Random Matrices, Spectral Statistics, and Ensemble Questions

At finite $N_\chi$, the Schwarzian controls a smooth low-energy density but not the positions of individual SYK levels. The narrower question here is statistical: after selecting one irreducible symmetry block and removing the smooth variation of the density, do the remaining level correlations match the Wigner–Dyson class fixed by that block’s antiunitary symmetries? Agreement is evidence for local universality in a declared sector, energy window, and averaging protocol. It does not identify a unique Hamiltonian, determine the full density of states, or prove a gravitational dual.

Required background. Matrix Eigenvalue Saddles, Loop Equations, and Phase Transitions supplies matrix-ensemble observables and large-$N$ densities.

Helpful background. Spectral Statistics, Form Factors, and Late-Time Evidence develops generic chaos diagnostics; Spectra, Resolvents, Spectral Measures, and Functional Calculus supplies spectral definitions; JT Topological Expansion and Weil–Petersson Volumes explains why a matrix integral appears in JT gravity.

Scope. The worked model is nonsupersymmetric $q=4$ Majorana SYK with even $N_\chi$, one fermion-parity block, and bulk spectral windows. Generic $q$, supersymmetric variants, and spectral-edge classes require the wider classification linked below. The argument moves from probability spaces, to the block symmetry, to local observables, to a finite-$N_\chi$ calculation and its adversarial controls.

Evidence reviewed: 29 August 2026.

Reading route. First separate the fixed-spectrum baseline from the five averaging operations, then resolve the symmetry block, unfold the smooth density, and compare local with longer-range observables. The reproducible calculation and adversarial controls show exactly how far the conclusion survives.

A fixed SYK spectrum and two comparison ensembles

Write one coupling realization as $\boldsymbol J$ and use the convention

$$H_{\boldsymbol J} =\sum_{1\le i<j<k<l\le N_\chi} J_{ijkl}\chi_i\chi_j\chi_k\chi_l, \qquad \{\chi_i,\chi_j\}=\delta_{ij},$$

with independent centered Gaussian couplings satisfying

$$\mathbb E_{\boldsymbol J}[J_{ijkl}^{2}] =\frac{3!J^2}{N_\chi^3}.$$

For even $N_\chi=2m$, the Hilbert space has dimension $2^m$. Fermion parity commutes with $H_{\boldsymbol J}$ and divides it into two blocks of dimension $2^{m-1}$. Fixing $\boldsymbol J$ and one block gives a definite finite matrix and a definite list of levels, so no ensemble randomness remains. A predeclared window can still define an empirical average by assigning equal weight to its eligible level indices; varying $\boldsymbol J$ defines the separate disorder average.

The SYK disorder measure is not a Gaussian invariant measure on all entries of that block. A Gaussian random-matrix-theory (RMT) reference instead has the schematic measure

$$d\mu_{\beta_{\rm D}}(M) \propto e^{-c\operatorname{Tr}M^2}\,dM, \qquad \beta_{\rm D}=1,2,4,$$

on real-symmetric, complex-Hermitian, or quaternion-self-dual matrices. The scale $c>0$ disappears after local unfolding. Universality says that very different matrix measures can share limiting local correlations when they have the same symmetry class. It does not say that their smooth densities or individual eigenvalues agree.

The word “average” therefore needs an object, not just a bar:

Operation	What varies	What is held fixed	What it can establish
One fixed realization	Nothing	$\boldsymbol J$, parity block, and all levels	A sample-specific statement
Spectral-window average	The level index inside a declared window	One $\boldsymbol J$ and one block	Correlations sampled within that spectrum, if the window is locally stationary
Disorder average $\mathbb E_{\boldsymbol J}$	The SYK couplings	$N_\chi$, $q$, normalization, sector rule, and analysis protocol	Typical behavior and sample-to-sample uncertainty for that SYK ensemble
RMT average $\mathbb E_{\rm RMT}$	The reference matrix	Matrix dimension, Dyson class, and normalization	A calibrated universal reference distribution
Time smoothing	A time bin or time window	Hamiltonian and spectral normalization	A stabilized real-time diagnostic, treated on the spectral-form-factor page
JT matrix-integral average	Matrix degrees of freedom; on the gravity side, the perturbative expansion sums allowed topologies	The specified completion and observable	Matrix-integral correlators, not automatically the spectrum of one fixed boundary theory

