Single-Theory Completion and Open Questions

A single-theory completion must preserve exact multiplication on disconnected boundaries while explaining why semiclassical wormhole saddles successfully predict some connected or averaged observables. Proposed additional data include fixed microscopic couplings, brane or eigenvalue sectors, baby-universe alpha states, missing charged states, null-state quotients, and a nonperturbative integration cycle. Each mechanism has controlled examples; no one mechanism is yet a universal construction of higher-dimensional quantum gravity.

Required background. Quantum Information supplies Hilbert-space tensor products, conditioning, positive inner products, and null states.

Helpful background. Semiclassical Gravity supplies wormhole saddles; Quantum Fields in Curved Spacetime supplies reflection positivity and partition functions; Black-Hole Information supplies the factorization problem; Quantum Gravity supplies microscopic and nonperturbative consistency criteria.

What a completion must reproduce

For one fixed boundary theory $H$, a candidate completion must give

$$Z_H(B_1\sqcup B_2)=Z_H(B_1)Z_H(B_2)$$

for independent disconnected boundaries, not merely up to powers of $e^{-S}$. At the same time it should reproduce the one-boundary spectrum, semiclassical expansion, black-hole thermodynamics, and any connected quantities whose definitions explicitly include ensemble, time, or sector averaging.

This creates a stringent matching problem. Cancelling a wormhole term cannot spoil the accurate spectral ramp it organizes under the appropriate smoothing. Declaring an ensemble cannot erase the evidence that a proposed holographic dual is one fixed theory. Selecting an alpha sector must specify how boundary conditions make the selection. A nonperturbative completion must also preserve unitarity and reflection positivity.

The perturbative JT genus series is matched by a matrix integral, but the asymptotic series alone does not uniquely specify a nonperturbative completion; see Saad, Shenker, and Stanford 2019, §§2–5. A chosen matrix integral solves its own definition, not automatically the fixed-theory problem for another gravitational model.

Fixed couplings, sectors, and extra saddles

In disordered microscopic models, ensemble-averaged wormholes can be reorganized at fixed couplings by additional “half-wormhole” contributions. In the controlled examples of Saad, Shenker, Stanford, and Yao 2021, §§2–5, the extra contributions restore fixed-instance factorization while averaging recovers the smooth wormhole answer. This demonstrates that a wormhole saddle can encode accurate statistics without being the complete saddle expansion of one sample.

Baby-universe alpha sectors provide another organization. An unconditioned state gives covariances of commuting boundary-creation operators; a sharp common eigenstate factorizes. Marolf and Maxfield 2020, §§2–5 make this conditional structure explicit. The missing datum is which state the microscopic boundary conditions prepare.

Brane or fixed-eigenvalue sectors can likewise condition a matrix integral on more detailed spectral data. Null states may remove apparent extra Hilbert-space factors when the gravitational inner product is imposed. These proposals change the state space or conditioning data, not the classical wormhole solution itself. Their adequacy must be checked on all allowed boundary sources and charged sectors.

Reflection-positive reconstruction

A 2026 result supplies a different, rigorous axis. Given the paper’s source-category setup and Axioms 4.1–4.5—finiteness, reality, multiplicativity, continuity, and reflection positivity—a closed-manifold partition function reconstructs a unitary compact functorial QFT, unique up to unitary isomorphism. This is parts (a) and (b) of McNamara and Wang 2026, §1.1, Theorem 1.1; §§4–5. Part (c) separately identifies the closed span of states preparable by the chosen manifolds and sources with the invariant subspace of a maximal compact Hausdorff symmetry that acts trivially on those sources. The charged-state and gravitational factorization interpretation is developed later in McNamara and Wang 2026, §§9 and 11.

Reflection positivity means that gluing sources across a reflection defines a positive semidefinite Gram matrix. Schematically,

$$\sum_{i,j}\overline{c_i}c_j Z(\overline M_i\cup M_j)\ge0.$$

All $M_i$ here prepare states on the same boundary data, and $\overline M_i\cup M_j$ denotes gluing along that common boundary. Positivity must hold for every finite collection and every set of coefficients, not only for the individual diagonal partition functions. The two-state exercise below is the smallest nontrivial test; passing it alone does not establish the theorem’s full positivity hypothesis.

The theorem itself reconstructs the functorial QFT and characterizes the preparable invariant subspace. In its later factorization interpretation, apparent Hilbert-space nonfactorization can arise when the chosen sources prepare only that invariant sector; enlarging the source or charged-state spectrum may then expose the missing tensor factors without assigning fundamental ensemble averaging to wormholes. This interpretive step is a proposed application of part (c), not an input to parts (a) and (b).

Its hypotheses are substantial. The reconstruction assumes the specified source category and a finite, real, multiplicative, continuous, reflection-positive partition function. It does not prove that a proposed gravitational path integral satisfies those axioms, identify its contour, determine whether the chosen sources generate the physically required charged sectors, or construct a UV-complete gravity theory. Full charged-spectrum completeness is thus a check on the gravitational factorization application, not a reconstruction axiom. The result is a strong competing framework, not a universal dismissal of wormhole physics.

