Fixed-Theory Factorization and Nonperturbative Completion Tests

A fixed-theory completion must define exact observables, not just their asymptotic saddle expansion. It must also say what is held fixed when a boundary is copied, conditioned, averaged, or coarse-grained. Matching every coefficient in a genus series can be decisive perturbative evidence while still leaving the exact spectrum, multiboundary factorization, and even the nonperturbative integration cycle undetermined.

Required background. Factorization, Ensembles, and the Gravitational Path Integral distinguishes a fixed copy from a shared ensemble draw. Nonperturbative Definition and Completion Criteria supplies the more general completion standard.

Helpful background. Nonperturbative Exponential Effects and Finite-N Sectors compares effects invisible to a large-$N$ series. Microscopic Black-Hole Entropy: Claim and Ensemble Contract gives an example in which the microscopic object and the averaging prescription must be declared separately.

Reading path. First separate topology powers from beyond-all-genus data. Then define one protocol for the three candidate objects, work through an exact two-level counterexample, and use the final evidence section and exercises to test what each result does and does not establish.

Topology powers do not determine the exact theory

For connected Euclidean JT amplitudes with $n$ asymptotic boundaries of lengths $\boldsymbol\beta=(\beta_1,\ldots,\beta_n)$, take

$$g_{\mathrm{top}}=e^{-S_0}, \qquad \beta_i>0,$$

and hold the $\beta_i$ fixed as $g_{\mathrm{top}}\to0^+$. The orientable connected expansion is

$$\left\langle\prod_{i=1}^{n}Z(\beta_i)\right\rangle_{c} \sim \sum_{g=0}^{\infty} g_{\mathrm{top}}^{\,2g+n-2} Z_{g,n}(\boldsymbol\beta).$$

The exponent is $-\chi$, where $\chi=2-2g-n$. It therefore gives $g_{\mathrm{top}}^{-1}$ for the disk and $g_{\mathrm{top}}^0$ for the genus-zero double trumpet. Full, possibly disconnected moments are assembled from these connected cumulants. This is the multiboundary power counting needed for a factorization test, not the boundary-free power $g_{\mathrm{top}}^{2g-2}$. The JT/matrix-model equality of these formal connected coefficients is developed in Saad, Shenker, and Stanford 2019, §3, especially Eq. (63).

Three small quantities that often appear in the same discussion must remain distinct:

• Topology powers are $g_{\mathrm{top}}^p=e^{-pS_0}$. When $S_0$ scales as $1/G_N$, positive powers have the familiar $e^{-O(1/G_N)}$ form.

• Beyond-all-genus terms such as $e^{-A/g_{\mathrm{top}}}=e^{-Ae^{S_0}}$, with $A>0$, are smaller than every power of $g_{\mathrm{top}}$:

  $$\lim_{g_{\mathrm{top}}\to0^+} \frac{e^{-A/g_{\mathrm{top}}}} {g_{\mathrm{top}}^M}=0 \qquad\text{for every fixed }M>0.$$

• Finite-$N$ exponentials such as $e^{-aN^q}$ have a theory-dependent relation to $S_0$ and $G_N$. They cannot be identified with the preceding scale without a parameter map.

The formal genus data consequently do not choose a global matrix potential, an eigenvalue contour or wall, its normalization, or its Stokes data. In JT gravity, different contour choices can share the same asymptotic series, and the naive real-contour model is nonperturbatively unstable Saad, Shenker, and Stanford 2019, §§5.5–5.6. A chosen positive Hermitian measure can define an exact ensemble completion; the genus series does not select that completion uniquely.

Exact observables that separate candidate completions

Let $\lambda$ label a Hamiltonian, a baby-universe sector, or another piece of completion data, and let $z_\lambda(\beta)$ denote the candidate’s declared one-boundary value. After normalizing its measure $d\mu(\lambda)$, define

$$M_n(\boldsymbol\beta) =\int d\mu(\lambda)\prod_{i=1}^{n}z_\lambda(\beta_i).$$

When the candidate supplies a self-adjoint Hamiltonian $H_\lambda$, it must additionally show, on a stated trace-class domain, that

$$z_\lambda(\beta)=\operatorname{Tr}e^{-\beta H_\lambda}.$$

An alpha eigenvalue can define the left-hand side without yet constructing the Hamiltonian on the right. Keeping the two notions separate prevents a spectral conclusion from being assumed before it is tested.