The comparison on this page is between disorder- and window-averaged statistics of $H_{\boldsymbol J}$ and the corresponding RMT average. In ordinary oriented bosonic JT gravity, the genus expansion maps to a double-scaled Hermitian matrix integral Saad, Shenker, and Stanford 2019, abstract and §§ 1–2. Time-reversal, spin, pin, and supersymmetric variants change the allowed gravity topologies and can realize other Dyson or Altland–Zirnbauer classes Stanford and Witten 2020, §§ 2 and 5. This gravity-side classification is not obtained by reading $N_\chi\bmod8$ from one finite SYK model. Whether the matrix integral should be interpreted as an ensemble of microscopic theories is a separate question, treated on Non-Unique JT Matrix-Integral Completion.

The symmetry block fixes the Dyson class

An antiunitary operator has the form $\mathcal A=UK$, with $K$ complex conjugation in a chosen basis. What matters for a block is not merely whether $\mathcal A$ commutes with the full Hamiltonian, but whether it maps the selected block to itself.

• If a block-preserving $\mathcal A$ obeys $\mathcal A^2=+1$, a basis exists in which the block is real symmetric: the orthogonal class, GOE, with $\beta_{\rm D}=1$.
• If no antiunitary acts within the block, it is in the unitary class, GUE, with $\beta_{\rm D}=2$.
• If a block-preserving $\mathcal A$ obeys $\mathcal A^2=-1$, every level is a Kramers doublet and the distinct multiplets follow the symplectic class, GSE, with $\beta_{\rm D}=4$.

For the model and convention fixed above, Clifford-algebra periodicity gives García-García and Verbaarschot 2016, § IV and Table I and You, Ludwig, and Xu 2017, Eqs. (2)–(3) and Table I:

$N_\chi\bmod 8$	Action of the antiunitary symmetry	Comparison for one parity block	Required handling
$0$	Preserves each block and squares to $+1$	GOE, $\beta_{\rm D}=1$	Analyze each parity block separately
$2$	Exchanges the two parity blocks	GUE, $\beta_{\rm D}=2$	Use one block; the two block spectra are symmetry-related copies
$4$	Preserves each block and squares to $-1$	GSE, $\beta_{\rm D}=4$	Replace every exact Kramers pair by one multiplet energy before taking gaps
$6$	Exchanges the two parity blocks	GUE, $\beta_{\rm D}=2$	Use one block; the two block spectra are symmetry-related copies

This table is periodic in $N_\chi$ and specific to nonsupersymmetric $q=4$ SYK. Changing $q\bmod4$ or adding supersymmetry can move the Hamiltonian into other Altland–Zirnbauer classes; Kanazawa and Wettig 2017, §§ 2–3 gives that broader classification.

Two errors that look similar have different causes. Concatenating the two $N_\chi\bmod8=2,6$ parity spectra inserts duplicate levels because a symmetry relates the blocks. Retaining both members of an $N_\chi\bmod8=4$ Kramers pair inserts duplicates within a block. By contrast, combining unrelated symmetry sectors creates unconstrained crossings between otherwise repelling sequences. All three operations manufacture small gaps, but the repair is determined by the symmetry, not by deleting small spacings after looking at the answer.

Unfolding separates the density from the fluctuations

For the ordered levels of one resolved block, define the empirical staircase and a smooth approximation by

$$\mathcal N_{\boldsymbol J}(E) =\sum_i\Theta(E-E_i), \qquad \overline{\mathcal N}_{\boldsymbol J}(E) =\int_{-\infty}^{E}\overline\rho_{\boldsymbol J}(E')\,dE'.$$

The unfolded levels and spacings are

$$\xi_i=\overline{\mathcal N}_{\boldsymbol J}(E_i), \qquad s_i=\xi_{i+1}-\xi_i, \qquad \langle s_i\rangle_{\rm win}\simeq1.$$

This transformation removes the smooth density and leaves dimensionless fluctuations. It is essential because the large-$N_\chi$ SYK density is not the Wigner semicircle even when its local bulk correlations are Wigner–Dyson García-García and Verbaarschot 2016, §§ III–IV.