Results and open questions as of 2026-08-19

This discussion reflects results available through 2026-08-19.

Evidence or proposal	Controlled result and assumptions	Limitation or open task
JT matrix completion, Saad, Shenker, and Stanford 2019, §§2–5	Matrix integral reproduces the perturbative JT topology expansion	Ensemble valued; asymptotic gravity data do not uniquely choose one fixed Hamiltonian
Fixed-coupling half-wormholes, Saad, Shenker, Stanford, and Yao 2021, §§2–5	Selected disordered models reorganize averaged wormholes into fixed-instance contributions	Model-specific and not a construction of generic gravitational UV data
Alpha-sector factorization, Marolf and Maxfield 2020, §§2–5	Sharp eigenstates of commuting baby-universe operators factorize conditionally	Requires a state-selection rule and a defined baby-universe algebra
Reflection-positive QFT reconstruction, McNamara and Wang 2026, §1.1, Theorem 1.1; §§4–5, 9, and 11	The source-category setup plus finiteness, reality, multiplicativity, continuity, and reflection positivity reconstruct a unique unitary compact functorial QFT	Part (c), not the reconstruction hypotheses, controls the preparable invariant sector; charged completion is a later gravitational interpretation, and no UV gravity construction is supplied

One position expects exact cancellation or reorganization within each fixed microscopic theory. A second treats ensemble averaging as fundamental in the solvable gravitational model. A third restores factorization by conditioning on brane or alpha data. The newest formal-QFT position emphasizes complete source and charged-state reconstruction. These competing positions can coexist in different models, and a general gravitational conclusion remains unresolved.

A September 2026 follow-up

After the August cutoff, Klinger 2026, Definition I.2, §II.1 Eq. (19), and §III.1, posted 17 September, proposed a conditional expectation: a projection from an enlarged observable algebra onto its smooth-gravity subalgebra. Compact-symmetry averaging supplies it in the multiplicative case.

The nonmultiplicative extension uses reducible QFTs, a potentially nontrivial empty-set state space, and relative tensor products. This differs from ordinary tensor-product factorization for an irreducible boundary theory. Discrete-sector results must be distinguished from the conjectured continuous or diffuse extension; uniqueness is up to Morita equivalence of representation structures. Finite, real, continuous, reflection-positive amplitudes and the category hypotheses remain necessary. This targeted follow-up supplies neither a gravitational contour nor a microscopic evaporation S-matrix, and does not update the whole August comparison.

Adverse checks and decision criteria

Every proposal should face four tests. Multiplicativity: do all disconnected closed-manifold amplitudes factorize? Positivity: are the reflected gluing matrices positive semidefinite after quotienting null states? Completeness: have charged states, brane sectors, and allowed sources been included? Microscopic matching: are the spectrum, multi-boundary observables, and nonperturbative errors reproduced in one definition?

An adverse case for the reconstruction theorem is a divergent infinite state sum or a nonpositive gluing form; then its hypotheses fail. An adverse case for alpha conditioning is an observable that changes alpha sectors. An adverse case for saddle cancellation is a residual connected term in an exactly computed fixed instance. These checks identify which premise fails rather than assuming one resolution.

The conclusions could change after a UV-complete gravitational path integral is shown to satisfy the 2026 reconstruction hypotheses, after fixed-coupling saddles are derived in a higher-dimensional holographic theory, or after exact multi-boundary data rule out a proposed alpha, brane, null-state, or cancellation mechanism.

Exercises

Two reflected boundary preparations have Gram matrix

$$G=\begin{pmatrix}1&c\\ c&1\end{pmatrix},$$

with real $c$. Determine the reflection-positivity condition and identify the null state at saturation.

Solution

The eigenvalues are $1+c$ and $1-c$, so positive semidefiniteness requires $-1\le c\le1$. At $c=1$, the antisymmetric combination of the two preparations has zero norm; at $c=-1$, the symmetric combination is null. A reconstructed Hilbert space must quotient such null vectors. If $\lvert c\rvert\gt1$, no quotient repairs the negative eigenvalue, so the proposed partition-function data violate reflection positivity.

References

• Klinger 2026, Marc S. Klinger, “Toward a Unique Filter for the Gravitational Path Integral,” arXiv:2609.20697v1 (17 September 2026).
• Marolf and Maxfield 2020, Donald Marolf and Henry 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.
• McNamara and Wang 2026, Jacob McNamara and Zhencheng Wang, “Wormholes as Red Herrings: Reflection Positivity and the Reconstruction of Unitary Quantum Field Theories,” arXiv:2607.01322 (2026).
• Saad, Shenker, and Stanford 2019, Phil Saad, Stephen H. Shenker, and Douglas Stanford, “JT Gravity as a Matrix Integral,” arXiv:1903.11115 (2019).
• Saad, Shenker, Stanford, and Yao 2021, Phil Saad, Stephen H. Shenker, Douglas Stanford, and Shunyu Yao, “Wormholes without Averaging,” Journal of High Energy Physics 09 (2021) 133.