The connected two-boundary quantity and its dimensionless defect are

$$\begin{aligned} \kappa_{12} &=M_2(\beta_1,\beta_2) -M_1(\beta_1)M_1(\beta_2),\\ f_{12} &=\frac{\kappa_{12}} {M_1(\beta_1)M_1(\beta_2)}. \end{aligned}$$

The normalized defect $f_{12}$ is defined only when $M_1(\beta_1)M_1(\beta_2)\ne0$.

For two decoupled copies of one fixed theory $T$, with no shared coupling, random draw, projection, or global constraint,

$$Z_{T\otimes T}[B_1,J_1;B_2,J_2] =Z_T[B_1,J_1]Z_T[B_2,J_2], \qquad \kappa_{12}^{T}=0.$$

All mixed connected source derivatives vanish as well. This exact boundary requirement is the factorization problem isolated in Harlow and Jafferis 2020, §4. Passing it is necessary for a fixed theory, but it does not by itself construct a positive Hilbert space, a unique completion, or a Lorentzian quantum field theory.

For a self-adjoint fixed Hamiltonian with a discrete finite-volume spectrum,

$$Z_T(\beta)=\sum_j d_j e^{-\beta E_j}, \qquad d_j\in\mathbb Z_{\ge0},$$

and the raw, unnormalized spectral form factor is

$$\begin{aligned} K_T(\beta,t) &=Z_T(\beta+it)Z_T(\beta-it)\\ &=\sum_{j,k}d_jd_k e^{-\beta(E_j+E_k)}e^{-it(E_j-E_k)}. \end{aligned}$$

A finite discrete spectrum is generally quasiperiodic and has arbitrarily accurate recurrences. It has an exact period $T$ only if every populated gap obeys $(E_j-E_k)T\in2\pi\mathbb Z$. An ensemble average, energy smoothing, time-window average, and smooth projection are four different operations; a ramp or plateau reported after one of them is not automatically a statement about the raw $K_T$ of one sample. The order of the fixed-size and late-time limits must also be reported.

Finally, a nonnegative mean spectral density is not enough to establish quantum-mechanical positivity. A completion should supply a normalized positive state on a $*$-algebra, reflection-positive Euclidean amplitudes for prepared states, the quotient by null states, a self-adjoint lower-bounded Hamiltonian, and strongly continuous unitary evolution. A recent reconstruction theorem obtains a unitary compact functorial QFT only after imposing a precise finite, multiplicative, reflection-positive framework; it does not prove that a gravitational path integral satisfies those assumptions McNamara and Wang 2026, Theorem 1.1 and §4.1.

JT ensemble, sharp alpha sector, and fixed CFT under one test

The comparison below uses the same fields for every candidate. Pass means the declared object supplies the item. Conditional means a stated extra hypothesis is essential. Fails fixed-theory test means the object can be well defined but is not one fixed Hamiltonian under this protocol. Not supplied means the cited construction does not yet provide the item; it is not a proof of impossibility.

The JT column evaluates the $\sigma=0$ hard-wall branch discussed on Fixed Theory, Disorder Average, and Ensemble Distinctions. Its published $k_{\max}=7$, $\hbar=1$ calculation truncates the string-equation sum and gives numerical evidence for a matrix-model candidate, not a proof of a normalized positive untruncated double-scaled measure Johnson 2022, §§II–IV, especially §IV.B, Eqs. (8)–(10), (20), (22)–(25), and Figs. 6–9. Exact ensemble-level cells are therefore marked conditional; only the formal order-by-order correspondence is scored independently of that existence question.

