Hollow-grams: Generalized Entanglement Wedges from the Gravitational Path Integral

Kyle Ritchie; Pratik Rath; Sami Kaya

REVIEW 3 major objections 5 minor 3 cited by

Hollow-grams: Generalized Entanglement Wedges from the Gravitational Path Integral

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Path integral derivation shows a bulk region's entropy is the generalized entropy of its smallest enclosing entanglement wedge.

desk verdict A genuinely new path-integral derivation of the Bousso-Penington proposal for bulk regions, worth serious referee time, but the central diagonal approximation is assumed rather than proven. read the letter →

arxiv 2506.10064 v4 pith:T6VFJVEV submitted 2025-06-11 hep-th gr-qcquant-ph

Sami Kaya , Pratik Rath , Kyle Ritchie This is my paper

classification hep-thgr-qcquant-ph

keywords holographicentanglemententropywedgegravitationalpathintegralrandomtensornetworksgeneralizedreplicatrickRenyiBousso-Peningtonproposal

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the Bousso-Penington (BP) proposal for the entanglement wedge of a gravitating bulk region follows from the gravitational path integral in time-reflection symmetric settings. The authors define the entropy of a bulk region by a hollow-graphic construction—removing the region to create new boundaries—and compute its Rényi entropy through fixed-geometry states. They show that in the n→1 limit the result equals the generalized entropy A(γ_a, h_ψ)/(4G) + S(ρ_{E(a)}), independent of how the bulk region is gauge-invariantly specified. This matters because it elevates the BP proposal from a geometric conjecture to a consequence of the gravitational path integral.

What carries the argument

The hollow-graphic state |ψ_a⟩: the state obtained by removing (hollowing out) the bulk region a, generating new open boundaries a±, and computing entropies of the resulting open legs. Combined with fixed-geometry states |h⟩, it reduces the bulk-region entropy computation to a random-tensor-network-style replica calculation where the only nontrivial saddles are two domains separated by a domain wall, whose area gives the generalized entropy.

What would settle it

Find a static holographic state and a gauge-invariant bulk region where replica symmetry breaking or off-diagonal fixed-geometry contributions dominate the Rényi entropy, and show the n→1 limit deviates from A(γ_a)/(4G) + S(ρ_{E(a)}). A concrete example would be a state with competing extremal surfaces of nearly equal area, where the diagonal approximation is known to fail for boundary regions, used as input for a bulk region.

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Extended reading notes

Core claim

The central claim is that for a gravitating bulk region a in a time-reflection symmetric holographic state, the von Neumann entropy S(a) computed from the hollow-graphic replica path integral equals the generalized entropy of the smallest region E(a) that contains a and shares its conformal boundary: S(a) = A(γ_a, h_ψ)/(4G) + S(ρ_{E(a)}). The paper derives this by expanding the state in fixed-geometry states, applying the hollow-graphic map, computing Rényi entropies via replica saddles with a diagonal approximation, and taking n→1. The n-dependence of the saddles depends on the gauge-invariant specification of a, but the dependence drops out in the n→1 limit, so the BP proposal is recovered universally.

Load-bearing premise

The diagonal approximation: off-diagonal terms in the fixed-geometry expansion are dropped, motivated by replica symmetry, without a proof that this holds for bulk regions defined by the hollow-graphic map.

Editorial extensions

If this is right

If the derivation holds, the BP proposal is not an additional postulate but a consequence of the gravitational path integral in static settings.
The generalized entanglement wedge of a bulk region is independent of how the region is gauge-invariantly prescribed, resolving potential ambiguities in defining bulk subregions.
Restricted entanglement wedges (computed in a spacetime with other regions treated as boundaries) provide the natural objects for entropy inequalities, unifying SSA and the holographic entropy cone.
The same hollow-graphic method extends to compute min and max entanglement wedges via Petz-map reconstruction for incompressible states where replica symmetry breaks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof's reliance on the diagonal approximation suggests that bulk-region entropies in states with strong replica symmetry breaking may deviate from the BP formula; testing such states in JT gravity is a concrete next step.
The gauge-invariance of the n→1 limit hints that the BP formula is a robust semiclassical statement, but the n-dependent saddles imply that Rényi entropies of bulk regions are not universal—they encode how the region is defined.
The restricted-wedge formulation points toward a fully holographic proof of the entropy cone for gravitating regions that does not require the stronger independence conditions used in earlier work.
A Lorentzian derivation would be needed to extend the hollow-graphic construction to time-dependent settings; the paper's static Euclidean method cannot capture the time-dependent BP proposal.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know: this is not another restatement of BP. It gives a concrete random-tensor-network construction—hollowing out a bulk region and computing the entropy of the opened legs—and then maps that construction onto a gravitational path integral using fixed-geometry states. The restricted entropy inequalities and the observation that the Rényi saddles depend on the gauge-invariant specification of the region are new and useful, and the JT and minisuperspace examples are worked out in enough detail to be checked.

