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arxiv: 2602.04888 · v2 · submitted 2026-01-26 · ✦ hep-th · gr-qc

Graph models for covariant holographic entropy I

Bowen Zhao This is my paper

Pith reviewed 2026-05-16 11:27 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords holographic entropy coneHRT surfacesgraph modelscovariant entropyno-short-cut theorementanglement horizonsAdS/CFT
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The pith

Under a condition on exposed regions, graph models exactly reproduce HRT entropies and equate the covariant and static entropy cones.

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

The paper builds graph models for holographic entropies in time-dependent spacetimes by removing the obstruction from partial HRT surfaces. It identifies the existence of exposed regions for each pair of HRT surfaces as the key geometric condition that permits construction of weight functions via projection along null generators of entanglement horizons. With this condition in place, a Conditional No-Short-Cut Theorem is proved showing that every graph cut is dominated by a surface made from complete HRT surfaces. The result establishes that the graph model reproduces the full set of HRT entropies, so the covariant and static holographic entropy cones coincide in this regime. The construction is further extended by grouping surfaces into timelike clusters when nesting prevents exposed regions.

Core claim

We identify a geometric condition—the existence of exposed regions for each pair of HRT surfaces—under which the obstruction from partial surfaces is removed. Under this condition, we construct weight functions by projecting along null generators of entanglement horizons and prove a Conditional No-Short-Cut Theorem: any graph cut is dominated by a surface composed of complete HRT surfaces. Consequently, the graph model reproduces HRT entropies, establishing the equivalence between the covariant and static holographic entropy cones in this regime.

What carries the argument

Weight functions constructed by projecting along null generators of entanglement horizons, which prove that graph cuts are dominated by complete HRT surfaces.

If this is right

  • The graph model computes exact HRT entropies whenever the exposed-region condition holds.
  • The covariant and static holographic entropy cones are identical in the exposed-region regime.
  • Nesting of interaction regions can be handled by grouping HRT surfaces into timelike clusters.
  • The same weight-function technique offers a route to a fully covariant graph model beyond the current regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may extend to proofs of other holographic inequalities in dynamical geometries if the exposed-region condition can be verified.
  • Explicit checks in Vaidya or other time-dependent AdS solutions could determine how often the exposed-region condition occurs.
  • If the condition turns out to be generic, graph models would simplify numerical extraction of time-dependent entanglement entropies.

Load-bearing premise

Exposed regions must exist for every pair of HRT surfaces so that partial surfaces cannot create short-cuts that undercut the entropy of complete surfaces.

What would settle it

A concrete spacetime configuration of HRT surfaces lacking exposed regions in which some graph cut built from partial surfaces has strictly smaller total length than any surface assembled only from complete HRT surfaces.

read the original abstract

We construct a graph model for holographic entropies in general time-dependent spacetimes. In static settings, such models arise from Ryu-Takayanagi surfaces on a common Cauchy slice and imply that the holographic entropy cone is polyhedral. Extending this construction to the covariant Hubeny-Rangamani-Takayanagi (HRT) setting is obstructed by the absence of a preferred time slice, raising the possibility of unphysical "short-cuts" built from partial HRT surfaces. We identify a geometric condition--the existence of exposed regions for each pair of HRT surfaces--under which this obstruction is removed. Under this condition, we construct weight functions by projecting along null generators of entanglement horizons and prove a Conditional No-Short-Cut Theorem: any graph cut is dominated by a surface composed of complete HRT surfaces. Consequently, the graph model reproduces HRT entropies, establishing the equivalence between the covariant and static holographic entropy cones in this regime. We further show that configurations in which exposed regions are absent due to nesting of interaction regions can be partially resolved by grouping HRT surfaces into timelike clusters. This provides evidence that the graph model extends beyond the exposed-region regime and suggests a path toward a complete covariant construction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper constructs a graph model for holographic entropies in general time-dependent spacetimes. It identifies a geometric condition—the existence of exposed regions for each pair of HRT surfaces—under which a Conditional No-Short-Cut Theorem is proved: any graph cut is dominated by a surface composed of complete HRT surfaces. This establishes equivalence between the covariant and static holographic entropy cones in this regime. For cases where exposed regions are absent due to nesting, the paper shows partial resolution by grouping into timelike clusters.