For reproducibility, the benchmark holds

$$\hbar=e^{-S_0}, \qquad \mu=\Gamma=0, \qquad \sigma=0, \qquad t_k=\frac{\pi^{2k-2}}{2k!(k-1)!}.$$

The perturbative comparison takes $\hbar\to0$ at fixed $E>0$ or fixed $\beta$; the spectral edge is treated nonperturbatively. These inputs identify the branch and limit without promoting the $k_{\max}=7$, $\hbar=1$ numerical truncation to an exact untruncated solution.

The alpha candidate is an exact simultaneous eigenstate of the complete commuting boundary-insertion algebra $\widehat Z[J]$. When the alpha spectrum is continuous, a point eigenstate can be generalized or delta-normalized rather than an ordinary normalized vector; a direct-integral prescription or a controlled sharp-window limit must then replace an unsupported normalizable-state assumption. The fixed-CFT benchmark is finite-$N$ $SU(N)$ $\mathcal N=4$ super-Yang–Mills theory on a fixed-radius $S^3$, with $N$, complex coupling $\tau$, spin structure, sources, and renormalization prescription fixed. The last benchmark uses the standard unitary-CFT definition; constructive continuum existence and equivalence remain separate mathematical questions, and the exact bulk duality is conditional.

The same nonperturbative completion tests applied to three candidate objects

Test field	JT hard-wall ensemble candidate	Exact sharp alpha sector	Named finite-N fixed CFT
Exact defining data and domain	Conditional candidate. The wall and formal branch are named, but a normalized positive untruncated measure and its double-scaling limit are not proved by the cited finite-truncation calculation.	Conditional. Requires a simultaneous eigenstate after quotienting null states, a normalization prescription appropriate to point or continuous spectrum, and a declaration of the complete commuting algebra.	Pass in the boundary-QFT definition. N, global gauge-group form, coupling, spatial manifold, sources, and renormalization data are fixed.
Controls, evidence, and residual uncertainty	Controls: string-equation truncation kmax, hbar, wall, beta range, regulator, and limit path. Evidence: formal all-genus match plus finite-truncation numerics at kmax = 7 and hbar = 1. Uncertainty: untruncated existence, positivity, normalization, and nonuniqueness.	Controls: algebra, sharpness or window, and allowed topology. Evidence: exact algebraic multiplication in the model, but approximate or restricted JT geometry. Uncertainty: preparation, completeness, positivity, residual cumulants, and sector preservation.	Controls: N, tau, radius, beta, sources, regulator, and copy protocol. Evidence: exact factorization in the boundary definition. Uncertainty: constructive continuum realization, exact duality, and the complete bulk topology rule.
One-boundary thermal object	Conditional. The candidate has finite-truncation spectral data; an exact untruncated ensemble mean requires the missing normalized measure. A smooth mean would not be one sample's spectrum.	Conditional. The eigenvalue of each declared boundary insertion is fixed, but those eigenvalues alone need not construct a Hamiltonian trace.	Pass as an exact object. The finite-volume trace is definite, although the full interacting spectrum is not generally known in closed form.
Two-boundary replication protocol	Fails the fixed-theory test if the ensemble exists. A shared-draw covariance is generally nonzero. Independent redraws factorize but answer a different question.	Pass within the declared algebra. Exact eigenstate expectation values multiply. Coarse or approximate conditioning can leave a residual connected term.	Pass. Two decoupled copies have zero mixed connected correlators when no shared projection or constraint is introduced.
Higher multiboundary observables	Pass for the formal coefficients; conditional exactly. Connected JT amplitudes match formal matrix cumulants order by order, while exact nonperturbative moments require the completed measure.	Conditional. Products factorize only for operators in the complete diagonal algebra and only in an exactly prepared sector.	Pass for disconnected copies. All mixed connected derivatives vanish by tensor-product factorization.
Spectrum and late-time diagnostic	Conditional. A regulated draw can have a discrete spectrum; the ensemble mean is smooth. Recurrences and the double-scaled late-time limit require the regulator and order of limits.	Not supplied. Alpha eigenvalues do not by themselves give energy levels, multiplicities, or the raw spectral form factor of one Hamiltonian.	Pass as a definite spectrum. The raw form factor is sample-specific and generally quasiperiodic; explicit levels remain a hard dynamical calculation.
Positive state and Hilbert space	Conditional. A supplied positive finite-regulator Hermitian measure would pass realization by realization, but positivity of the cited untruncated hard-wall candidate is not established here. An ensemble would still not be one Hilbert space with one Hamiltonian.	Not supplied in general. A positive baby-universe inner product, null quotient, boundary spectral measure, and gluing consistency must be shown.	Pass under the unitary-CFT axioms. Reflection positivity and a positive Hilbert space are part of the declared boundary theory; constructive rigor is a separate caveat.
Lorentzian evolution	Conditional per realization. A supplied self-adjoint regulated Hamiltonian evolves unitarily; the cited evidence does not construct one exact untruncated shared Hamiltonian.	Not supplied unless time evolution and every allowed observable preserve the sector.	Pass under the unitary-CFT axioms. The generator is the fixed theory's Hamiltonian on the cylinder.
Nonperturbative topology or sector rule	Conditional candidate. The wall supplies extra data beyond the genus coefficients, but existence, positivity, and the exact untruncated limit still require proof.	Conditional. Exact factorization concerns the selected operator algebra; preparation, completeness, and topology-by-topology cancellation require more input.	Not supplied by the boundary definition. A complete bulk topology sum and its cancellation or reinterpretation remain to be derived.
Uniqueness and strongest supported statement	Not unique from genus data. The named branch is a nonperturbative ensemble candidate consistent with the formal JT expansion, not an established unique exact completion.	Conditional algebraic result. Exact sharp conditioning factorizes the declared commuting algebra, not automatically the full Hilbert space or local QFT.	Exact boundary factorization; conditional bulk interpretation. The boundary theory does not by itself provide an independently completed bulk topology sum.
Decisive downgrade	An unspecified or nonpositive contour, or treating a same-draw covariance as one fixed copy, invalidates the claimed fixed-theory completion.	A residual conditional cumulant invalidates exact factorization or completeness of the label for those observables. An allowed operator that mixes labels invalidates superselection for the enlarged algebra.	A nonzero mixed connected correlator of genuinely decoupled boundary copies would contradict the boundary definition; an unmatched bulk saddle instead diagnoses an incomplete bulk representation.