There are at least three non-equivalent choices: unfold with the disorder-averaged staircase $\mathbb E_{\boldsymbol J}\mathcal N_{\boldsymbol J}$, fit a smooth staircase separately in each realization, or use a local energy window as an ergodic estimator. Realization-dependent long-wavelength modes, especially scale fluctuations, make these choices differ in SYK Jia and Verbaarschot 2020, §§ 1 and 7.

Unfolding is therefore an inference step, not a cosmetic rescaling. An under-flexible fit leaves density curvature in a long-range statistic. An over-flexible fit follows the individual levels and subtracts the fluctuations being measured. A reproducible analysis declares the retained window, fits only scales longer than the statistic of interest, and repeats the calculation with independently motivated smoothers. The conclusion is restricted to the range on which those choices agree.

The spectral edge is not governed by the bulk sine kernel. Its correct edge class and scaling variable must be established separately; one must not impose regular soft-edge Airy scaling on an arbitrary SYK edge. Recent results determine the leading $q=4$ edge scale He 2026, Theorem 1.1 and prove a mesoscopic bulk local law—a density estimate on shrinking energy windows that still contain many levels—Benigni and Cipolloni 2026, Theorem 1.1. Neither result turns microscopic finite-$N_\chi$ SYK edge statistics into a classical soft-edge theorem.

Short- and long-range observables answer different questions

The nearest-neighbor distribution $P(s)$ tests the shortest unfolded scale. Level repulsion appears as $P(s)\sim s^{\beta_{\rm D}}$ near $s=0$, whereas independent Poisson levels have $P(s)=e^{-s}$.

An especially robust local statistic uses raw gaps $\delta_i=E_{i+1}-E_i$:

$$\widetilde r_i =\frac{\min(\delta_i,\delta_{i+1})} {\max(\delta_i,\delta_{i+1})}, \qquad 0\le\widetilde r_i\le1.$$

It is invariant under a local affine rescaling $E\mapsto aE+b$ with $a>0$ and therefore needs no explicit unfolding. It still requires a pure symmetry sector and a window over which the density changes slowly compared with a few gaps.

For independent exponential gaps, the unbounded ratio $u=\delta_{i+1}/\delta_i$ has density $(1+u)^{-2}$. Folding $u$ to $\widetilde r=\min(u,u^{-1})$ gives

$$p_{\rm P}(\widetilde r) =\frac{2}{(1+\widetilde r)^2}, \qquad \langle\widetilde r\rangle_{\rm P} =2\ln2-1 =0.386294\ldots.$$

For the three Gaussian ensembles, the accurate three-level surmise for the unbounded ratio is

$$p_{\beta_{\rm D}}^{(3)}(u) =\frac{1}{Z_{\beta_{\rm D}}} \frac{(u+u^2)^{\beta_{\rm D}}} {(1+u+u^2)^{1+3\beta_{\rm D}/2}}, \qquad 0\le u<\infty,$$

and the folded density on $0\le\widetilde r\le1$ is $2p_{\beta_{\rm D}}^{(3)}(\widetilde r)$. The surmise is not the exact large-matrix distribution. Atas et al. 2013, Eqs. (1)–(7) and Table I distinguish the two benchmarks:

Reference	$\beta_{\rm D}$	Three-level surmise for $\langle\widetilde r\rangle$	Large-matrix benchmark
Poisson	—	$0.386294\ldots$ (exact)	$0.386294\ldots$
GOE	$1$	$0.53590$	$\simeq0.5307$
GUE	$2$	$0.60266$	$\simeq0.5996$
GSE	$4$	$0.67617$	$\simeq0.6744$

The finite matrix dimension, energy window, and estimator should be calibrated against simulated reference matrices rather than hidden inside extra digits of these asymptotic values.

