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Simple Holography in General Spacetimes

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Pith's one-line read This paper defines the 'simple wedge' of holographic reconstruction in arbitrary spacetimes via a zigzag of antinormal lightsheets, and proves it is unique and contained in every other accessible throat.

desk verdict A novel, clearly presented construction of the simple wedge in general spacetimes, but a real gap in the uniqueness proof (Theorem 27) makes the central claim unproven as written. read the letter →

arxiv 2505.00695 v2 pith:6AUUFZDG submitted 2025-05-01 hep-th gr-qc

Raphael Bousso , Elisa Tabor This is my paper · ORCID
classification hep-thgr-qc
keywords holographysimplewedgeoutermostentanglementzigzaglightsheetsgeneralizedentropyAdS/CFTDiscreteMax-Focusing
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

In AdS/CFT, the simple (or outermost) wedge is the part of the bulk that can be reconstructed from boundary data with only polynomial effort, while the rest of the entanglement wedge is exponentially hard to reconstruct. This paper extends that notion to arbitrary spacetimes, with no assumed boundary field theory. It defines the simple wedge $z(a)$ of an input wedge $a$ as the infinite limit of a zigzag: a sequence of alternating future and past antinormal lightsheet wedges, each step growing the region outward. The paper proves that $z(a)$ is a throat accessible from $a$, that it is contained in every other such throat and is therefore unique, and that it lies inside the generalized entanglement wedge. For AdS boundary regions the construction reduces to the familiar outermost-wedge prescription while adding a preferred piecewise-null Cauchy slice that makes the wedge manifestly accessible.

What carries the argument

The load-bearing object is the zigzag: alternating future (zig) and past (zag) steps, where each step is the wedge union of all lightsheet wedges of the current wedge that are antinormal — nonexpanding in both future and past outgoing null directions — on their newly added edge and contain no proper noncontracting lightsheet wedge. Each step is antinormal on its newly added edge, and the construction carries a preferred piecewise-null Cauchy slice $Z_n=H(z_n)\setminus I(z_{n-1})$ on which the generalized conditional max entropy of the final wedge relative to any intermediate wedge is non-positive; this is what makes each $z_n$ accessible from $a$. The proofs run on Discrete Max-Focusing — the conjecture that two nested future (or past) lightsheet wedges of the same wedge have non-positive generalized max entropy difference — together with discrete subadditivity and a chain rule for the generalized max entropy, and on the fact that the intersection of two noncontracting wedges is again noncontracting (Lemma 16).

What would settle it

Find a spacetime with an input wedge $a$ for which two different accessible throats exist, neither containing the other; even one such example would refute Theorem 27. More directly, any pair of nested future lightsheet wedges of the same wedge with positive $H_{\mathrm{max,gen}}$ would violate Discrete Max-Focusing and break the step in Theorem 22 that proves accessibility.

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Extended reading notes

Core claim

The central claim is that the simple wedge admits a purely geometric, spacetime-covariant definition. Starting from an input wedge $a$, the zig is the wedge union of all future lightsheet wedges of $a$ that are antinormal on their newly added edge, contain no proper noncontracting future lightsheet wedge, and stay within the complement of $a$'s fundamental complement; the zag is the time-reversed step. Iterating with $z_0=a$ and $z_n=z_+(z_{n-1})$ for odd $n$, $z_n=z_-(z_{n-1})$ for even $n$, the simple wedge is defined as $z(a)=\lim_{n\to\infty} z_n$. The paper proves that every finite zigzag $z_n$ is accessible from $a$ through the preferred Cauchy slice $Z_n=H(z_n)\setminus I(z_{n-1})$, and that the limit $z(a)$ is a throat accessible from $a$ and is contained in every other throat accessible from $a$, hence unique and contained in the max-hologram $e_{\max}(a)$. In the AdS/CFT case, applying the construction to the causal wedge of a boundary region reproduces the traditional simple/outermost wedge.

