ΔE, which converts q!/N^q into the physical 1/N^{q-2} suppression claimed for high moments.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the moments of the spectral form factor in the SYK model and in a sparse variant. The authors identify saddle-point configurations that describe the power-law ramp of ⟨|Z(iT)|^{2k}⟩, show that at leading order in large N the moments reduce to the RMT values k! (or (2k−1)!! for q=2 mod 4), and compute a perturbative 1/N correction around these saddles. Their central result, Eq. (3.37), states that the ratio ⟨|Z|^{2k}⟩/⟨|Z|²⟩^k equals k! (or (2k−1)!!) times a factor 1 + k(k−1) q! T²|ΔE|²/(4N^q) + …, and by assuming ΔE is extensive in N they convert this into a 1/N^{q−2} correction that becomes important when k ∼ N^{q/2−1}. The paper also analyzes the q=2 model, where an enhanced U(2k) symmetry produces an exponential ramp with heavy-tailed moments, and presents exact diagonalization results for sparse SYK, where the k(k−1) correction is observed numerically and grows as p is decreased.","tokens_in":46372,"tokens_out":5697,"duration_ms":56665,"significance":"If the central result holds, it gives a concrete, falsifiable statement about how SYK departs from random matrix universality: high moments deviate when k reaches a fixed fraction of N^{q/2−1}, with the correction controlled by the number of independent random couplings. The saddle-point framework is natural and the factorization argument that produces the k! factor is clearly presented. The paper contains several genuine strengths: an exact treatment of the one-time-point SYK model, a diagrammatic identification of the leading 1/N term in section 2, extensive numerics for sparse SYK, and an interesting separate analysis of the q=2 model. However, the central quantitative scaling rests on properties of ΔE that are assumed rather than derived, and the numerical support does not currently isolate the N-dependence of the correction. The significance is therefore high if the ΔE assumption can be justified, but the paper is not yet self-contained at this load-bearing point.","major_comments":[{"comment":"The central claim that the leading correction is ∼ k(k−1) q! T²|ΔE|²/(4N^q), and hence ∼ k(k−1)/N^{q−2} when ΔE is extensive, rests on two unproved inputs. First, ΔE(β_aux) is never computed; it is defined by Eq. (3.35) and averaged in Eq. (3.38). Second, the saddle-point solution imported from [24] that is used to evaluate the relevant correlators is explicitly stated in §3.2 to be accurate only for β_aux ≪ T, whereas the integral defining ΔE is dominated by β_aux ≳ T, near the spectral edge. The sentence 'since ΔE is extensive in N' in §3.4 is therefore an assumption, not a derivation. Because the headline scaling k ∼ N^{q/2−1} and the claimed departure from RMT follow entirely from this extensivity, the paper needs either a direct estimate of the edge contribution to ΔE, a consistency argument, or a substantial reformulation of the quantitative claim.","section":"§3.4, Eqs. (3.34)–(3.38)"},{"comment":"The numerical evidence for the parametric dependence in Eq. (3.37) is incomplete. The fit B = α k(k−1) in Fig. 6 confirms the k-dependence, which is a useful check. However, Fig. 7 gives α ∼ p^{−1.2} rather than the expected α ∼ p^{−1}, and no N-dependence of α is reported. Since the crucial physical statement is the 1/N^{q−2} suppression (α ∼ 1/N² for q=4), a scan over N at fixed p and k is needed before the numerics can be said to support the extensive-ΔE mechanism. In addition, because the sparse-SYK variance in Eq. (5.3) is rescaled by 1/p, changing p changes both the number of couplings and their strength, so the interpretation 'correction ∝ 1/(number of random parameters)' is not isolated by these data.","section":"§5, Figs. 6–7"},{"comment":"The claim that the correction is approximately independent of T is not directly demonstrated. Eq. (3.37) contains the combination T²|ΔE|², which could have nontrivial time dependence; the numerical quantity B in Eq. (5.5) is time-averaged over the ramp, so the flat plateaus in Fig. 