Significance. If the exposed-region condition holds more generally or the construction is extended rigorously, this would advance the field by showing that graph models apply to covariant HRT entropies, implying the holographic entropy cone remains polyhedral in time-dependent spacetimes. The proof of the conditional theorem via null projections and the geometric constructions are clear strengths, providing a concrete path toward a complete covariant construction.

major comments (2)
  1. [Abstract and §4 (theorem statement and proof)] The Conditional No-Short-Cut Theorem (stated in the abstract and proved via weight-function construction) is load-bearing on the existence of exposed regions for every HRT pair to enable null-generator projections and block partial-surface short-cuts. The manuscript supplies no general criterion guaranteeing this condition for arbitrary collections of HRT surfaces in time-dependent geometries, limiting the theorem's applicability beyond the stated regime.
  2. [§5 (timelike clustering discussion)] The timelike-clustering approach for nested configurations (described as providing partial evidence for extension beyond exposed regions) is only sketched and lacks a full proof that the graph model reproduces HRT entropies without the exposed-region hypothesis. This makes the suggestion of a path toward a complete covariant construction weaker than claimed.
minor comments (2)
  1. [§3 (construction)] The notation for 'exposed regions' and 'weight functions obtained by null projection' would benefit from an explicit equation or diagram in the construction section to improve readability.
  2. [Abstract and introduction] A few sentences in the abstract and introduction repeat the conditional nature of the result; tightening this language would clarify the scope without altering the claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment below, clarifying the scope of our results while acknowledging the limitations noted.

read point-by-point responses

  1. Referee: [Abstract and §4 (theorem statement and proof)] The Conditional No-Short-Cut Theorem (stated in the abstract and proved via weight-function construction) is load-bearing on the existence of exposed regions for every HRT pair to enable null-generator projections and block partial-surface short-cuts. The manuscript supplies no general criterion guaranteeing this condition for arbitrary collections of HRT surfaces in time-dependent geometries, limiting the theorem's applicability beyond the stated regime.

    Authors: We agree that the Conditional No-Short-Cut Theorem is explicitly conditional on the exposed-region hypothesis for each HRT pair, as stated in the abstract and developed in §4. The manuscript does not supply (nor claim to supply) a general criterion guaranteeing the existence of exposed regions for arbitrary collections of HRT surfaces in time-dependent spacetimes; determining such a criterion is a distinct geometric question outside the scope of this work. Our contribution is to prove the theorem under the stated condition and to establish the resulting equivalence of the covariant and static entropy cones in that regime. We will add a brief remark in the introduction and conclusions noting that the generality of the exposed-region condition remains an open question. revision: partial

  2. Referee: [§5 (timelike clustering discussion)] The timelike-clustering approach for nested configurations (described as providing partial evidence for extension beyond exposed regions) is only sketched and lacks a full proof that the graph model reproduces HRT entropies without the exposed-region hypothesis. This makes the suggestion of a path toward a complete covariant construction weaker than claimed.

    Authors: We acknowledge that the timelike-clustering construction in §5 is presented as a sketch that furnishes partial evidence rather than a complete proof. The manuscript already qualifies the discussion in precisely these terms, stating that it 'provides evidence that the graph model extends beyond the exposed-region regime and suggests a path toward a complete covariant construction.' We do not assert a full resolution without the exposed-region hypothesis. To address the concern, we will expand §5 with additional geometric details on the clustering procedure and explicitly restate its provisional character, while deferring a rigorous proof to future work. revision: partial

Circularity Check

0 steps flagged

Conditional theorem under explicit geometric hypothesis; no reduction to self-definition or fitted inputs

full rationale

The derivation proceeds by stating the exposed-region condition as an external geometric hypothesis, then constructing weight functions via null-generator projection and proving that any graph cut is dominated by a union of complete HRT surfaces. The Conditional No-Short-Cut Theorem is explicitly conditional on this hypothesis and does not derive the hypothesis from the graph model or from any fitted parameter. No self-citation supplies the central uniqueness or existence claim, no ansatz is smuggled, and no known empirical pattern is merely renamed. The argument is therefore self-contained as a proof under stated assumptions rather than a tautology or statistical fit.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on the HRT prescription as the definition of covariant holographic entropy and on the new geometric notion of exposed regions.

axioms (1)
  • domain assumption The Hubeny-Rangamani-Takayanagi (HRT) prescription defines holographic entanglement entropy via extremal surfaces in time-dependent spacetimes.
    This is the foundational input taken from prior literature and used to define the target quantities the graph model must reproduce.
invented entities (2)
  • Exposed regions on pairs of HRT surfaces no independent evidence
    purpose: Geometric condition that prevents partial-surface shortcuts in the graph model
    Introduced to enable the weight-function construction and the no-short-cut theorem.
  • Graph weight functions obtained by null projection no independent evidence
    purpose: Assign edge weights so that graph cuts reproduce HRT areas
    Constructed in the paper by projecting along null generators of entanglement horizons.

pith-pipeline@v0.9.0 · 5503 in / 1313 out tokens · 28802 ms · 2026-05-16T11:27:55.705479+00:00 · methodology

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Reference graph

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