Load-bearing premise

Everything rests on Discrete Max-Focusing, the assumption that generalized conditional max entropy never increases between two nested future (or past) lightsheet wedges of the same wedge, together with higher-order strong subadditivity and a chain rule for the generalized max entropy that the paper itself flags as proven only at leading order in Newton's constant $G$.

Editorial extensions

If this is right

  • The simple wedge is unique: the zigzag and the time-reversed zagzig have the same limit $z(a)$.
  • $z(a)$ is contained in the max-hologram $e_{\max}(a)$, so the zigzag never extends into the Python's lunch region of the entanglement wedge.
  • For an AdS boundary region $B$, the prescription $z(B)=z[c(B)]$ recovers the traditional simple/outermost wedge and adds a preferred piecewise-null Cauchy slice that exists even where a time-symmetric-slice construction fails.
  • The accessibility criterion gives a covariant, slice-based notion of which wedges can be reconstructed from an input wedge, the missing ingredient for formulating reconstruction complexity in spacetimes without a known dual field theory.
  • In semiclassical gravity the quantum version applies to evaporating black holes: before the Page time the simple wedge ends at a quantum extremal surface near the horizon, leaving the interior as a Python's lunch, and after the Page time it agrees with the max-hologram.

Reading between the lines

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

  • Because the zigzag is null rather than spacelike, the paper's own examples suggest (but do not prove) that tensor-network models of holography should be built from broken null hypersurfaces rather than from a single time-symmetric Cauchy slice; this is an extension beyond the paper's explicit results.
  • The use of Discrete Max-Focusing in place of the full Quantum Focusing Conjecture implies the construction should remain valid at caustics and corners where smooth quantum expansions are undefined, a regime where conventional quantum extremal surface prescriptions become difficult to state.
  • If uniqueness of the accessible throat holds generally, the edge of the simple wedge could serve as a covariant quasi-local horizon in cosmological or evaporating spacetimes where no global extremal surface exists; this would be a new diagnostic, not derived in the paper.
  • A direct numerical test in a simple 2+1 spacetime (e.g., comparing the zigzag limit with the outermost extremal surface on a time-symmetric slice) would either corroborate the claim that the null construction is the right covariant generalization or show where accessibility fails.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes a covariant definition of the 'simple wedge' (outermost wedge) in arbitrary globally hyperbolic spacetimes. Starting from an input wedge a, it defines a 'zigzag' sequence of antinormal lightsheet wedges, alternating future and past lightsheets, and defines the simple wedge z(a) as the infinite limit of this sequence. The authors claim that z(a) is accessible from a, that it is a throat (future- and past-marginally accessible), that it is contained in every other throat accessible from a (Theorem 27), and hence that it is unique and contained in the generalized entanglement wedge (max-hologram). They further claim that in the AdS boundary case the construction reproduces the standard simple/outermost wedge prescription, and that the zigzag supplies a preferred piecewise-null Cauchy slice not previously identified. The proofs rely on a 'Discrete Max-Focusing' conjecture and on higher-order conjectural entropy inequalities, as the paper acknowledges.

Significance. The conceptual contribution is substantial: a definition of the simple wedge outside AdS, with an explicit preferred Cauchy slice, and a candidate 'outermost' property proved modulo stated conjectures. The paper is unusually explicit about its definitions and about its reliance on conjectural input, and it identifies a genuinely new structure (the piecewise-null zigzag slice) even in the AdS setting. If Theorem 27 can be repaired and the limit step made rigorous, this would be a valuable step toward extending the holographic dictionary to general spacetimes and toward tensor-network models of broken null hypersurfaces. At present, however, the central containment/uniqueness claim is not established as written, so the significance is conditional on a repaired proof.

