Non-Unique JT Matrix-Integral Completion: Fixed-Theory, Disorder-Average, and Ensemble Distinctions

The JT genus expansion does not determine a unique exact Hamiltonian or even a unique exact matrix ensemble. It determines a formal series of connected amplitudes. A concrete pair of hard-wall matrix models makes the gap visible: the choices $\sigma=0$ and $\sigma=u_{\rm t}<0$ have the same JT expansion to every order in $\hbar=e^{-S_0}$, but only the second permits negative-energy levels. No amount of genus data can decide between them.

That completion question is separate from the ensemble question. Choosing an exact matrix measure, choosing one Hamiltonian from it, deciding whether two boundaries share the same draw, and selecting a baby-universe sector are four different operations. This page constructs the two completions, gives an observable that separates them, and then applies the factorization test.

Required background. JT Topological Expansion and Weil–Petersson Volumes supplies the perturbative amplitudes and their large-genus limitation; Random Matrices, Spectral Statistics, and Ensemble Questions distinguishes a fixed spectrum from a comparison ensemble.

Helpful background. Fixed-Theory, Ensemble, and Superselection Claims develops the general factorization grammar; JT/SYK Spectral Form Factors and Universality Windows shows how the same distinction enters a late-time observable.

Evidence cutoff: 29 August 2026.

Fixed Hamiltonians, matrix measures, and JT amplitudes

Let $r$ label one spectral realization. Its Hamiltonian $H_r$ has definite thermal trace and exact density

$$Z_r(\beta)=\operatorname{Tr}e^{-\beta H_r}, \qquad \rho_r(E)=\sum_j\delta(E-E_{rj}).$$

A probability measure $d\mu(r)$ produces ensemble moments and the smooth ensemble-mean density

$$\overline{Z(\beta_1)\cdots Z(\beta_n)} =\int d\mu(r)\prod_{i=1}^n Z_r(\beta_i), \qquad \overline{\rho(E)}=\int d\mu(r)\rho_r(E).$$

The integral includes a discrete disorder sum as a special case. It is important that $\overline{\rho(E)}$ is not the ordered spectrum of any one realization.

At a finite regulator, a positive normalized Hermitian matrix model supplies such a probability measure. A formal double-scaled loop equation, an unstable real-contour integral, or a complex-contour prescription need not do so until its contour, normalization, and positivity properties are specified. Saad, Shenker, and Stanford first establish the JT relation order by order and explicitly call the associated exact completion nonunique Saad, Shenker, and Stanford 2019, § 1, pp. 2–6, and § 5.5, pp. 50–51.

Write

$$\hbar=e^{-S_0}.$$

A connected surface of genus $g$ with $n$ asymptotic boundaries has Euler characteristic $\chi=2-2g-n$ and weight

$$e^{S_0\chi}=\hbar^{2g+n-2}.$$

In the standard JT matrix-integral dictionary,

$$\left\langle\prod_{i=1}^n Z(\beta_i)\right\rangle_c \sim \sum_{g=0}^{\infty} \hbar^{2g+n-2}\mathcal Z_{g,n}(\beta_1,\ldots,\beta_n).$$

The subscript $c$ denotes an ensemble cumulant. For two boundaries,

$$\overline{Z_1Z_2} =\overline{Z_1}\,\overline{Z_2} +\overline{Z_1Z_2}_{\!c}.$$

At leading topology, two disks supply the leading product term and the double trumpet supplies the genus-zero connected covariance; higher-genus one- and two-boundary surfaces complete the respective series. This is a precise perturbative equality between topological amplitudes and matrix cumulants, not a statement that a fixed $Z_r(\beta)$ is random Saad, Shenker, and Stanford 2019, § 3, especially Eq. (63).