Longer ranges require unfolding. At unit unfolded density, the distinct-level GUE bulk correlation is

$$R_2(s)=1-Y_2(s), \qquad Y_2(s)=\left(\frac{\sin\pi s}{\pi s}\right)^2,$$

so that the connected part, excluding the self-correlation contact term, is $R_{2,c}(s)=-Y_2(s)$. If one instead correlates the density operator $\widehat\rho(x)=\sum_i\delta(x-\xi_i)$ with itself, the connected correlation contains the additional contact term $\delta(x-y)$. The scalar sine-kernel formula is specific to the GUE bulk; GOE and GSE require their Pfaffian matrix kernels.

The sine kernel is an exact scaling-limit statement for the GUE bulk and a much wider Wigner universality class under stated hypotheses Erdős et al. 2010, Theorems 1.1–1.2. It is a comparison target, not a theorem about finite SYK.

The corresponding number variance counts levels in an unfolded interval of mean length $L$:

$$\Sigma^2(L) =L+2\int_0^L(L-s)R_{2,c}(s)\,ds =L-2\int_0^L(L-s)Y_2(s)\,ds.$$

Poisson gives $\Sigma^2(L)=L$, while GUE grows only logarithmically at large $L$. Number variance is fixed by the connected two-point function. By contrast, the nearest-neighbor distribution is a gap probability and depends on the full hierarchy of correlations. Agreement can therefore hold for $P(s)$ and $\widetilde r$ yet fail at larger $L$. That failure locates a universality window; it does not erase the shorter-range result.

Worked finite-size SYK comparison

Protocol

The repository checker scripts/benchmark-random-matrix-spectral-ensembles.py makes the comparison reproducible with a Python 3 interpreter and NumPy. Run python scripts/benchmark-random-matrix-spectral-ensembles.py; it writes no files and prints one machine-readable JSON record. The fixed protocol is:

• PCG64 master seed $20260829$, IEEE-754 binary64/complex128 arithmetic, and $64$ independent SYK coupling realizations per $N_\chi$;
• the Hamiltonian and variance displayed above with $J=1$, exact Jordan–Wigner Majorana operators, and the even-parity block;
• the central $50\%$ of distinct levels, with exact Kramers pairs collapsed before windowing;
• the raw $\widetilde r$ statistic, so no fit-dependent unfolding enters this finite-SYK comparison;
• a normal-approximation $95\%$ Monte Carlo interval computed as the mean plus or minus $1.96s_{\rm real}/\sqrt{64}$, where $s_{\rm real}$ is the standard deviation of the realization means. Individual ratios are not treated as independent samples.

The reported record was reproduced with Python $3.12.13$ and NumPy $2.3.5$. The script SHA-256 is 43f8a50362a79cf4bb3fc90ef5c2d3fccee8941f73319b4a2b162badcf4f463b. Its canonical scientific payload—$120{,}445$ UTF-8 bytes after excluding the two runtime-version fields—has SHA-256 39031ee9219559242d5a7dda494e0f06b90d538216be91a86a716fb1dce87d72.

Finite-SYK result

The complete default run gives:

The parity-block dimensions for $N_\chi=12,14,16$ are $32,64,128$. The central window retains $8,32,64$ distinct levels and therefore supplies $6,30,62$ ratios per realization, respectively.

$N_\chi$	Predicted class	$\langle\widetilde r\rangle$ and realization-level $95\%$ interval	Large-matrix reference
$12$	GSE	$0.68863$; $[0.66667,0.71058]$	$0.6744$
$14$	GUE	$0.59676$; $[0.58331,0.61021]$	$0.5996$
$16$	GOE	$0.53221$; $[0.52320,0.54121]$	$0.5307$

Each measured mean is closest to the class predicted by $N_\chi\bmod8$. The $N_\chi=12$ interval is widest because Kramers reduction leaves only six ratios per realization; it is a useful finite-size check, not a precise GSE determination.

Independently trained unfolding

The bounded ratio does not test the unfolded spacing distribution or a longer-range two-level statistic. For that comparison, the checker focuses on the $N_\chi=16$ GOE case and keeps training separate from evaluation. Independent sets of $256$ SYK spectra and $256$ dimension-$128$ GOE spectra determine frozen monotone density maps for the two ensembles. The maps send the mean ordered energies to half-integer ranks, but use knots only every eight levels; they therefore vary on a scale longer than the largest tested interval, $L=4$, and never use an evaluation level to define its own rank. Separate sets of $64$ SYK and $64$ GOE spectra are then unfolded and restricted to fixed order indices $32$ through $95$, the central $64$ levels.