What is solid: the RTN part is rigorous in the large-bond-dimension limit, and it cleanly reproduces the BP entanglement wedge for bulk regions. The fixed-geometry technology is used carefully, and the paper is honest about the approximate nature of the fixed-geometry basis and about the difference between its construction and earlier related work in Refs. [12], [16], [38]. The examples showing that the n→1 limit is independent of how the region is gauge-invariantly specified are concrete and plausible.

The load-bearing step is Eq. (4.4), the diagonal approximation for bulk regions. For boundary regions this was justified in [33]. For bulk regions, the justification would have to pass through the hollow-graphic map Va. The paper proves that Va preserves norms, but it does not prove that Va is an isometry, and nothing in Secs. 4 or 5 shows that off-diagonal fixed-geometry overlaps are suppressed after hollowing. The examples only evaluate the diagonal saddle, so they cannot detect a failure of this assumption. This makes the derivation conditional rather than wrong. The paper explicitly says the diagonal approximation is motivated by replica symmetry, and for sharply peaked semiclassical states one might expect such suppression, but it remains an assumption—and it is the same assumption that does the work in reaching Eq. (4.9). The stress-test note is therefore on target: unless Va is isometric, the boundary-region suppression mechanism does not automatically transfer.

A second soft spot is that gamma_a in Eq. (4.6) is already defined as the surface homologous to partial a that minimizes the generalized entropy of E(a). So Eq. (4.9) has some circularity built in; the independent content comes from the RTN calculation and the explicit saddle-point examples. I would not call this fatal, but the paper could be more upfront that the minimization is encoded in the definition.

Net: this is a serious, worthwhile paper. The derivation is important enough that the bulk-region diagonal approximation should be either proven or clearly marked as a conjecture. I would send it to a serious referee, and I would cite it in my own work with a qualifier.

Referee Report

3 major / 5 minor

Summary. This paper proposes a derivation of the Bousso-Penington (BP) proposal for generalized entanglement wedges of gravitating bulk regions using the gravitational path integral, in time-reflection symmetric settings. The authors first formulate the proposal in random tensor networks by 'hollowing out' a bulk region a, i.e., removing the tensors and internal edges in a and computing the entropy of the resulting open legs; this yields S(a)=A(γ_a)/(4G)+S(ρ_{E(a)}). They then use the connection between RTNs and fixed-geometry states in gravity, decompose a general holographic state as a superposition over fixed-geometry states, and invoke a diagonal approximation to compute Rényi entropies. They show that the saddles computing the Rényi entropies depend on how the bulk region is gauge-invariantly specified, but claim that the dependence drops out in the n→1 limit and that the BP proposal is universally recovered. Explicit computations are given in JT gravity for four different gauge-invariant specifications of an interval, and in Einstein gravity in a minisuperspace approximation for an annulus.