The alpha qualification follows from the law of total covariance. With $m_i(\alpha)=\mathbb E[X_i\mid\alpha]$,

$$\operatorname{Cov}(X_1,X_2) =\int p(d\alpha)\operatorname{Cov}(X_1,X_2\mid\alpha) +\operatorname{Cov}_{p}\!\left(m_1(\alpha),m_2(\alpha)\right).$$

Sharp conditioning removes the between-alpha term. The within-alpha term is guaranteed to vanish for every pair in the declared diagonal algebra when the label is complete for that algebra and the conditioned state is an exact simultaneous eigenstate. Without that guarantee it can remain and must be checked; one chosen covariance can also vanish accidentally in an incomplete sector. Marolf and Maxfield prove the multiplication rule for their $\widehat Z[J]$ algebra Marolf and Maxfield 2020, §2.3, Eqs. (2.16)–(2.23). Their gluing map need not be surjective Marolf and Maxfield 2020, §2.4, around Eq. (2.33), so this algebraic factorization is not automatically full Hilbert-space factorization. JT calculations with approximate alpha states have restricted topological control and residual errors Saad, Shenker, and Yao 2024, §§4.2–4.3 and 5.3. The construction and the covariance decomposition are developed on Baby Universes, Alpha Parameters, and Proposed Superselection Sectors.

Same genus asymptotics, different exact spectra and factorization

Here is a complete finite-dimensional test in which positivity, normalization, and every trace are explicit. Set $g\equiv g_{\mathrm{top}}$ so that “same perturbative series” has the same bookkeeping meaning as in the genus expansion above. The fixture is a logical discriminator, not a claim that either object is selected by JT dynamics.