Two predeclared sensitivity changes guard against an analysis-dependent answer. The coarse-knot stride is changed from $8$ to $12$. In addition to the primary ensemble-density map, which retains realization-to-realization scale fluctuations, a second protocol first aligns each realization’s center and central-half width to the training mean; this affine mode asks the narrower question about local correlations conditional on that scale. Only features stable across both strides and both width treatments are interpreted.

For the primary stride-$8$ ensemble-density protocol, the mean unfolded spacing is $0.99721$ with interval $[0.99071,1.00370]$ for SYK and $1.00203$ with interval $[0.99853,1.00552]$ for GOE. The independent-ensemble difference, SYK minus GOE, is $-0.00482$ with interval $[-0.01219,0.00256]$. The full spacing histogram is:

$s$ interval	SYK mass ($95\%$)	GOE mass ($95\%$)
$[0,0.25)$	$0.04812$; $[0.04284,0.05339]$	$0.04712$; $[0.04062,0.05362]$
$[0.25,0.5)$	$0.13492$; $[0.12419,0.14565]$	$0.13294$; $[0.12413,0.14174]$
$[0.5,0.75)$	$0.18328$; $[0.17159,0.19498]$	$0.18824$; $[0.17818,0.19831]$
$[0.75,1)$	$0.18229$; $[0.17011,0.19447]$	$0.16468$; $[0.15193,0.17743]$
$[1,1.25)$	$0.16270$; $[0.15112,0.17427]$	$0.17411$; $[0.16443,0.18378]$
$[1.25,1.5)$	$0.11607$; $[0.10799,0.12415]$	$0.12376$; $[0.11431,0.13321]$
$[1.5,2)$	$0.12302$; $[0.11327,0.13277]$	$0.11979$; $[0.10942,0.13016]$
$[2,3)$	$0.04886$; $[0.04281,0.05491]$	$0.04812$; $[0.04212,0.05411]$
$[3,\infty)$	$0.00074$; $[-0.00008,0.00157]$	$0.00124$; $[0.00019,0.00229]$

Each row first averages the bin mass within a realization; only the $64$ realization summaries enter its standard error. The negative lower endpoint in the sparse final bin is an artifact of the unbounded normal approximation, not a negative probability. All nine primary SYK-minus-GOE intervals contain zero. The largest positive contrast is $0.01761$ with interval $[-0.00002,0.03524]$ for $s\in[0.75,1)$; the largest negative contrast is $-0.01141$ with interval $[-0.02649,0.00368]$ for $s\in[1,1.25)$. The pooled empirical-CDF sup distance is $D=0.02108$; its delete-one-realization sensitivity scale is $0.00736$. Because the supremum is non-smooth and positively biased near equal distributions, the checker reports no p-value or confidence interval for $D$.

For the two-level statistic, fixed interval origins have phases $0.17$ and $0.61$ relative to the unfolded unit grid and remain at least $L+4$ ranks from the retained-window boundary. Origins are averaged within each realization, never treated as independent observations. The primary result is:

$L$	SYK $\Sigma^2$ ($95\%$)	GOE $\Sigma^2$ ($95\%$)
$1$	$0.44216$; $[0.42581,0.45851]$	$0.44276$; $[0.42665,0.45886]$
$2$	$0.58259$; $[0.55503,0.61014]$	$0.55692$; $[0.53224,0.58160]$
$4$	$0.70749$; $[0.66525,0.74972]$	$0.73201$; $[0.68686,0.77716]$

The corresponding SYK-minus-GOE number-variance differences are $-0.00060$ with interval $[-0.02355,0.02235]$, $0.02567$ with interval $[-0.01132,0.06266]$, and $-0.02453$ with interval $[-0.08635,0.03730]$. The mean counts are $1.00646$, $2.00542$, and $4.00509$ for SYK and $1.00165$, $1.99665$, and $4.00164$ for GOE; every count-difference interval contains zero. Across all four unfolding configurations, $D$ ranges from $0.01587$ to $0.02108$. The SYK/GOE ranges for $\Sigma^2(L)$ are $0.44171$–$0.44742$/$0.43900$–$0.44276$ at $L=1$, $0.58179$–$0.58530$/$0.55692$–$0.57669$ at $L=2$, and $0.70403$–$0.70839$/$0.73183$–$0.75872$ at $L=4$.