Object	Operation	What it can determine
One fixed $H_r$	No spectral average	Ordered levels, exact recurrences, and the number $Z_r(\beta)$
A normalized $d\mu(r)$	Average over realizations	Mean densities, moments, cumulants, and distributions
The JT genus series	Expand connected amplitudes in $\hbar$	Every coefficient $\mathcal Z_{g,n}$ in its perturbative domain
An exact completion	Add a contour, wall, Stokes data, or equivalent boundary condition	Beyond-all-orders spectral information

Exponentially small data escape every genus coefficient

A function $f(\hbar)$ is beyond all perturbative orders when it vanishes faster than every power of $\hbar$. For $A>0$ and every fixed nonnegative integer $m$,

$$\lim_{\hbar\to0^+} \frac{e^{-A/\hbar}}{\hbar^m} =\frac{1}{A^m} \lim_{y\to\infty}y^m e^{-y} =0, \qquad y=\frac{A}{\hbar}.$$

Consequently, adding a term of order $e^{-A/\hbar}=e^{-Ae^{S_0}}$ changes none of the coefficients in a power series in $e^{-S_0}$. The statement applies at fixed energy or fixed inverse temperature in the perturbative regime. It need not be uniform in a moving edge window or at times and temperatures that themselves scale exponentially with $1/\hbar$.

Different exact inputs can carry this missing information. Saad, Shenker, and Stanford discuss normalized combinations of eigenvalue contours with the same asymptotic series. Hard spectral walls, trans-Fermi boundary data, and transseries Stokes constants are related examples of nonperturbative input, but they are not literally the same construction. A recent KdV transseries calculation recovers the leading instanton sectors yet leaves its Stokes data undetermined; the authors conclude that the KdV equation alone is incomplete as a full nonperturbative definition Hatsuda, Matsumoto, and Okuyama 2025, § 7, p. 25.

Two hard-wall completions with the same JT expansion

The promised pair comes from a constrained family of double-scaled Hermitian matrix models. Fix the JT couplings, the Fermi level $\mu=0$, and $\Gamma=0$. Define

$$\mathcal R[u] =\sum_{k=1}^{\infty}t_k R_k[u]+x, \qquad t_k=\frac{\pi^{2k-2}}{2k!(k-1)!},$$

where the Gel’fand–Dikii polynomial $R_k[u]$ begins with $u^k$ and contains derivative terms. The hard-wall string equation is

$$(u-\sigma)\mathcal R^2 -\frac{\hbar^2}{2}\mathcal R\mathcal R'' +\frac{\hbar^2}{4}(\mathcal R')^2 =0,$$

with primes denoting $x$ derivatives. Its perturbative Fermi-sea branch is

$$\mathcal R=0, \qquad x<0.$$

Crucially, this branch does not contain $\sigma$. It gives the JT disk density

$$\rho_{\rm disk}(E) =\frac{\sinh(2\pi\sqrt E)}{4\pi^2\hbar}, \qquad E>0,$$

and the same recursive higher-genus coefficients for every allowed finite $\sigma$. The parameter enters when the solution is continued through and beyond the Fermi surface, where perturbation theory no longer fixes it Johnson 2022, §§ II–IV, especially Eqs. (8)–(10), (20), and (22)–(25).

The exact ensemble-mean density associated with a completed solution $u_\sigma(x)$ is obtained from the auxiliary Schrödinger problem

$$\left[-\hbar^2\partial_x^2+u_\sigma(x)\right] \psi_\sigma(E,x)=E\psi_\sigma(E,x),$$ $$\rho_\sigma(E) =\int_{-\infty}^{0} \left\lvert\psi_\sigma(E,x)\right\rvert^2dx.$$

This $\rho_\sigma$ is an exact density of the proposed ensemble; a single matrix draw still has delta-function levels.