Significance. If the derivation is correct, this is an important result: it gives a gravitational path-integral rationale for the BP proposal and connects it to tensor-network constructions. The paper has several concrete strengths: a clear RTN derivation of the BP formula, a new notion of restricted entanglement wedges with a discussion of entropy inequalities, explicit worked examples in JT gravity and in a minisuperspace Einstein-gravity model, and a demonstration that the n>1 Rényi saddles depend on the gauge-invariant specification while the n→1 result does not. The main unresolved point is the bulk diagonal approximation, which is assumed rather than proven; this is a load-bearing step for the claimed derivation.

major comments (3)

[Sec. 4, Eq. (4.4)] The diagonal approximation is the load-bearing step of the gravitational derivation, but it is not justified for bulk regions. For boundary subregions, the suppression of off-diagonal terms follows from the analysis of the fixed-geometry basis in Ref. [33]. Here, the hollowing map V_a is shown only to preserve the norm of each fixed-geometry state, ⟨h_a|h_a⟩=⟨h|h⟩, not to preserve the inner products ⟨h_a|h'_a⟩=⟨h|h'⟩ for h≠h'. Therefore the boundary-case argument does not automatically transfer: the new boundaries ∂a are not traced over in the same way as the boundary complement, and one must show that off-diagonal hollowed overlaps are suppressed at order e^{-c/G}. The examples in Sec. 5 evaluate only the diagonal saddle and cannot detect such contributions. Please provide an estimate or proof of off-diagonal suppression in the hollowed Hilbert space, or state this as an explicit assumption with its consequences.
[Sec. 4, after Eq. (4.9)] The claimed universality of the n→1 limit depends on the assumption that the gauge-invariant prescription used to define a on off-shell geometries does not affect the limit. The paper argues that all prescriptions agree on the leading geometry h_ψ, so h_1=h_ψ; however, the off-diagonal terms in Eq. (4.3), if not negligible, can depend on the extension of a to geometries h≠h', and the n→1 limit of the full expression need not be independent of the prescription. Please show that off-diagonal contributions are subleading before taking n→1, or give a separate argument that the n→1 limit commutes with the diagonal approximation.
[Sec. 3 and Sec. 4] The paper acknowledges, in Sec. 3 (footnote 11) and Sec. 4 after Eq. (4.2), that the fixed-geometry states form only an approximate basis because of wormhole corrections. This is particularly important for the hollowing construction, since wormhole overlaps in the hollowed Hilbert space could be of the same order as the diagonal contributions to Tr(ρ_a^n). In JT gravity, fixed-dilaton states have non-zero wormhole overlaps; the examples in Sec. 5 do not compute these. Please quantify the size of wormhole corrections to the hollowed inner products in the regimes considered, or explain why they are subleading in the n→1 limit.

minor comments (5)

[Appendix B, Eq. (B.6)] The factor (m−1) in Eq. (B.6) should presumably be (n−1); the symbol m is not defined and the surrounding equations use n.
[Sec. 5.1, Eq. (5.9)] The displayed limit contains the stray notation 'ℓ − − − →'; this should be a single arrow or phrased as a limit.
[References] Reference [36] appears to duplicate Reference [32]; the two entries for the modified cosmic brane proposal should be consolidated.
[Sec. 6.2, Fig. 12 caption] The caption contains the phrase 'null like', which should be 'null'.
[Sec. 4, after Eq. (4.3)] The density matrix ρ_a is not manifestly normalized; please state the normalization convention used in the Rényi trace, as is done for the boundary case in Eq. (3.5).

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the BP formula emerges from a replica-saddle calculation, not from a fit or a self-citation chain; the only minor concern is a coauthored diagonal-approximation reference and an explicitly assumed bulk analogue.

full rationale

The paper's central claim is that the hollow-graphic replica calculation yields S(a) = A(γ_a, h_ψ)/(4G) + S(ρ^bulk_{E(a)}(h_ψ)) in the n→1 limit. This is not a restatement of the BP proposal: the Rényi entropy on the left is defined independently by cutting open the region a (Eqs. (2.8), (4.2)–(4.5)), and the right-hand side is obtained from the saddle-point evaluation of the replica path integral (Eqs. (4.6)–(4.9)). The surface γ_a is not a fitted parameter; it is the location of the minimal domain wall selected by the action, and the n-dependence of the saddle is computed in the examples rather than assumed. The JT and minisuperspace Einstein-gravity calculations are worked saddles, not fits to the target formula. The main caveats are non-circular: (i) the bulk-region diagonal approximation in Eq. (4.4) is explicitly assumed ('Motivated by replica symmetry, we will again assume'), not derived from Ref. [33]; Ref. [33] (coauthored by P. Rath) establishes the boundary analog, and the extension to bulk regions is an acknowledged assumption, so the citation is not load-bearing for the bulk claim. (ii) The fixed-geometry basis is approximate because of wormhole corrections, acknowledged in Sec. 3. (iii) The phrase in Eq. (4.6) characterizing γ_a as the surface 'that minimizes the generalized entropy' is the n→1 description of the saddle, and the calculation supplies the area and matter contributions; this is the standard structure of holographic entropy derivations rather than a definitional identity. These gaps are correctness/rigor concerns, not circular reductions. Hence a low circularity score is appropriate.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