5 do not by themselves show T-independence. A direct plot or fit of B(T), or of the coefficient |ΔE(T)|, would be needed to substantiate the statement that the correction is approximately constant in time.","section":"§3.4, §3.5, Eq. (5.5)"}],"minor_comments":[{"comment":"The footnote quotes a negative correction for the CUE plateau, −k(k−1)/(4L), while the sparse-SYK correction in Fig. 6 is positive; if 'similar behaviour' refers only to the k(k−1) growth, the sign difference should be stated explicitly.","section":"Footnote 11"},{"comment":"There are several typographical errors, e.g., 'behvaiour' in §3.4 and 'seciton' in §5 and in footnote 11; these should be corrected in a final version.","section":"Throughout"},{"comment":"The quantity α is presented without error bars, and the text calls the k(k−1) curve an 'interpolation' when it is in fact a fit; please clarify the fitting procedure and report uncertainties.","section":"Fig. 7"},{"comment":"The comparison of Eq. (4.37) with numerics is shown only for N=50, k=2, and the large-k formula (4.52) is explicitly derived without the full one-loop determinant; a sentence stating the expected size of the missing one-loop effects would help the reader judge the accuracy of the k≫N extrapolation.","section":"§4.2–4.3"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine step forward on SYK moments, and the leading-order saddle point part is solid, but the headline 1/N claim (Eq. 3.37) leans on an assumption the paper does not back up. The reader's conditional verdict is the right one.\n\nWhat is new and worth keeping: the pairing-saddle construction for the moments, the one-loop factorization that gives k! (or (2k-1)!!) with the correct replica-symmetry breaking, and the perturbative framework that isolates delta-G_perp fluctuations as the source of the 1/N correction. The q=2 analysis with its U(2k) zero-mode manifold and the large-k duality is a nice separate result, and the sparse SYK numerics in Sec. 5—especially the k(k-1) scaling and the microcanonical filter test—are useful evidence. I have no quarrel with the leading-order statement: SYK's low moments match RMT to leading order, and the combinatorial factors come out right.\n\nThe soft spot is exactly where the stress-test note points. Eq. (3.37) contains Delta-E, defined in (3.35)-(3.38). The paper states it is \"difficult to compute,\" and the saddle-point correlators from [24] used to evaluate it are, by the paper's own account (Sec. 3.2), only accurate for beta_aux << T. But Delta-E is dominated by beta_aux >~ T, the spectral edge. The conversion of q!/N^q into the physical 1/N^{q-2} suppression then relies on Delta-E being extensive in N. That is asserted, not derived. If Delta-E scales differently, the threshold k ~ N^{q/2-1} shifts. The sparse numerics do not rescue this: the fit gives alpha ~ p^{-1.2}, not p^{-1}, and there is no N-scaling check of alpha. Time-independence of the correction is only tested in sparse SYK. So the central quantitative formula is not under control, even though I suspect the structural conclusion is right.\n\nNone of this is fatal to the paper's value. The q=2 section and the saddle-point machinery are worth having, and the sparse numerics will interest people building tabletop SYK simulators. But the authors should either evaluate Delta-E (at least its scaling) or explicitly reframe (3.37) as a conjecture with numerical support. For peer review: send it out. A good referee can sort out whether extensivity can be proven from the [24] solution or needs new input. I would not desk-reject this; I also would not accept it as is.","headline":"Solid saddle-point analysis of SYK spectral form factor moments; the leading-order result is right, but the headline 1/N correction relies on an unevaluated edge quantity whose extensivity is assumed, so the paper is conditionally acceptable.","tokens_in":46927,"tokens_out":2886,"would_cite":true,"duration_ms":26139,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["05.45.Mt"],"model":"deepseek-v4-flash","headline":"This paper shows