Choose the two endpoints of the continuous, single-valued, one-cut family:

Field	Completion A	Completion B
Hard-wall datum	$\sigma_A=0$	$\sigma_B=u_{\rm t}$
Semiclassical continuation	$\mathcal R_0=0$ up to $x=0$, then $u_0=0$	$\mathcal R_0=0$ to its first turning point, then $u_0=u_{\rm t}$
Exact support in the proposal	$E\ge0$	$E\ge u_{\rm t}$
Shared data	All JT genus coefficients at fixed perturbative kinematics	The same coefficients
Distinguishing data	No negative levels	A nonperturbatively small negative-energy population
Logical status	Stable hard-wall matrix-model candidate	Inequivalent candidate in the same constrained family

The $\sigma=0$ endpoint was first proposed as a stable low-energy completion with the same high-energy perturbation theory Johnson 2020, abstract and §§ III.C–IV. The later construction embeds it in the continuous family above; neither paper derives a preferred member from a microscopic fixed boundary theory.

The second endpoint is analytic. If $j_{0,1}=2.404825557\ldots$ is the first positive zero of $J_0$, then

$$u_{\rm t} =-\left(\frac{j_{0,1}}{2\pi}\right)^2 =-0.1464898117\ldots .$$

The wall does not merely delete levels from Completion A: changing $\sigma$ changes the self-consistent potential, wavefunctions, density, kernel, and ground-state distribution. Johnson’s $k=7$, $\hbar=1$ numerical truncation displays these near-edge differences; it is evidence for the proposed infinite-family construction, not an exact numerical solution of the untruncated infinite-$k$ problem Johnson 2022, § IV.B, Figs. 6–9.

A semiclassical scale for the difference follows from the one-eigenvalue effective potential, for $E<0$,

$$V_{\rm eff}(E) =\frac{ \sin(2\pi\sqrt{-E}) -2\pi\sqrt{-E}\cos(2\pi\sqrt{-E}) }{4\pi^3\hbar}.$$

At the second wall,

$$V_{\rm eff}(u_{\rm t}) =\frac{0.0197783992\ldots}{\hbar},$$

so a representative instanton correction scales as $e^{-0.0197783992/\hbar}$. The coefficient is completion- and observable-dependent; its role here is to show explicitly why the difference is invisible to every genus coefficient, not to claim a universal pointwise error estimate.

Reproducibility declaration. The conventions are $\hbar=e^{-S_0}$, $\mu=0$, $\Gamma=0$, and the fixed $t_k$ above. The primary observables are the ensemble-mean density and the negative-level count below. The perturbative comparison holds at fixed $E>0$ or fixed $\beta$ as $\hbar\to0$; the edge itself is treated nonperturbatively. The control changes only $\sigma$ between the two stated endpoints. Analytic support and common formal coefficients are distinct from the published finite-truncation numerical evidence. The family assumes a one-cut solution and a continuous, single-valued $\hbar\to0$ seed; discontinuous and multi-cut extensions are outside this comparison.

The negative-level test defeats unique-completion inference

Now ask the genus expansion to predict the spectral observable

$$N_-(H)=\operatorname{Tr}\mathbf 1_{(-\infty,0)}(H).$$

Completion A gives

$$N_-(H)=0$$

for every realization. Completion B instead has

$$\overline{N_-} =\int_{u_{\rm t}}^0\rho_{u_{\rm t}}(E)\,dE>0,$$

although the expectation is beyond all orders as $\hbar\to0$. Both models receive the same complete perturbative input $\{\mathcal Z_{g,n}\}$. They nevertheless disagree on a binary spectral question. This is a direct counterexample to the inference

$$\text{all JT genus amplitudes} \quad\Longrightarrow\quad \text{one unique exact completion}.$$

The counterexample does not prove that no microscopic principle could select one model. Positivity, a wall or contour, Stokes data, a UV construction, or a specified boundary Hamiltonian may do so. It proves that each such selector is additional information.