These finite samples therefore support a deliberately descriptive statement: after independently trained unfolding, the central $N_\chi=16$ SYK spacing distribution is close to the identically processed GOE reference, and the number-variance point estimates remain comparable at $L=1,2,4$. In the primary protocol, none of the predeclared spacing-bin or number-variance contrasts is resolved away from zero by its normal interval. Across the three sensitivity protocols, every number-variance contrast and all but one spacing-bin contrast still contain zero; the stride-$8$ affine protocol gives $0.01711$ with interval $[0.00014,0.03408]$ for $s\in[0.75,1)$. That isolated, unadjusted bin-wise interval is not a global distribution test, and non-resolution elsewhere is not a proof of equivalence. The intervals condition on the frozen $256$-spectrum training maps; training-map uncertainty, finite-size effects, window choice, and model systematics are not included. This calculation does not determine a Thouless scale, edge statistics, the low-energy Schwarzian regime, a large-$N_\chi$ limit, or a gravity ensemble.

Reference calibration

A separate reference calibration in the same checker uses $128$ samples. The GOE and GUE matrices have dimension $192$ and retain their $96$ central eigenvalues. GSE uses a $192$-dimensional complex representation, reduces it to $96$ distinct Kramers multiplets, and retains the central $48$; the Poisson control uses $95$ independent exponential gaps. The calculation obtains $0.38261$ for Poisson, $0.52715$ for GOE, $0.59782$ for GUE, and $0.67504$ for Kramers-collapsed GSE. For GUE, the signed raw-minus-analytic-unfolded change in $\langle\widetilde r\rangle$ is only $-1.21\times10^{-5}$, as expected for this local affine-invariant statistic. Direct quadrature of the sine-kernel formula predicts

$$\Sigma^2(1)=0.344163, \qquad \Sigma^2(2)=0.415672, \qquad \Sigma^2(4)=0.486326,$$

while the calibrated GUE spectra give $0.34498$ with interval $[0.33897,0.35100]$, $0.41174$ with interval $[0.40402,0.41945]$, and $0.47958$ with interval $[0.46906,0.49010]$, respectively. Each sine-kernel target lies inside the displayed realization-level interval.

Implementation checks and published scale

For finite SYK, the checker independently verifies the Majorana algebra to $1.2\times10^{-16}$, parity preservation, Hermiticity, tracelessness, and Kramers pairing. The separate Gaussian GSE calibration also verifies the self-duality of its projected matrices. In the $N_\chi=12$ SYK run, the mean of each realization’s maximum relative Kramers-pair splitting is $1.00\times10^{-14}$, with interval $[8.93\times10^{-15},1.11\times10^{-14}]$. These are implementation checks; the statistical conclusion remains finite-$N_\chi$, $q=4$, one-sector exact-diagonalization evidence.

The stronger published calculation diagonalized parity blocks through $N_\chi=36$: $N_\chi=28$ followed GSE, $N_\chi=32$ followed GOE, and $N_\chi=34$ followed GUE at short range, while the number variance eventually departed from RMT at larger $L$ García-García and Verbaarschot 2016, § IV.1–IV.2 and Figures 7–10.

Adversarial spectral controls locate the inference boundary

The same checker runs three controls designed to fail for known reasons, followed by a realization-paired window-sensitivity test.

Combine unresolved sectors. Two individually GOE reference blocks have the equal-weight mean $\langle\widetilde r\rangle=0.52715$. Merging their equally scaled spectra lowers it to $0.42395$, with a paired shift of $-0.10320$ and interval $[-0.10887,-0.09753]$. The fraction $\Pr(\widetilde r<0.1)$ rises from $0.03079$ to $0.14280$. This protocol-specific control models unresolved independent sectors; the mixed sequence is not asserted to be exactly Poisson. It must not be confused with the symmetry-related parity spectra at $N_\chi\bmod8=2,6$.