the SYK spectral form factor matches random-matrix statistics only for low-order moments, the leading correction growing as $k^2/N^{q-2}$ from spectral-edge fluctuations and amplified by sparsification.","keywords":["spectral form factor","SYK model","random matrix theory","quantum chaos","moments","large-N expansion","spectral edge","sparse SYK"],"falsifier":"One could settle the claim numerically: for fixed $q=4$, solve the SYK saddle-point equations by iteration at inverse temperatures $\\beta_{\\rm aux}\\gtrsim T$, compute $\\Delta E$ from (3.35), and test whether it is extensive in $N$ and matches the conformal-image approximation; if it does not, equation (3.37) is not established. A complementary check is exact-diagonalization data on unsparsified $q=4$ SYK at two values of $N$, asking whether the time-averaged moment correction $B/k(k-1)$ scales as $N^{-2}$ across the ramp plateau as predicted.","tokens_in":45766,"feed_emoji":"🎲","tokens_out":20962,"duration_ms":171124,"temperature":0.7,"pith_summary":"This paper sets out to show that the higher moments of the spectral form factor, the statistic that captures the erratic noise around the universal ramp, can be computed in the SYK model by a family of pairing saddle points, and that the result agrees with random-matrix theory only for low order. The central quantitative claim is a leading correction to the moment ratio $\\langle |Z|^{2k}\\rangle/\\langle |Z|^2\\rangle^k$ that grows as $k^2/N^{q-2}$, is driven by fluctuations near the edge of the spectrum, and is inversely proportional to the number of independent random couplings in the Hamiltonian. If correct, it means the SYK model mimics a random matrix only up to $k\\sim N^{q/2-1}$, and that sparser realizations of the model deviate from universality even earlier; numerical study of sparse SYK supports this. The $q=2$ free-fermion case is shown to be sharply different, with an exponential ramp and noise that grows exponentially in time. This matters because the moments are a stricter test of quantum chaos than the ramp alone, and because the deviations delimit how faithfully SYK and its approximations can stand in for random-matrix chaos.","feed_headline":"SYK moments break random-matrix universality at high order","feed_subtitle":"The leading 1/N correction grows with k², and sparser Hamiltonians deviate from random-matrix statistics even faster.","key_machinery":"The carrying mechanism is the collective-field path integral (3.4) for $\\langle |Z(iT)|^{2k}\\rangle$ with $2k$ replicas, written with a $2k\\times 2k$ antisymmetric matrix of bilocal fields $(G,\\Sigma)$. The relevant saddle points are pairing configurations in which $(G,\\Sigma)$ are block diagonal, with each block a copy of the two-replica ramp saddle point built by summing images of thermofield-double correlators over relative time shifts $\\Delta$ and auxiliary inverse temperatures $\\beta_{\\rm aux}$; the spontaneous breaking of the $k$ relative time translations supplies the ramp power $T^k$, and the breaking of the discrete replica symmetry $S_k\\times S_k\\to S_k$ (or $S_{2k}\\to S_k\\times S_2^k$ for $q=2\\bmod 4$) supplies the combinatorial factor $k!$ (or $(2k-1)!!$). The one-loop determinant factorizes over the blocks, and the leading $1/N$ correction comes exclusively from the 'perpendicular' fluctuations $\\delta G_\\perp,\\delta\\Sigma_\\perp$: every candidate Feynman diagram vanishes except those with one $\\delta G_\\perp^q$ and one $\\delta G^q$ vertex, producing $\\frac{q!}{N^q}T^2|\\Delta E|^2$ with $\\Delta E(\\beta_{\\rm aux})=\\frac{iN}{q}\\partial_t(G_{LL}+G_{RR})|_{t\\to0^+}$. In the $q=2$ free-fermion limit the symmetry enhances to $U(2k)$ acting on each Fourier mode, and the zero-mode manifold with volume ${\\rm vol}(U(2k)/U(k)^2)$ per mode inside $|\\omega_n|<2J$ generates the exponential ramp.","core_discovery":"In the large-$N$ limit the ratio of moments takes the random-matrix value, $k!$ for $q=0 \\bmod 4$ and $(2k-1)!!