Demanding one fixed theory is stronger still. One must request a single ordered spectrum in a regulated window and receive the same spectrum on every repetition. Returning a ground-state distribution, an ensemble covariance, or a fresh matrix draw answers an ensemble question instead. The double-scaled theory is an $L\to\infty$ edge limit, so this test should not be misstated as a demand for one particular finite-dimensional matrix.

Shared draws create covariance; fixed copies factorize

Consider two genuinely decoupled boundary systems in one fixed realization $r$. Their partition functions on a disconnected union factorize:

$$Z_r(B_1\sqcup B_2)=Z_r(B_1)Z_r(B_2).$$

This statement assumes independent tensor factors and no shared constraint, coupling, gauge projection, or random label. It does not say that all operator correlators inside one connected quantum system vanish.

It is specifically a boundary-theory statement. Canonical quantization of pure Lorentzian JT instead gives a nonfactorizing two-boundary Hilbert space; Harlow and Jafferis use this result to argue that standalone pure JT is not itself a standard fixed two-copy CFT dual Harlow and Jafferis 2019, § 4, pp. 22–24.

An ensemble can use the same draw for both boundaries. For a two-point distribution with probabilities $p$ and $1-p$, write $Z_a^{(i)}=Z_a(\beta_i)$ and $Z_b^{(i)}=Z_b(\beta_i)$. The shared-draw covariance is

$$C_{\rm shared} =p(1-p) \left(Z_a^{(1)}-Z_b^{(1)}\right) \left(Z_a^{(2)}-Z_b^{(2)}\right).$$

It is generally nonzero. If the two boundaries receive independent draws, the joint measure is $d\mu(r_1)d\mu(r_2)$ and

$$C_{\rm independent}=0.$$

In the standard JT matrix-integral interpretation, the connected double trumpet computes the shared-ensemble covariance. It does not by itself show that one fixed boundary theory violates factorization Saad, Shenker, and Stanford 2019, § 6.2, pp. 59–61.

An August 2026 preprint proposes a different map: a smooth filter acting on one erratic microscopic spectrum can yield a nonfactorizing macroscopic projection even though the underlying fixed partition function factorizes. The filter is therefore neither a disorder average nor the unfiltered fixed observable, and its general holographic construction is explicitly left open Liu 2026, § I.A, especially Eqs. (1.1)–(1.7). This proposal broadens the possible interpretation of a wormhole amplitude; it does not choose between the two exact hard-wall completions above.

Baby-universe language gives a conditional alternative. If the relevant boundary operators commute, one may choose simultaneous eigenstates $\lvert\alpha\rangle$. A Hartle–Hawking superposition produces ensemble moments, whereas conditioning on one $\alpha$ eigenstate makes those amplitudes factorize Marolf and Maxfield 2020, § 2.3, Eqs. (2.20)–(2.23). But selecting that sector is extra state data, and exact geometric calculations in JT $\alpha$-states are poorly controlled; controlled treatments use approximate states and restricted topology Saad, Shenker, and Yao 2024, abstract and §§ 1.1 and 5.3.

One explicit fixed-spectrum construction adds correlated branes, equivalently a nonlocal deformation, whose tuned interaction cancels connected wormholes and localizes the matrix integral on a supplied spectrum $H_0$ Blommaert, Iliesiu, and Kruthoff 2022, §§ 3.1 and 4.1–4.2. It is a deformed theory with $H_0$ as input, not a derivation of a preferred Hamiltonian from undeformed JT. Later work uses this construction to study conditional closed-universe factorization Usatyuk and Zhao 2025, §§ 2.1–2.3 and § 3.1.