No data are fitted. The derivation carries standard semiclassical assumptions: a fixed-geometry basis, replica symmetry and the diagonal approximation, gauge-invariant region specifications, and the hollowing map as a valid norm-preserving operation.

assumptions (6)

domain assumption Fixed-geometry states |h> form an approximate basis of the Hilbert space; wormhole corrections are neglected.
Used to expand |psi> in Eq. (3.1), to linearly extend V_a in Eq. (4.2), and to justify the diagonal approximation; the paper admits this is not strictly true.
domain assumption Diagonal approximation: only replica-symmetric saddles contribute; off-diagonal psi(h) psi*(h') terms in Tr(rho_a^n) are dropped.
Invoked in Eq. (4.4) for bulk regions, following Ref. [33] for boundary regions; no proof is given for the hollow-graphic case.
domain assumption The leading geometry and all fixed-geometry histories preserve time-reflection symmetry on Sigma.
The entire derivation is restricted to time-reflection symmetric settings; Section 6.2 leaves time dependence open.
domain assumption A gauge-invariant specification of the bulk region a exists on every geometry h and agrees with the desired region on h_psi.
Needed before Eq. (4.1); the claimed n to 1 independence of the prescription depends on this agreement.
ad hoc to paper The hollowing map V_a, defined by cutting an infinitesimal slit around a, is a well-defined and norm-preserving operation.
Introduced in Section 4, Eq. (4.1); its consistency with the gravitational path integral is assumed.
domain assumption Standard fixed-geometry Renyi formula Tr(rho_A(h)^n) = exp(-(n-1)A/4G) Tr((rho_bulk)^n) from prior literature.
Used in Eqs. (3.7) and (4.6); relies on Refs. [6, 26, 27].

invented entities (2)

Hollow-graphic state |psi_a> and hollowing map V_a
purpose: Defines the entropy of a bulk region by removing the tensors or geometry in a and computing the entropy of the resulting open legs or new boundaries.
Core new construction of the paper; it is a calculational device with no independent falsifiable prediction outside the derived entanglement wedge formula.
Restricted entanglement wedge E(p)|_{q'}
purpose: Candidate notion for entropies of spacelike separated bulk regions and for restricted entropy inequalities in Section 2.3.
Introduced in Section 2.3 as a bookkeeping device; its empirical content is indirect.

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Pith. "Pith review of Hollow-grams: Generalized Entanglement Wedges from the Gravitational Path Integral." pith.science (2026). https://pith.science/paper/T6VFJVEV

@misc{pith2026250610064,
  author       = {Pith},
  title        = {Pith review of: Hollow-grams: Generalized Entanglement Wedges from the Gravitational Path Integral},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T6VFJVEV}},
  note         = {Machine review of arXiv:2506.10064}
}

abstract

Recently, Bousso and Penington (BP) made a proposal for the entanglement wedge associated to a gravitating bulk region. In this paper, we derive this proposal in time-reflection symmetric settings using the gravitational path integral. To do this, we exploit the connection between random tensor networks (RTNs) and fixed-geometry states in gravity. We define the entropy of a bulk region in an RTN by removing tensors in that region and computing the entropy of the open legs thus generated in the "hollowed" RTN. We thus derive the BP proposal for RTNs and hence, also for fixed-geometry states in gravity. By then expressing a general holographic state as a superposition over fixed-geometry states and using a diagonal approximation, we provide a general gravitational path integral derivation of the BP proposal. We demonstrate that the saddles computing the R\'enyi entropy $S_n$ depend on how the bulk region is gauge-invariantly specified. Nevertheless, we show that the BP proposal is universally reproduced in the $n\to1$ limit.

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Forward citations

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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