Choose $A,\Lambda,\Delta>0$, $0<g\le g_0$, and

$$\epsilon(g)=\Lambda e^{-A/g}, \qquad \epsilon(g_0)<\Delta.$$

Completion $F$ is the fixed two-level Hamiltonian

$$H_0=\operatorname{diag}(0,\Delta), \qquad Z_0(\beta)=1+e^{-\beta\Delta}.$$

Completion $E$ is the normalized equal-probability ensemble of two positive, self-adjoint Hamiltonians

$$H_{\sigma}=\operatorname{diag}(0,\Delta+\sigma\epsilon), \qquad Z_{\sigma}(\beta) =1+e^{-\beta(\Delta+\sigma\epsilon)}, \qquad \sigma=\pm1.$$

The two realizations have different exact gaps, yet

$$\begin{aligned} Z_+(\beta)-Z_-(\beta) &=-2e^{-\beta\Delta}\sinh(\beta\epsilon)\\ &=-2\beta\Lambda e^{-\beta\Delta}e^{-A/g} +O\!\left(e^{-3A/g}\right). \end{aligned}$$

Thus their power-series expansions in $g$ agree coefficient by coefficient. Their raw spectral form factors are nevertheless different:

$$K_{\sigma}(\beta,t) =1+e^{-2\beta(\Delta+\sigma\epsilon)} +2e^{-\beta(\Delta+\sigma\epsilon)} \cos\!\left[t(\Delta+\sigma\epsilon)\right],$$

with exact periods $2\pi/(\Delta+\sigma\epsilon)$. Times of order $1/\epsilon$ resolve a difference invisible at every fixed order in $g$; this is why the asymptotic statement is not uniform in exponentially late time.

The ensemble mean agrees with the fixed completion up to a still smaller flat term,

$$\begin{aligned} M_1(\beta) &=1+e^{-\beta\Delta}\cosh(\beta\epsilon),\\ M_1(\beta)-Z_0(\beta) &=\frac{\beta^2\Lambda^2}{2}e^{-\beta\Delta}e^{-2A/g} +O\!\left(e^{-4A/g}\right). \end{aligned}$$

But using the same random label $\sigma$ on both boundaries gives

$$\begin{aligned} \kappa_{12}^{E} &=e^{-(\beta_1+\beta_2)\Delta} \sinh(\beta_1\epsilon)\sinh(\beta_2\epsilon)\\ &=\beta_1\beta_2\Lambda^2 e^{-(\beta_1+\beta_2)\Delta}e^{-2A/g} +O\!\left(e^{-4A/g}\right)>0. \end{aligned}$$

For $\Delta=2$, $A=\Lambda=1$, $g=1/2$, $\beta_1=1$, and $\beta_2=2$, the exact value is

$$\epsilon=e^{-2}\approx0.135335, \qquad \kappa_{12}^{E}\approx9.2193463\times10^{-5}.$$

The leading expression gives $9.0799860\times10^{-5}$; the exact value is $1.535\%$ above it. Completion $F$ has $\kappa_{12}^{F}=0$, as does completion $E$ when the two boundaries receive independent redraws. Therefore $F$ and the same-draw $E$ have identical expansions to every algebraic order in $g$ but different exact spectra and factorization data. This is the requested adversarial pair, with every assumption visible.

For a gravity-specific counterpart, the two hard-wall JT completions share the formal JT branch while differing in nonperturbative spectral data. That example is evidence for nonuniqueness of candidate JT completions, not proof that either branch is uniquely selected by the local JT action.

The adversarial test, carried out

An adversarial test changes one hidden assumption, recomputes the same observable, and records exactly which inference fails.