The two axes must therefore remain separate:

Question	Choice being made	What does not answer it
Which exact completion?	Wall $\sigma$, contour, Stokes data, or another nonperturbative definition	Knowing every genus coefficient
Which microscopic theory?	One realization $r$ or independently supplied boundary Hamiltonian	Specifying only the ensemble measure
Which replication protocol?	Shared draw or independent draws	Writing an overbar without its measure
Which baby-universe sector?	Hartle–Hawking state, one $\alpha$ eigenstate, or another state	The existence of a connected saddle alone

The strongest supported claim is consequently narrow and useful:

Connected JT amplitudes reproduce the topological cumulants of a specified double-scaled random-matrix class. Perturbative JT data alone select neither a unique exact matrix completion nor a unique spectral realization. A specified fixed realization factorizes on the boundary; representing that fixed theory with controlled bulk variables requires additional, model-dependent sector, boundary, or deformation data.

Common pitfalls

An all-orders series is an exact definition. An asymptotic series can leave terms of order $e^{-A/\hbar}$ undetermined. State the nonperturbative input separately.

A smooth exact density is one Hamiltonian’s spectrum. $\rho_\sigma(E)$ is an ensemble mean. One realization has a discrete delta-function density and sample-specific phases.

A hard wall, a contour, and a Stokes constant are interchangeable names. Each can encode data invisible to the same perturbative series, but they define different mathematical completion problems.

A connected wormhole disproves fixed-theory factorization. In JT it matches a connected ensemble cumulant. The fixed-theory test instead uses two decoupled copies with one specified Hamiltonian.

An $\alpha$-state follows automatically from topology. Selecting an eigenstate of the commuting boundary algebra is additional state information, and its exact gravitational realization is model-dependent.

Exercises

1. Topology and the two-boundary cumulant

Write the moment–cumulant decomposition for two thermal boundaries. What power of $\hbar$ multiplies a connected genus-$g$ surface with two boundaries, and what is the leading power of the disconnected pair of disks?

Solution

The decomposition is

$$\overline{Z(\beta_1)Z(\beta_2)} =\overline{Z(\beta_1)}\,\overline{Z(\beta_2)} +\overline{Z(\beta_1)Z(\beta_2)}_{\!c}.$$

For a connected surface, $\chi=2-2g-2=-2g$, so its weight is $\hbar^{2g}$. A disk has $g=0$, $n=1$, and weight $\hbar^{-1}$; two disconnected disks therefore begin at $\hbar^{-2}$. The cylinder or double trumpet begins at $\hbar^0$. This power counting is why the connected term is subleading relative to the disconnected product without being zero.

2. Beyond all orders

Prove that $e^{-A/\hbar}$ is smaller than every power $\hbar^m$ for $A>0$. Why does the proof not guarantee uniform smallness at inverse temperature $\beta\sim e^{A/\hbar}$?

Solution

Set $y=A/\hbar$. Then

$$\frac{e^{-A/\hbar}}{\hbar^m} =\frac{y^m e^{-y}}{A^m}\longrightarrow0$$

because an exponential defeats every fixed polynomial as $y\to\infty$. The statement keeps all external kinematics fixed. If an observable contains a compensating factor that itself grows like $e^{A/\hbar}$, as can happen after taking an exponentially large time or temperature scaling, the product need not remain small. One must analyze that new double-scaling limit separately.

3. A spectral discriminator for the two walls

Using $j_{0,1}=2.404825557\ldots$, compute $u_{\rm t}$. Then explain why $N_-(H)$ distinguishes the two completions although their genus expansions agree.

Solution

Substitution gives

$$u_{\rm t} =-\left(\frac{2.404825557\ldots}{2\pi}\right)^2 =-0.1464898117\ldots .$$

The $\sigma=0$ wall forbids $E<0$, so $N_-=0$ realization by realization. The $\sigma=u_{\rm t}$ ensemble has support between $u_{\rm t}$ and zero and therefore $\overline{N_-}>0$. The negative population is exponentially suppressed in the perturbative limit, so it changes no coefficient of the common $\hbar$ series. It is nevertheless an exact spectral distinction.

4. One disorder draw or two?

Derive the shared-draw covariance for a two-point ensemble with weights $p$ and $1-p$. Repeat the calculation when each boundary receives an independent draw.

