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Holographic Entropy Cone Beyond AdS/CFT

T0 review · 2 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper proves that every holographic entropy cone inequality that can be certified by a contraction map also holds for generalized entanglement wedges of bulk regions in arbitrary static spacetimes, provided the regions satisfy a…

desk verdict A real generalization of the holographic entropy cone to bulk regions, with two repairable gaps (tile-boundary definition and wedge nesting) that should not block refereeing. read the letter →

arxiv 2502.03516 v1 pith:5LL3IHPH submitted 2025-02-05 hep-th

Raphael Bousso , Sami Kaya This is my paper · ORCID
classification hep-th
keywords holographicentropyconegeneralizedentanglementwedgecontractionmapmutualindependenceconditionmonogamyofinformationstrongsubadditivityquantumgravity
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

The paper extends the holographic entropy cone—the set of universal area inequalities obeyed by Ryu-Takayanagi surfaces in AdS/CFT—to generalized entanglement wedges of bulk regions in arbitrary static spacetimes. The main theorem states that any inequality certified by an $\alpha,\beta$-contraction map also holds for the areas of generalized entanglement wedges, provided the input regions satisfy a mutual independence condition: each region lies outside the entanglement wedge of the union of all the others. Because every holographic entropy cone inequality corresponds to such a contraction map, the generalized entropy cone coincides with the standard one in static settings. This matters because it carries the geometric entanglement structure of AdS/CFT into spacetimes without an asymptotic boundary, and it identifies when bulk degrees of freedom can be treated as independent despite holographic nonlocality.

What carries the argument

The load-bearing construction is the tile decomposition of $\Sigma$ into $2^L$ regions $s(x)=\bigcap_l E(v_l)^{x_l}$, where $E(v_l)^1=E(v_l)$ and $E(v_l)^0=E(v_l)'$; each tile is labeled by which left-hand-side entanglement wedges contain it. Their shared boundaries $\gamma(x,y)=\partial s(x)\cap \partial s(y)$ carry the areas, and the left-hand side of any candidate inequality becomes $\sum_{x<y} d(\alpha;x,y)\,\mathrm{Area}[\gamma(x,y)]$ with $d(\alpha;x,y)=\sum_l \alpha_l |x_l-y_l|$. An $\alpha,\beta$-contraction map $f$ defines candidate regions $c_r=\bigcup_{x:f_r(x)=1} s(x)$, whose total area is bounded by the left-hand side because $d(\alpha;x,y)\ge d(\beta;f(x),f(y))$. The independence condition guarantees each input region $a_i$ lies entirely in the single tile $s(x_i)$, so the map's values on the $x_i$ put exactly the right regions into each $c_r$ and give $c_r$ the same conformal boundary as $w_r$; minimality of $E(w_r)$ then yields the claimed inequality.

What would settle it

Compute generalized entanglement wedges for two concentric wedges $A\subset B$ on the time-reflection-symmetric slice of a Schwarzschild or de Sitter spacetime and check whether $E(A)\subset E(B)$; a failure would break the proof's tiling step. Alternatively, search numerically for bulk regions that satisfy the independence condition but violate monogamy of mutual information or another contraction-proven inequality, which would disprove the theorem.

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Extended reading notes

Core claim

The paper's central claim is Theorem 12. Let $a_1,\ldots,a_n$ be open bulk regions (wedges) on a time-reflection-symmetric Cauchy slice $\Sigma$, with $a_0=\emptyset$, satisfying $a_i \subset [E(\curlyvee_{j\neq i} a_j)]'$ for each $i$. For any collections $v_l$ and $w_r$ of wedge unions of the $a_i$ and any positive-integer coefficient strings $\alpha$ and $\beta$, if there exists an $\alpha,\beta$-contraction map $f: \{0,1\}^L \to \{0,1\}^R$ with $f(x_i)=y_i$ for all $i$, then $\sum_l \alpha_l \, \mathrm{Area}[E(v_l)] \geq \sum_r \beta_r \, \mathrm{Area}[E(w_r)]$. The proof tiles $\Sigma$ by the regions $s(x)$ determined by which of the $E(v_l)$ contain each point, writes left-side areas as sums over tile boundaries weighted by $\alpha$-distances, uses $f$ to assemble candidate right-side regions $c_r$ as unions of tiles, and then invokes the definition of $E(w_r)$ as the smallest-area wedge with the correct conformal boundary. This establishes that every holographic entropy cone inequality admitting a contraction map survives for generalized entanglement wedges.

Load-bearing premise

The proof relies on the unstated assumption that generalized entanglement wedges respect containment: if a bulk region $A$ lies inside $B$, then $E(A)$ lies inside $E(B)$, a property that holds for Ryu-Takayanagi surfaces but is not proven for generalized wedges in arbitrary spacetimes.

