Fixed-Theory Factorization and Nonperturbative Completion Tests

A nonperturbative completion of a fixed theory must define exact observables beyond an asymptotic saddle expansion and must pass factorization, positivity, spectral, state-space, and Lorentzian-consistency tests. Reproducing a genus series is valuable evidence, but different completions can share every perturbative coefficient while differing by $e^{-O(1/G_N)}$ effects that decide these tests.

Required background. Factorization, Ensembles, and the Gravitational Path Integral formulates the central tension. Nonperturbative Definition and Completion Criteria supplies the general acceptance standard.

Helpful background. Nonperturbative Exponential Effects and Finite-N Sectors explains the missing scale. Microscopic Black-Hole Entropy: Claim and Ensemble Contract gives a protected microscopic comparison.

Completion data and observable tests

Suppose semiclassics determines

$$Z_{\mathrm{pert}}(g_s) \sim\sum_{h=0}^{\infty}g_s^{2h-2}Z_h, \qquad g_s\sim e^{-S_0}.$$

This is generally asymptotic. Two exact functions may have the same coefficients but differ by

$$\delta Z(g_s)=C\,e^{-A/g_s}\left(1+O(g_s)\right).$$

Hence a completion must specify the integration contour or exact spectral object, not merely the $Z_h$. For a fixed theory it should provide:

• a Hilbert space or algebra with a positive state;
• finite normalized partition functions and correlators in a stated domain;
• a discrete or otherwise precisely defined spectrum and its multiplicities;
• exact factorization for independent copies;
• a unitary Lorentzian continuation or a precise substitute;
• a rule for topology, baby-universe sectors, and all nonperturbative ambiguities.

Application: three candidates on one checklist

Candidate	Exact object	Two-boundary result	Spectrum	What remains
JT matrix integral	A specified double-scaled matrix ensemble	Ensemble covariance may be nonzero	Random spectrum with a defined measure	Not one fixed Hamiltonian
Alpha-conditioned proposal	One proposed baby-universe sector	Factorizes if all covariance came from alpha averaging	Must be constructed sector by sector	Inner product, preparation, and completeness
Fixed holographic CFT	One Hamiltonian and operator algebra	Independent copies factorize exactly	Definite finite-volume spectrum	Derive the bulk topology prescription nonperturbatively

The JT matrix integral exactly realizes the perturbative topological expansion and supplies nonperturbative spectral observables Saad, Shenker, and Stanford 2019. It therefore passes the “defined exact object” test as an ensemble. It does not, without further conditioning, pass the “one fixed boundary Hamiltonian” test.

For a candidate fixed theory, a useful diagnostic is the spectral form factor

$$K_T(t)=Z_T(\beta+it)Z_T(\beta-it).$$

An ensemble prediction $\overline{K(t)}$ can reproduce a smooth ramp and plateau, while an individual finite spectrum has erratic fluctuations and exact recurrences. Agreement after time averaging does not determine the individual phases or levels.

Same asymptotics, different exact spectra

Consider two positive spectral densities

$$\rho_\pm(E)=\rho_{\mathrm{smooth}}(E) \pm e^{-A/g_s}f(E),$$

with $\int dE\,e^{-\beta E}|f(E)|$ finite and amplitudes small enough to preserve positivity. Their partition functions have identical expansions in powers of $g_s$, yet differ nonperturbatively:

$$Z_+(\beta)-Z_-(\beta) =2e^{-A/g_s}\int dE\,e^{-\beta E}f(E).$$

With suitable discrete realizations they can have different level spacings and late-time recurrences. This elementary construction shows why genus asymptotics cannot select a unique finite spectrum.

Adversarial acceptance test

Run the same observables in three domains: the ensemble, a conditioned sector, and a named fixed theory. Check one- and two-boundary partition functions, a positive spectral density or state, higher moments, and a Lorentzian correlator. Then perturb the proposed completion by an allowed exponentially small term that leaves all known $Z_h$ unchanged. If factorization, positivity, or the spectrum changes, existing perturbative evidence does not yet fix the completion.

A claimed alpha sector must additionally show that every allowed asymptotic observable preserves the sector. A claimed fixed CFT must identify how connected semiclassical wormholes are cancelled, reinterpreted, or excluded. Failure to supply that mechanism is an open problem, not evidence that factorization is approximate.

Controlled conclusion

The strongest established statement in JT is an exact matrix-integral completion of the gravitational genus expansion, with ensemble observables. Higher-dimensional fixed-CFT holography supplies exact factorizing boundary theories but not a universally accepted bulk topology-sum definition. These successes constrain one another without being the same completion. The evaporation consequences are assessed in What Island Calculations Establish—and What They Do Not.

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

• McNamara, J., and C. Vafa. “Baby Universes, Holography, and the Swampland.” arXiv:2004.06738.
• Saad, P., S. H. Shenker, and D. Stanford. “JT Gravity as a Matrix Integral.” arXiv:1903.11115.
• Stanford, D., and E. Witten. “JT Gravity and the Ensembles of Random Matrix Theory.” Advances in Theoretical and Mathematical Physics 24 (2020): 1475–1680. DOI.