Solution

For one shared draw,

$$\overline{Z_1Z_2} =pZ_a^{(1)}Z_a^{(2)} +(1-p)Z_b^{(1)}Z_b^{(2)}.$$

Subtracting

$$\overline{Z_1}\,\overline{Z_2} =\left[pZ_a^{(1)}+(1-p)Z_b^{(1)}\right] \left[pZ_a^{(2)}+(1-p)Z_b^{(2)}\right]$$

and collecting terms yields

$$C_{\rm shared} =p(1-p) \left(Z_a^{(1)}-Z_b^{(1)}\right) \left(Z_a^{(2)}-Z_b^{(2)}\right).$$

With independent draws, the joint expectation already factors into the product of the two means, so $C_{\rm independent}=0$. The change is in the sampling protocol, not in either fixed Hamiltonian.

5. Classify the claim

Suppose a calculation reproduces every connected JT genus amplitude and a linear ensemble ramp. Which of the following follow: an exact matrix completion, a preferred $\sigma$, one fixed Hamiltonian, fixed-copy factorization, or the perturbative matrix-cumulant dictionary?

Solution

Only the perturbative matrix-cumulant dictionary follows. An ensemble ramp is one consequence of the connected two-level correlations under the declared average. The two hard-wall models show that the same genus data do not choose an exact completion or a preferred $\sigma$. An ensemble does not choose one realization, and fixed-copy factorization is a separate test after a Hamiltonian and replication protocol have been specified.

Scope and continuation

This page used low-dimensional JT matrix models as a controlled nonuniqueness laboratory. The general wormhole problem is developed in Factorization, Ensembles, and the Gravity Path Integral; the sector proposal is treated in Baby Universes, $\alpha$-Parameters, and Proposed Superselection Sectors; and contour and resurgence questions continue in Conformal-Factor Contours and Resurgent Completion Proposals.

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

• Blommaert, Andreas, Luca V. Iliesiu, and Jorrit Kruthoff. “Gravity Factorized.” Journal of High Energy Physics 2022, 080 (2022). DOI. Open PDF.
• Harlow, Daniel, and Daniel Jafferis. “The Factorization Problem in Jackiw–Teitelboim Gravity.” arXiv:1804.01081 [hep-th] (2019). arXiv.
• Hatsuda, Yasuyuki, Takaki Matsumoto, and Kazumi Okuyama. “Non-Perturbative Effects in JT Gravity from KdV Equations.” arXiv:2505.16433 [hep-th] (2025). arXiv.
• Johnson, Clifford V. “Nonperturbative Jackiw–Teitelboim Gravity.” Physical Review D 101, 106023 (2020). DOI. Open PDF.
• Johnson, Clifford V. “Consistency Conditions for Non-Perturbative Completions of JT Gravity.” arXiv:2112.00766 [hep-th] (2022). arXiv.
• Liu, Hong. “Ramp, Plateau, and Wormholes without Averaging, and Hyper-Non-Perturbative Structures in Gravity.” arXiv:2608.02743 [hep-th] (2026). arXiv.
• Marolf, Donald, and Henry Maxfield. “Transcending the Ensemble: Baby Universes, Spacetime Wormholes, and the Order and Disorder of Black Hole Information.” Journal of High Energy Physics 2020, 044 (2020). DOI. Open PDF.
• Saad, Phil, Stephen H. Shenker, and Douglas Stanford. “JT Gravity as a Matrix Integral.” arXiv:1903.11115 [hep-th] (2019). arXiv.
• Saad, Phil, Stephen H. Shenker, and Shunyu Yao. “Comments on Wormholes and Factorization.” Journal of High Energy Physics 2024, 076 (2024). DOI. Open PDF.
• Usatyuk, Mykhaylo, and Ying Zhao. “Closed Universes, Factorization, and Ensemble Averaging.” Journal of High Energy Physics 2025, 052 (2025). DOI. Open PDF.