How exact observables respond when one completion assumption is changed

Starting object and intervention	Recomputed result	Hypothesis that fails	Statement that survives
Replace fixed completion F by the same-label ensemble E above	The one-boundary trace changes only beyond all orders, but the exact two-boundary connected term becomes positive.	The assumption that the genus data specify one fixed replicated object.	Both objects share every algebraic coefficient in g; exact spectral and replication data remain independent input.
Keep the formal JT series but change the wall, contour, or Stokes data	The formal genus coefficients can remain unchanged while the exact spectrum, stability, or positivity changes.	Uniqueness of the nonperturbative completion from perturbative JT data.	The order-by-order JT/matrix-cumulant correspondence.
Replace a sharp alpha eigenstate by a finite window or coarse label	A distribution over several alpha values generally reintroduces between-alpha covariance; residual within-alpha conditional covariance can also remain.	Exact factorization from conditioning alone.	An approximate statement with an explicit window and residual-error bound may survive.
Enlarge the alpha-sector observable algebra by an intertwining operator	The new operator can mix labels even though the original commuting insertions remain diagonal.	Superselection for the enlarged algebra.	Superselection for the original restricted algebra, if its preservation proof remains valid.
Compare two decoupled copies of the fixed CFT with a truncated bulk sum containing a connected wormhole	The boundary connected term is exactly zero; an uncancelled bulk connected term disagrees with that observable.	Completeness of the truncated bulk topology prescription.	Exact boundary factorization and any bulk results that do not depend on the unmatched term.
Apply a nonmultiplicative smooth filter or trace out a shared sector	The transformed or reduced object can have a connected term while the raw fixed-theory partition function still multiplies.	Identification of the transformed observable with the raw microscopic one.	A filtered or effective-theory interpretation, provided the map and its domain are specified.

Evidence status and strongest supported conclusions

Evidence cutoff: 30 August 2026. The literature supports several precise but inequivalent statements.

• JT connected amplitudes match the formal connected cumulants of a double-scaled matrix model order by order, while extra global data are needed for an exact and nonunique ensemble completion Saad, Shenker, and Stanford 2019, §§1, 3, and 5.5–5.6.
• Exact sharp-alpha eigenstates multiply observables in their complete commuting boundary-insertion algebra. Current JT calculations of approximate alpha states do not upgrade this to a general construction of one exact Hamiltonian or a full local QFT Marolf and Maxfield 2020, §§2.3–2.4; Saad, Shenker, and Yao 2024, §§4.2–4.3 and 5.3.
• Correlated branes and related nonlocal deformations can cancel wormholes and localize on a supplied Hamiltonian in controlled two-dimensional models. These are explicit modified models, not a universal completion of undeformed gravity Blommaert, Iliesiu, and Kruthoff 2022, §§3.1 and 4.1–4.2; Usatyuk and Zhao 2025, §§2.1, 3.1, 3.3, and Appendix A.
• Exact integration over a sector can produce wormhole-like nonlocality and nonfactorizing reduced or replicated observables even when the underlying complete partition function factorizes. This supplies an alternative effective-theory mechanism, but changes the observable being tested Hernández-Cuenca 2025, §§2.3 and 3.2.
• An August 2026 AdS$_3$ proposal organizes topology change with a third-quantized Hamiltonian. Its effective ensemble interpretation requires CFT-realizable sectors, and the known two-torus amplitude constrains but does not uniquely determine that Hamiltonian; reconstructing the CFT-realizable locus remains open Hirano 2026, §§2.5–2.6, 3.1, and 5.
• A 2026 preprint reconstructs unitary compact functorial QFTs from multiplicative reflection-positive partition functions under explicit axioms. It sharpens the sufficiency question but does not establish those axioms for a proposed gravitational integral McNamara and Wang 2026, Theorem 1.1 and §4.1.
• A 2026 proposal instead extracts ramps, plateaux, and wormhole contributions from a smooth projection of one erratic fixed spectrum. Its minimal holographic structure is postulated, so it is a live alternative interpretation rather than an established general construction Liu 2026, §I.1, Eqs. (1)–(7), §§II.2–II.3, and §IV.1.

The strongest conclusion is therefore narrower than “JT defines one fixed boundary theory.” All-orders JT genus data determine a formal connected matrix-cumulant expansion, not a unique exact completion. A named positive ensemble can pass ensemble-level spectral and positivity tests. A complete sharp-alpha eigenstate can factorize its declared commuting algebra. A named fixed CFT has exact tensor-copy factorization in its boundary definition. None of those statements alone supplies a unique, positive, Lorentzian bulk topology sum for one fixed higher-dimensional theory.