Editorial extensions

If this is right

  • Monogamy of mutual information, the five-party cyclic inequality, and every other holographic entropy cone inequality proven by a contraction map hold for generalized entanglement wedges of mutually independent bulk regions.
  • The static generalized holographic entropy cone is identical to the standard holographic entropy cone: no additional universal inequalities arise from using bulk regions as inputs.
  • Any future holographic entropy inequality that is proven via a contraction map automatically applies to generalized entanglement wedges, since the theorem only assumes the map exists.
  • The mutual independence condition is the operative physical criterion for treating bulk regions as carrying distinct degrees of freedom; configurations that violate it can violate even monogamy of mutual information.
  • The area inequalities now apply to spacetimes without an AdS boundary, so they can serve as consistency checks for entanglement-wedge proposals in cosmological and black-hole settings.

Reading between the lines

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

  • The proof's tiling step silently assumes the nesting property $E(A)\subset E(B)$ for generalized entanglement wedges; testing monotonicity on a Schwarzschild or de Sitter static slice would determine whether this hidden hypothesis restricts the theorem's scope.
  • A covariant generalization may be possible by replacing the static tiling with a maximin or quantum-extremal construction and adapting the independence condition, but the paper proves the static case only.
  • The independence condition may admit a precise interpretation in tensor-network models of holography, where it would correspond to which bulk legs can be independently varied without affecting the others' entanglement wedges; examples in such models could turn the condition into a concrete diagnostic.
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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

2 major / 4 minor

Summary. The paper claims to extend the entire known static holographic entropy cone (HEC) of AdS/CFT boundary regions to static generalized entanglement wedges of arbitrary bulk regions in general spacetimes. The main result, Theorem 12, states that if an α,β-contraction map f maps prescribed bit strings x_i to y_i, then the corresponding weighted sum of generalized entanglement wedge areas satisfies the associated HEC inequality, under a mutual independence condition a_i ⊂ [E(⋓_{j≠i} a_j)]'. The proof partitions the Cauchy slice into tiles by inclusion/exclusion in the left-hand-side wedges, uses the contraction map to define candidate regions c_r for the right-hand-side wedges, and invokes the minimal-area property of the generalized entanglement wedge to conclude the inequality. The paper also argues that this makes the generalized holographic entropy cone identical to the standard holographic entropy cone.

Significance. If the proof is corrected, this is a substantial and interesting result: it would carry the complete graph-theoretic contraction-map technology for HEC inequalities out of the AdS/CFT setting to generalized entanglement wedges, with a precise independence condition that has physical significance for gravitating bulk regions. The work builds on the authors' earlier construction of generalized entanglement wedges and would subsume their previous MMI result. The contraction-map formulation is elegant, parameter-free, and uses only the minimal-area definition of the generalized wedge. The paper is clearly written and the tile construction is intuitive. However, two technical gaps in the proof need to be repaired before the central claim can be regarded as established.

major comments (2)
  1. [Section 2, Eq. (2.18) and (2.19)] The regions c_r are defined as ordinary unions of the open tiles s(x). Since each s(x) defined in Eq. (2.13) is an intersection of open sets (E(v_l) and their complements), each tile is open. For an ordinary union of open tiles, a boundary segment γ(x,y) shared by two selected adjacent tiles is not contained in c_r, and therefore lies in ∂c_r even when |f_r(x)-f_r(y)|=0. Consequently Eq. (2.19) does not compute Area[c_r]; it computes the area of the wedge union of the selected tiles, which fills in those internal boundary segments. The inequality chain then fails to bound Σβ Area[E(w_r)], because the true area of c_r is larger than the value used in Eq. (2.20). The fix is straightforward: define c_r = ⋓_{x:f_r(x)=1} s(x), i.e., the wedge union of the selected tiles. With that definition Eq. (2.19) is correct, the containment c_r ⊃ w_r also holds (because each a_i lies in a selected tile and wedge union is monotone), and the minimality argument for E(w_r) goes through. This correction needs to be stated explicitly in the proof.
  2. [Section 2, paragraph after Eq. (2.21)] The claim that the independence condition (2.10) implies that each a_i lies in exactly one tile s(x_i) relies on the nesting/monotonicity property E(a) ⊂ E(b) whenever a ⊂ b. This property is not stated, proved, or cited for the generalized entanglement wedge of Def. 7. It is standard for RT surfaces in AdS/CFT, but Def. 7 alone (smallest-area open set with given conformal boundary) does not make monotonicity immediate, since E(b) need not be a candidate for a when the conformal boundaries of a and b differ. Without this lemma, an input region a_i could intersect E(v_l) even when a_i is not part of v_l, and then a_i would not be contained in a single tile. Please add a lemma establishing nesting for the generalized entanglement wedge, or give a precise reference that covers the static generalized case, and use it explicitly in the proof.
minor comments (4)
  1. [Section 3, first paragraph] The sentence 'The proof of the traditional ... differs ... chiefly through' is missing a preposition; it should read 'differs from the present proof ... chiefly through' or similar.
  2. [Section 3, paragraph beginning 'This can be made more precise'] There is a typo: 'more precise is if' should be 'more precise if'.
  3. [Section 3, paragraph discussing Eq. (2.10)] The phrase 'the inclusion condition of Def. 6' appears to be a misreference; the defining containment and conformal-boundary conditions for an entanglement wedge are in Def. 7, not Def. 6.
  4. [References] Reference [24] is listed as 'To appear' with no year or identifier; since the Discussion cites it as the source for the deeper origin of the independence condition, please provide an arXiv number or remove the reference if it is not yet available.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the generalized entropy cone proof is a self-contained contraction-map argument that does not fit parameters or reduce to its inputs.