major comments (4)
  1. [4.1, Theorem 27] The contradiction step in the proof of Theorem 27 does not follow from the definitions. The wedge j ≡ k ∩ z_N(a) is claimed, via Lemma 16, to be a PNC/FNC proper subwedge of z_N(a). But Definition 18 only forbids z_N(a) from containing a proper PNC/FNC lightsheet wedge of z_{N-1}(a); it imposes no restriction on arbitrary PNC/FNC proper subwedges. The proof never shows that j is a lightsheet wedge of z_{N-1}(a): in general the new boundary of j will include points on H(k) rather than on Z_N(a), so Definition 13 is not met. In addition, the chosen index N need not exist, since H(k) may intersect z(a) only in the infinite limit, with no finite Z_N(a) intersecting H(k). Thus the central containment/uniqueness theorem is unproven even conditional on Conjecture 14.
  2. [4.1, Definition 25 and Corollary 26] The infinite limit z(a) ≡ lim_{n→∞} z_n(a) is asserted to 'share all properties of the finite-n zigzags,' but no definition of the limit is given and no proof is provided that the limit is a wedge, that it is accessible from a, that it is FNC and PNC, or that Z(a) is a Cauchy slice. These properties are load-bearing for Corollary 26 and for Theorem 27; the limit step needs a lemma with explicit hypotheses ensuring convergence before the main claims are established.
  3. [3.2, Conjecture 14 and Theorems 8-9; also Theorem 22] The paper's main theorems are conditional on unproved input: Discrete Max-Focusing (Conjecture 14) and the higher-order (beyond leading order in G) versions of strong subadditivity and the chain rule for Hmax,gen are described as conjectural. Since Theorem 22, Corollary 26, and Theorem 27 all invoke these ingredients, the abstract's unconditional wording ('We show...') overstates the present status. I do not object to conjectural input in a physics paper, but the final theorems should be labeled as conditional on these conjectures, and the precise set of conjectures used by each theorem should be stated.
  4. [4.1, Theorem 22 proof, Eqs. (4.5)-(4.8)] The telescoping argument that derives Hmax,gen[z_n|h] ≤ 0 is not fully justified. The first inequality applies Conjecture 14 to h ⋓ z_{n-1} as a null deformation of z_{n-1}; subsequent inequalities involve h ⋓ z_i, but these are not shown to be lightsheet wedges of z_{i-1}, and the text switches to h ∩ z_{n-1} in an intermediate step without explaining how Discrete Subadditivity is applied to the union. The chain rule (Theorem 9) is then used on a sequence whose nesting is not explicitly verified. This step needs a detailed proof; as written it is a gap in the accessibility claim.
minor comments (5)
  1. [3.2, Definition 11, condition II] The statement 'k is antinormal at points p ∈ ðf\ða' uses the symbol ðf, which is not defined; it should presumably read ðk\ða.
  2. [4.2, Eq. (4.13)] The notation z[c(B)] is used without comment; either define square brackets as function application or write z(c(B)) consistently.
  3. [4.1, Definition 18, property B] The phrase 'proper subset' should be 'proper subwedge' to match the rest of the paper; a subset of a wedge need not itself be a wedge.
  4. [References] Reference [36] is listed as 'To appear' with no title or arXiv number, yet Definitions 6 and 7 are attributed to it; a full citation is needed. Also, Corollary 48 of the companion paper Ref. [35] is used in Lemma 19 and should be stated or quoted so the proof is self-contained.
  5. [2.1, footnote 4] The footnote says 'we will use "extremal" instead of "stationary" below,' but the surrounding text continues to use 'stationary'; the wording should be reconciled.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the zigzag is defined independently and the uniqueness claim is proved rather than assumed; the main caveats are a load-bearing same-author conjecture and a non-circular proof gap in Theorem 27.