$ for $q=2 \\bmod 4$, reproduced by saddle points that pair the $2k$ replicas into $k$ blocks. Around these saddles the leading correction is $$\\frac{\\langle |Z(iT)|^{2k}\\rangle}{\\langle |Z(iT)|^2\\rangle^k}=\\Bigl(1+\\frac{k(k-1)}{4}\\frac{q!}{N^q}\\,$T^{2}$|\\$\\Delta$ E|^2+\\cdots\\Bigr)\\times\\begin{cases} k!, & q=0\\bmod 4,\\\\ (2k-1)!!, & q=2\\bmod 4,\\end{cases}$$ where $\\Delta E$ is a replica energy imbalance inherited from the thermofield-double construction, is dominated by the spectral-edge regime $\\beta_{\\rm aux}\\gtrsim T$, and is extensive in $N$, so the correction effectively scales as $k^2/N^{q-2}$. The paper argues this is the earliest departure from random-matrix universality and ties it to the count of independent random couplings $N^q/q!$; numerics on sparsified SYK confirm the $k(k-1)$ law and a coefficient growing roughly as $p^{-1.2}$ in the sparsification probability, while a microcanonical filter that removes the spectral edges restores random-matrix behavior. For $q=2$ the effective symmetry enhances to a $U(2k)$ action on each Fourier mode, producing a zero-mode volume that grows with every mode inside $|\\omega_n|<2J$: an exponential ramp, moments growing exponentially in $k$ and in time, and a plateau at $JT\\sim 2N$ with $\\langle |Z|^{2k}\\rangle=\\binom{2k}{k}^{N/2}$; a dual $N\\times N$ matrix integral controls the $k\\gg N$ regime.","pith_inferences":["A natural generalization the paper leaves implicit is a criterion for any disordered Hamiltonian: the order $k$ at which moment universality breaks should scale with the square root of the number of independent random couplings times an edge-fluctuation factor, a prediction that could be tested in Sachdev-Ye and spin-glass models without changing the method.","Because the deviation appears while the ramp is unchanged, the moments act as a stricter chaos diagnostic than the spectral form factor; experimental quantum-simulation claims for SYK should therefore verify the moment ratio, not just the two-point ramp, before concluding random-matrix behavior.","If the edge-dominated correction has a gravitational counterpart, it should appear as a non-perturbative, edge-sensitive effect beyond the double-cone wormhole, giving a concrete target for matter or multi-boundary corrections in the dual dilaton-gravity description rather than genus-suppressed contributions.","The $q=2$ zero-mode volume mechanism suggests an organizing principle: the ramp shape, linear versus exponential, is set by the dimension and growth of the spontaneously broken zero-mode manifold, which could serve as a diagnostic separating single-particle from many-body scrambling in other free or weakly interacting models."],"forward_implications":["For $q>2$ SYK the normalized moments $\\langle |Z|^{2k}\\rangle/\\langle |Z|^2\\rangle^k$ equal $k!$ or $(2k-1)!!$ at leading order, so the noise statistics match random-matrix theory for low order, with the first deviation appearing near $k\\sim N^{q/2-1}$.","The leading correction is inversely proportional to the number of independent random couplings, so the deviation is amplified in sparsified SYK; numerics confirm the $k(k-1)$ dependence and a coefficient growing roughly as $p^{-1.2}$ with the sparsification probability $p$, in a regime where the linear ramp is still intact.","A microcanonical filter that removes the spectral edges restores the random-matrix values of the moments, confirming that the non-universal correction originates from edge fluctuations rather than the bulk of the spectrum.","In $q=2$ SYK the moments grow exponentially in $k$ and in time, with noise of order $e^{T\\log(N/T)}$, and the plateau value is $\\langle |Z|^{2k}\\rangle=\\binom{2k}{k}^{N/2}$, far from the $k!