full rationale

The paper's derivation is self-contained conditional on Definition 7 and on the externally established contraction-map formulation of the holographic entropy cone. The proof of Theorem 12 partitions the Cauchy slice into tiles built from the entanglement wedges E(v_l), uses a given contraction map f to select candidate regions c_r, and then invokes the defining minimality of E(w_r) to bound each Area[E(w_r)] by Area(c_r). The target inequality is obtained directly from the contraction-map condition; no parameter is fitted to the output, and the independence condition (2.10) is a hypothesis rather than a fitted or predicted quantity. The authors' prior work [18,19] supplies the definition of generalized entanglement wedges, and [22] supplies the MMI example, but Theorem 12 does not use any of those results as a load-bearing premise; it uses only the definition of E and the standard HEC contraction-map machinery. The proof does contain technical gaps unrelated to circularity, notably that Eq. (2.18) defines c_r as an ordinary union of open tiles while Eq. (2.19) counts wedge-union boundary terms, and that monotonicity of generalized entanglement wedges is assumed without proof when asserting each a_i lies in a unique tile. These are correctness concerns, not circular reductions, and they do not make the derivation equivalent to its inputs by construction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the definition and basic properties of generalized entanglement wedges from prior work, plus an implicit monotonicity/nesting property that is not justified in the paper. No free parameters are fitted. No new physical entities are introduced.

assumptions (3)
  • domain assumption Existence and defining properties of the generalized entanglement wedge E(a) (Definition 7), including existence of a smallest-area open set.
    The theorem assumes E(a) exists for every wedge a and that candidate regions c_r can be compared with E(w_r) by area minimality. This comes from Ref. [18] but is not re-proven here.
  • domain assumption Monotonicity (nesting) of the generalized entanglement wedge: if A ⊂ B, then E(A) ⊂ E(B).
    Used implicitly in the proof of Theorem 12 when asserting that each input region a_i lies in exactly one tile s(x_i); specifically, that a_i is disjoint from E(v_l) whenever a_i is not part of v_l. No proof or citation is given in this paper.
  • standard math Standard area measure theory: the tile decomposition yields additive boundary areas with shared segments γ(x,y), and area is well-defined and regularized at the conformal boundary.
    Used in Eqs. (2.16)-(2.20); the paper does not discuss the conformal-boundary divergence subtraction, which is standard in AdS/CFT.

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Pith. "Pith review of Holographic Entropy Cone Beyond AdS/CFT." pith.science (2026). https://pith.science/paper/5LL3IHPH

@misc{pith2026250203516,
  author       = {Pith},
  title        = {Pith review of: Holographic Entropy Cone Beyond AdS/CFT},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5LL3IHPH}},
  note         = {Machine review of arXiv:2502.03516}
}
read the original abstract

We extend all known area inequalities obeyed by Ryu-Takayanagi surfaces of AdS boundary regions -- the holographic entropy cone -- to static generalized entanglement wedges of bulk regions in arbitrary spacetimes. The generalized holographic entropy cone is subject to a mutual independence condition on the bulk regions: each bulk input region must be outside the entanglement wedge of the union of all others. The condition captures when gravitating regions involve fundamentally distinct degrees of freedom despite the nonlocality inherent in the holographic principle.

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Forward citations

Cited by 2 Pith papers

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

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

    hep-th 2025-06 conditional novelty 7.0 of 10

    The entropy of a bulk region in holographic states equals the generalized entropy of the smallest wedge containing it, derived from a replica path integral via a hollow-graphic construction.

  2. Entanglement Entropy of Quantum Corners

    hep-th 2025-07 conditional novelty 6.0 of 10

    For a two-dimensional corner symmetry algebra, coherent corner states give an entanglement entropy that scales with the area when mapped to near-extremal Reissner-Nordström black holes.

Reference graph

Works this paper leans on

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