full rationale

The simple wedge z(a) is defined operationally in Definitions 18, 21, and 25 as an infinite limit of zigzags, not as 'the largest throat' or 'the wedge contained in all throats.' Accessibility (Theorem 22) and containment (Theorem 27) are then argued from that definition. The proof of Theorem 22 uses Conjecture 14 (Discrete Max-Focusing) and Corollary 48 from the same authors' companion paper [35], with the paper itself noting that the strong-subadditivity and chain-rule ingredients are only proven at leading order in G and are conjectural beyond that; this is a load-bearing self-citation dependency and a genuine correctness risk, but it is not a circular reduction, because the conjecture is an independent (if unproven) input rather than a restatement of the target result. Theorem 27's uniqueness argument is not circular, but it appears to contain a nontrivial gap: the intersection j = k ∩ z_N(a) is declared to be a PNC/FNC proper subwedge of z_N(a) that contradicts the zigzag definition, yet Definition 18 only forbids a proper PNC future lightsheet wedge of a, and j is not shown to be a lightsheet wedge of a; nor is the existence of a finite N established. The AdS boundary reduction invokes an external result [42] and does not rename the target. No derivation step is equivalent to its input by construction, so the paper has no significant circularity, but its reliance on unverified same-author conjectures justifies a nonzero caution score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper is transparent about its main input: Discrete Max-Focusing is labeled a conjecture, and the higher-order entropy inequalities are also conjectural outside the leading semiclassical approximation. The proofs additionally import Corollary 48 from the authors' companion paper and assume the infinite zigzag limit has the properties of its finite truncations. No numerical parameters are fitted.

assumptions (5)
  • ad hoc to paper Conjecture 14 (Discrete Max-Focusing): for future or past lightsheet wedges b⊂c of a, Hmax,gen(c|b)≤0.
    This is the central unproved input; it is used in Theorem 22 to prove the accessibility of each zigzag and in Lemma 20. The paper labels it a conjecture, so the central theorems are conditional on it.
  • domain assumption Strong subadditivity and chain rule for Hmax,gen hold at all orders in G.
    Theorems 8 and 9 are proved at leading order from known entropy inequalities, but the paper states that at higher orders they must be restated as conjectures. The proof of Theorem 22 adds inequalities via these properties.
  • domain assumption Corollary 48 of Ref. [35]: the wedge union of antinormal lightsheet wedges is antinormal on the new edge.
    Used in Lemma 19 and Theorem 22 to show that the zigzag is antinormal. It is not proved in this paper and comes from a companion paper by the same authors.
  • ad hoc to paper The infinite zigzag limit z(a)=lim_n z_n(a) exists as a wedge and inherits the finite-n properties.
    Definition 25 asserts this without a convergence or regularity proof; Theorem 27 and Corollary 28 depend on the limit having the same properties as finite zigzags.
  • standard math M is globally hyperbolic and wedges satisfy a=a''.
    Stated in Sec. 3.1 and needed for all causal constructions, including the wedge complement and causal development operations.

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Pith. "Pith review of Simple Holography in General Spacetimes." pith.science (2026). https://pith.science/paper/6AUUFZDG

@misc{pith2026250500695,
  author       = {Pith},
  title        = {Pith review of: Simple Holography in General Spacetimes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6AUUFZDG}},
  note         = {Machine review of arXiv:2505.00695}
}
abstract

The simple or "outermost" wedge in AdS is the portion of the entanglement wedge that can be reconstructed with sub-exponential effort from CFT data. Here we furnish a definition in arbitrary spacetimes: given an input wedge $a$ analogous to a CFT boundary region, the simple wedge $z(a)$ is the largest wedge accessible by a "zigzag," a certain sequence of antinormal lightsheets. We show that $z(a)$ is a throat, and that it is contained in every other throat. This implies that $z(a)$ is unique; that it is contained in the generalized entanglement wedge; and that it reduces to the AdS prescription as a special case. The zigzag explicitly constructs a preferred Cauchy slice that renders the simple wedge accessible from $a$; thus it adds a novel structure even in AdS. So far, no spacelike construction is known to reproduce these results, even in time-symmetric settings. This may have implications for the modeling of holographic encoding by tensor networks.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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