Keep exact duplicates. If both members of every GSE Kramers pair are retained, $\langle\widetilde r\rangle$ falls from $0.67504$ after pair reduction to $2.3\times10^{-14}$. In the $N_\chi=12$ SYK run the analogous uncollapsed value is $2.9\times10^{-15}$. Concatenating the two symmetry-related GUE parity spectra at $N_\chi\bmod8=2,6$ creates the same zero-gap pathology. The correct operation is fixed by the symmetry before the data are inspected.

Fit the staircase itself. The maximally overfit map $\xi_i=i+\tfrac12$ produces a perfectly rigid lattice and forces the integer-$L$ number variance to zero. It is not legitimate unfolding: it has used every fluctuating level to define the “smooth” density. A result that exists only after such a fit is manufactured by the analysis.

Move the window or include the edge. Using the same diagonalized realization, the checker compares the central $50\%$ result with the central $75\%$ and with all distinct levels. The wider-window means are $0.67852$ and $0.67269$ for $N_\chi=12$, $0.59747$ and $0.59681$ for $N_\chi=14$, and $0.52716$ and $0.52838$ for $N_\chi=16$. The paired changes relative to the central half are:

$N_\chi$	Class	Central $75\%$ minus central $50\%$	All levels minus central $50\%$
$12$	GSE	$-0.01011$; $[-0.02540,0.00519]$	$-0.01593$; $[-0.03619,0.00432]$
$14$	GUE	$0.00070$; $[-0.00723,0.00864]$	$0.00005$; $[-0.00865,0.00874]$
$16$	GOE	$-0.00505$; $[-0.01049,0.00040]$	$-0.00383$; $[-0.00987,0.00220]$

Every paired interval contains zero, so this sample resolves no window shift in the mean bounded ratio. Including all levels is nevertheless only a sensitivity test: it neither establishes an edge universality class nor replaces the edge-specific scaling analysis discussed above.

Pool gaps as independent observations. Neighboring ratios share gaps, and all ratios from one disorder realization share the same couplings. Treating individual ratios as independent understates uncertainty. The checker instead forms one mean per coupling realization and estimates the standard error from the variation of those $64$ means. Its normal-approximation intervals quantify Monte Carlo realization scatter only; they exclude finite-size, window, unfolding, and model systematics. The paired window calculation above is a separate sensitivity check, not an extra contribution folded into those intervals.

The checker therefore licenses two deliberately narrow conclusions. For the finite $q=4$, even-parity blocks at $N_\chi=12,14,16$, the mean bounded adjacent-gap ratio in the declared central window is consistent with the class predicted by $N_\chi\bmod8$. For $N_\chi=16$, the independently unfolded central spacing distribution is descriptively close to the matched GOE result and the tested number variances have unresolved contrasts at $L=1,2,4$. The broader short-range Wigner–Dyson statement rests on the published spacing distributions and number-variance calculation through larger sizes García-García and Verbaarschot 2016, § IV.1–IV.2 and Figures 7–10. Neither result licenses the stronger claims that the SYK density is semicircular, that one realization equals an ensemble average, that the SYK coupling measure is the Gaussian invariant matrix measure, or that a JT matrix integral uniquely selects a microscopic boundary Hamiltonian.

Energy correlations lead to time correlations

The connected two-level correlation has a Fourier transform in the spectral form factor. Sine-kernel correlations become the universal ramp after the required normalization and averaging, while the finite number of levels produces a plateau. The time-domain statement needs its own connected subtraction, thermal weighting, time smoothing, and finite-size windows. Those operations—and the distinction between a visible ramp and a complete universality claim—belong to JT/SYK Spectral Form Factors and Universality Windows, with the SYK calculation developed by Cotler et al. 2017, §§ 3, 5–6, and 8.

Common pitfalls

“GOE, GUE, or GSE” describes the whole Hamiltonian. It describes the local correlation class of a resolved block in a stated regime. The smooth density and nonuniversal long-range modes retain model-specific information.

Any antiunitary symmetry implies GOE or GSE. The antiunitary must act within the selected block. If it exchanges two blocks, each block can be GUE even though the full Hamiltonian has an antiunitary symmetry.