$ form of a complex Gaussian.","Since deviations show up at $k\\sim N^{q/2-1}$, much earlier than the exponential-in-$N$ scale expected in random-matrix ensembles, SYK belongs to a different, 'sparse' universality class of chaotic systems whose higher moments reveal the difference."],"supporting_citations":[{"why":"Constructs the continuous family of ramp saddle points in SYK from image-summed thermofield-double correlators; the k-replica pairing saddles are built from these blocks.","marker":"[24]"},{"why":"Identifies the same pairing-saddle structure for spectral form factor moments in a quantum spin glass and supplies the exponential decay of correlators that justifies the image sum.","marker":"[29]"},{"why":"Provides the sparse-versus-dense classification of chaotic systems and the edge-of-spectrum fluctuation scalings used to interpret the correction's inverse dependence on the number of random parameters.","marker":"[30]"},{"why":"Introduces the zero-dimensional one-time-point SYK whose exact moments give the benchmark $1/N^{q-2}$ correction structure that the SYK computation mirrors.","marker":"[39]"},{"why":"Establishes the exponential ramp and time-dependent symmetry breaking in $q=2$ SYK that the moments analysis extends to all $k$.","marker":"[36]"},{"why":"Defines the sparse SYK model and its holographically relevant sparsification regime, the setting for the numerical moment analysis.","marker":"[34]"},{"why":"Shows that the linear ramp persists in the holographic phase of sparse SYK, fixing the regime where the paper's moment deviations appear.","marker":"[35]"},{"why":"Supplies the random-matrix perturbative computation of the moments, the $k!$ ramp, that constitutes the universality baseline the SYK result is compared against.","marker":"[27]"},{"why":"Computes the $k=1$ connected correction around the disconnected saddle point and frames the multi-trace correlator context for the edge fluctuations.","marker":"[33]"}],"fun_headline_variants":["SYK spectral form factor moments reveal deviation from random matrix universality","High-order SYK moments deviate from RMT, correction scales as k^2/N^{q-2}","SYK spectral form factor moments leave RMT at high order; sparsification amplifies","Exponential ramp in q=2 SYK from zero-mode volume growth","Random matrix universality breaks for SYK spectral form factor moments at high k"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the ramp saddle-point solution remains valid in the spectral-edge regime that dominates the correction, and that the energy imbalance it defines grows with $N$; if the edge region is not captured by the solution, the predicted $k^2/N^{q-2}$ scaling of the deviation is unsupported.","fun_headline_variants_meta":{"raw":{"variants":["SYK spectral form factor moments reveal deviation from random matrix universality","High-order SYK moments deviate from RMT, correction scales as k^2/N^{q-2}","SYK spectral form factor moments leave RMT at high order; sparsification amplifies","Exponential ramp in q=2 SYK from zero-mode volume growth","Random matrix universality breaks for SYK spectral form factor moments at high k"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001019,"raw_usage":{"total_tokens":4404,"prompt_tokens":1156,"completion_tokens":3248,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":772,"completion_tokens_details":{"reasoning_tokens":3140}},"tokens_in":772,"tokens_out":3248,"duration_ms":25095,"temperature":1.0,"reasoning_tokens":3140,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T04:31:18.590370+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One could settle the claim numerically: for fixed $q=4$, solve the SYK saddle-point equations by iteration at inverse temperatures $\\beta_{\\rm aux}\\gtrsim T$, compute $\\Delta E$ from (3.35), and test whether it is extensive in $N$ and matches the conformal-image approximation; if it does not, equation (3.37) is not established. A complementary check is exact-diagonalization data on unsparsified $q=4$ SYK at two values of $N$, asking whether the time-averaged moment correction $B/k(k-1)$ scales as $N^{-2}$ across the ramp plateau as predicted.","supporting_citations":[{"cited_title":"Spectral form factor of a quantum spin glass","cited_arxiv_id":null,"evidence_quote":"Identifies the same pairing-saddle structure for spectral form factor moments in a quantum spin glass and supplies the exponential decay of correlators that justifies the image sum."}],"review_version":1}