If the proposed AdS/CFT duality is exact, the independently defined fixed boundary theory supplies a nonperturbative bulk definition Witten 1998, §§2.3 and 3.2. The exact-duality premise is an assumption here, not a conclusion of the completion tests. The evaporation consequences of semiclassical replica calculations are assessed separately in What Island Calculations Establish—and What They Do Not.

Common pitfalls

Treating the genus series as an exact function. An asymptotic series does not specify the exponentially small sectors, contour, or Stokes data. State which exact object realizes the series.

Calling an exact ensemble a fixed theory. An ensemble measure can be perfectly well defined and still have a nonzero same-draw covariance. Say whether two boundaries share a draw or receive independent redraws.

Equating sharp and approximate alpha conditioning. Exact multiplication requires a complete simultaneous eigenstate for the relevant algebra. A finite window, coarse label, or restricted topology calculation needs a residual-error estimate.

Reading a smooth ramp as one raw spectrum. Ensemble averaging, energy smoothing, time averaging, and smooth filtering discard different information. Declare the operation before interpreting the result.

Using factorization as the whole completion test. Exact copy factorization does not establish reflection positivity, a complete operator algebra, self-adjoint evolution, or uniqueness. Test these properties independently.

Exercises

1. Check the topological powers

Use $\chi=2-2g-n$ to find the $g_{\mathrm{top}}$ power for a disk, a double trumpet, and a one-boundary surface with one handle.

Solution

The weight is $g_{\mathrm{top}}^{-\chi}=g_{\mathrm{top}}^{2g+n-2}$. A disk has $(g,n)=(0,1)$ and weight $g_{\mathrm{top}}^{-1}$. A double trumpet has $(0,2)$ and weight $g_{\mathrm{top}}^0$. A one-boundary, one-handle surface has $(1,1)$ and weight $g_{\mathrm{top}}^1$. The double trumpet is one power below a single disk, but the factorization comparison is with two disconnected disks of weight $g_{\mathrm{top}}^{-2}$, so the connected term is two powers down from their product.

2. Prove that the spectral pair is invisible to every power

For fixed $A,M>0$, prove $e^{-A/g}=o(g^M)$ as $g\to0^+$. Then use the expansion of $\sinh(\beta\epsilon)$ to derive $Z_+-Z_-$ through order $e^{-3A/g}$.

Solution

Set $x=A/g$. Then

$$\frac{e^{-A/g}}{g^M} =A^{-M}x^M e^{-x}\longrightarrow0$$

because an exponential dominates every polynomial as $x\to\infty$. With $\epsilon=\Lambda e^{-A/g}$,

$$\sinh(\beta\epsilon) =\beta\Lambda e^{-A/g} +\frac{\beta^3\Lambda^3}{6}e^{-3A/g} +O\!\left(e^{-5A/g}\right).$$

Multiplying by $-2e^{-\beta\Delta}$ gives

$$\begin{aligned} Z_+(\beta)-Z_-(\beta) &=-2\beta\Lambda e^{-\beta\Delta}e^{-A/g}\\ &\quad-\frac{\beta^3\Lambda^3}{3} e^{-\beta\Delta}e^{-3A/g} +O\!\left(e^{-5A/g}\right). \end{aligned}$$

Every coefficient in an algebraic power series in $g$ is therefore identical for the two traces.

3. Find the recurrence time of the two-level fixture

Starting from the two energy levels $0$ and $\Delta+\sigma\epsilon$, derive $K_\sigma(\beta,t)$ and find its smallest positive exact period.

Solution

Expanding the product of the two traces gives

$$K_\sigma =1+e^{-2\beta(\Delta+\sigma\epsilon)} +2e^{-\beta(\Delta+\sigma\epsilon)} \cos\!\left[t(\Delta+\sigma\epsilon)\right].$$

Only one nonzero gap occurs, so the smallest positive period is

$$T_\sigma=\frac{2\pi}{\Delta+\sigma\epsilon}.$$

For more than two levels, one exact period exists only when all populated gaps are commensurate.

4. Compare a shared draw with independent redraws

Derive $\kappa_{12}^{E}$ for the equal ensemble of $H_+$ and $H_-$. Then repeat the experiment when the two boundaries draw their signs independently.