The ratio statistic makes window choices irrelevant. $\widetilde r$ removes a local scale, not a rapidly changing density, an edge crossover, an unresolved symmetry, or a small-sample bias.

A three-level surmise is an exact large-matrix value. It is an unusually accurate approximation. Use the large-matrix benchmark—or better, the finite-dimension reference generated by the same analysis pipeline—when quoting a numerical comparison.

RMT agreement proves a gravity dual. It is evidence for spectral universality. Many nonholographic chaotic systems share the same local correlations.

Exercises

1. GUE level repulsion from the sine kernel

Expand the distinct-level GUE correlation at small $s$ and find its leading power.

Solution

Using

$$\frac{\sin\pi s}{\pi s} =1-\frac{\pi^2s^2}{6}+O(s^4),$$

one finds

$$Y_2(s)=1-\frac{\pi^2s^2}{3}+O(s^4), \qquad R_2(s)=\frac{\pi^2s^2}{3}+O(s^4).$$

The quadratic zero is the $\beta_{\rm D}=2$ level-repulsion exponent. This is a two-level statement; the nearest-neighbor distribution contains the additional condition that no level lies between the pair.

2. Why the bounded ratio needs no explicit unfolding

Show that $\widetilde r_i$ is invariant under $E_i\mapsto aE_i+b$ for $a>0$. Why does that not justify using a broad window with a rapidly varying density?

Solution

Every gap transforms as $\delta_i\mapsto a\delta_i$, so the common factor cancels between the minimum and maximum. A nonlinear unfolding is locally affine only when its derivative changes negligibly across the three levels entering the ratio. Across a broad nonstationary window, different local densities and possibly different regimes are pooled, so the empirical ratio distribution need not represent one statistical process.

3. Derive the Poisson benchmark

Starting from independent unit-mean exponential gaps $x$ and $y$, derive the density and mean of $\widetilde r=\min(x,y)/\max(x,y)$.

Solution

For the unbounded ratio $u=y/x$,

$$p(u)=\int_0^\infty x e^{-x}e^{-ux}\,dx =\frac{1}{(1+u)^2}.$$

The regions $u<1$ and $u>1$ give equal folded contributions, so

$$p_{\rm P}(\widetilde r)=\frac{2}{(1+\widetilde r)^2}.$$

Therefore

$$\langle\widetilde r\rangle =2\int_0^1\frac{r}{(1+r)^2}\,dr =2\ln2-1.$$

4. Resolve the symmetry before measuring gaps

For $q=4$, compare $N_\chi=26$ and $N_\chi=28$. Identify the target ensemble and explain what goes wrong if all apparent eigenvalues are used without resolving the degeneracy structure.

Solution

$26\bmod8=2$, so each parity block is GUE. The antiunitary exchanges the blocks and relates their spectra; concatenating both blocks inserts duplicate levels. $28\bmod8=4$, so each parity block is GSE and has Kramers doublets. One must first reduce each exact doublet to one multiplet energy. In both cases, keeping the duplicates creates zero gaps and falsely weakens the measured level repulsion, but the symmetry explanation and correct preprocessing differ.

5. Classify the claim, not just the statistic

Suppose one fixed $N_\chi=32$ parity block has $\langle\widetilde r\rangle=0.53$ in its central half. The mean over many coupling realizations is $0.531$ with a normal-approximation interval $[0.525,0.537]$ computed from the realization means. Under a declared unfolding protocol, the number variance agrees with GOE through $L\simeq25$ and then departs. State the strongest licensed conclusion.

Solution

The fixed sample’s mean bounded ratio is consistent with the GOE benchmark in that block and window. The disorder result supports the same mean-ratio consistency for the declared finite-$N_\chi$ SYK ensemble, with the displayed sampling uncertainty; one scalar ratio mean alone does not establish the full spacing distribution. The number variance separately supports GOE two-point behavior through roughly $L\simeq25$ for that unfolding protocol, while its subsequent departure limits the longer-range claim. None of these observations determines the smooth density, proves all realizations agree, identifies the SYK disorder measure with GOE, or establishes a gravity dual.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

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