Solution

For a shared sign,

$$\mathbb E_\sigma \!\left[e^{-\beta_1(\Delta+\sigma\epsilon)} e^{-\beta_2(\Delta+\sigma\epsilon)}\right] =e^{-(\beta_1+\beta_2)\Delta} \cosh\!\left[(\beta_1+\beta_2)\epsilon\right].$$

Subtracting the product of the means and using $\cosh(a+b)-\cosh a\cosh b=\sinh a\sinh b$ yields

$$\kappa_{12}^{E} =e^{-(\beta_1+\beta_2)\Delta} \sinh(\beta_1\epsilon)\sinh(\beta_2\epsilon).$$

With independent signs, the expectation of the product is the product of the expectations, so $\kappa_{12}=0$. The Hamiltonian family is unchanged; only the replication protocol differs.

5. Diagnose incomplete alpha conditioning

Suppose a conditioned label $a$ contains several exact sectors $\alpha$. Apply the law of total covariance at fixed $a$. Give two separately checkable sufficient conditions that guarantee factorization of $X_1$ and $X_2$ without relying on cancellations.

Solution

At fixed coarse label $a$,

$$\operatorname{Cov}(X_1,X_2\mid a) =\mathbb E\!\left[\operatorname{Cov}(X_1,X_2\mid\alpha,a)\mid a\right] +\operatorname{Cov}\!\left(m_1(\alpha),m_2(\alpha)\mid a\right).$$

A sufficient condition for the first term to vanish is that every exact sector in the support factorizes the chosen pair. A sufficient condition for the second is that the sector-dependent means have zero covariance—for example, because the coarse label fixes both means. These conditions are deliberately checked separately: the two signed terms could otherwise cancel accidentally. Exact alpha-eigenstate factorization for an entire declared algebra uses the stronger guarantee that all its operators belong to the complete commuting algebra diagonalized by the exact state.

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, A., L. V. Iliesiu, and J. Kruthoff. “Gravity Factorized.” Journal of High Energy Physics 09 (2022): 080. DOI.
• Harlow, D., and D. Jafferis. “The Factorization Problem in Jackiw–Teitelboim Gravity.” Journal of High Energy Physics 02 (2020): 177. DOI.
• Hernández-Cuenca, S. “Wormholes and Factorization in Exact Effective Theory.” Journal of High Energy Physics 05 (2025): 024. DOI.
• Hirano, S. “A Third-Quantized Description of Spacetime Wormholes in AdS/CFT.” arXiv:2608.23111 [hep-th] (2026). arXiv.
• Johnson, C. V. “Consistency Conditions for Non-Perturbative Completions of JT Gravity.” arXiv:2112.00766 [hep-th] (2022). arXiv.
• Liu, H. “Ramp, Plateau, and Wormholes without Averaging, and Hyper-non-perturbative Structures in Gravity.” arXiv:2608.02743 [hep-th] (2026). arXiv.
• Marolf, D., and H. Maxfield. “Transcending the Ensemble: Baby Universes, Spacetime Wormholes, and the Order and Disorder of Black Hole Information.” Journal of High Energy Physics 08 (2020): 044. DOI.
• McNamara, J., and Z. Wang. “Wormholes as Red Herrings: Reflection Positivity and the Reconstruction of Unitary Quantum Field Theories.” arXiv:2607.01322 [hep-th] (2026). arXiv.
• Saad, P., S. H. Shenker, and D. Stanford. “JT Gravity as a Matrix Integral.” arXiv:1903.11115 [hep-th] (2019). arXiv.
• Saad, P., S. H. Shenker, and S. Yao. “Comments on Wormholes and Factorization.” Journal of High Energy Physics 10 (2024): 076. DOI.
• Usatyuk, M., and Y. Zhao. “Closed Universes, Factorization, and Ensemble Averaging.” Journal of High Energy Physics 02 (2025): 052. DOI.
• Witten, E. “Anti-de Sitter Space and Holography.” Advances in Theoretical and Mathematical Physics 2 (1998): 253–291. DOI.