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REVIEW 3 major objections 3 minor 111 references

After the Page time, the correct Page curve for an eternal black hole appears precisely when either two exterior regions lose all mutual information or the island and radiation become infinitely entangled; the two conditions are equivalent

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-11 14:49 UTC pith:DHV2YGNC

load-bearing objection Clean incremental observation: the same scrambling-time locus that kills I(B+:B−) makes I(I:R) diverge, plus a usable closed form for the tripartite I(I:R+:R−), all inside pure eternal JT. the 3 major comments →

arxiv 2607.04706 v1 pith:DHV2YGNC submitted 2026-07-06 hep-th

New insights on mutual information in the island approach to the Page curve

Anirban Roy Chowdhury , Souvik Paul , Sunandan Gangopadhyay This is my paper
classification hep-th
keywords Page curveisland formulamutual informationtripartite informationeternal black holeJackiw-Teitelboim gravityscrambling timegeometric entanglement
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper argues that the post-Page-time Page curve of an eternal black hole is controlled by how mutual information is shared among regions on a single Cauchy slice. Earlier work showed that the mutual information between the two exterior intervals B+ and B- vanishes at scrambling time, producing a time-independent matter entropy and thereby the flat Page curve. The new observation is that the identical time condition forces the mutual information between the island and the full radiation region to diverge, so the island and radiation become maximally entangled. The authors interpret this as a conservation of geometric entanglement: when one bipartition disconnects, another becomes infinitely correlated. They also derive an explicit formula for the tripartite mutual information among the island and the two radiation halves, finding it always negative and therefore monogamous.

Core claim

Reproducing the expected Page curve after the Page time requires that either I(B+:B-)=0 or I(I:R) o infinity; these two conditions are mathematically equivalent under the pure-state property of the Cauchy slice and together rewrite the island formula so that the matter-entropy term becomes time-independent.

What carries the argument

The rewritten island formula that replaces the matter entropy S(I\cup R) by the entropy of B+\cup B- evaluated either at vanishing I(B+:B-) or at divergent I(I:R). The explicit expression for I(I:R) diverges exactly when ta-tb equals the absolute difference of tortoise coordinates of the island and radiation endpoints.

Load-bearing premise

The full Cauchy slice that contains the island, the radiation, and the complementary exterior regions must be in a pure quantum state, so the entropy of island-plus-radiation equals the entropy of the complementary pair of exterior intervals.

What would settle it

In any concrete model where the global state on the Cauchy slice is mixed (or the island and radiation do not exhaust a pure slice), compute both mutual informations at the putative scrambling time and check whether vanishing of one still forces divergence of the other and still yields a flat post-Page entropy.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The Page curve can be obtained by demanding infinite island-radiation mutual information instead of vanishing exterior mutual information.
  • Geometric correlations on a Cauchy slice are conserved: loss of correlation in one bipartition is compensated by maximal entanglement in another.
  • The tripartite information I(I:R+:R-) is always negative, so island degrees of freedom are encoded nonlocally across both radiation halves rather than redundantly in either half alone.
  • The same pair of equivalent conditions may determine the Page curve for evaporating black holes once the calculation is extended beyond the eternal case.

Where Pith is reading between the lines

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

  • If the conservation relation survives in dynamical evaporating geometries, one could track the Page curve solely through the growth of I(I:R) without repeatedly locating quantum extremal surfaces.
  • The observed monogamy of I(I:R+:R-) implies that partial access to only one radiation half cannot reconstruct the island, a multipartite security feature that may be testable in toy qubit models of black-hole evaporation.
  • The divergence of I(I:R) at scrambling time offers a diagnostic for the onset of replica-wormhole dominance that is independent of the area term in the island formula.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper studies mutual information for subsystems on a Cauchy slice in the island approach to the Page curve of an eternal black hole in JT gravity with a flat thermal bath. Building on the authors’ earlier results that I(B+:B−) vanishes at the scrambling time ta−tb=|r∗(a)−r∗(b)| (yielding a time-independent matter entropy and the correct post-Page Page curve), it shows that the identical geometric condition makes I(I:R) diverge. This is interpreted as conservation of geometric correlation among regions on the slice. Using free-CFT entropy formulas and purity of the full Cauchy slice, the authors also derive a closed-form expression for the tripartite mutual information I(I:R+:R−), demonstrate that it is always negative (monogamous/superextensive), and plot its dependence on island and observer times.

Significance. The dual conditions I(B+:B−)=0 or I(I:R)→∞ give an alternative phrasing of the island contribution that produces the Page curve and may be useful for evaporating black holes (as the authors suggest). The explicit tripartite formula and its monogamy property are a concrete new result in the island literature, obtained cleanly from standard Calabrese–Cardy distance formulas and Cauchy-slice purity. The algebraic steps are elementary and transparent. The work is incremental relative to the authors’ prior series, so overall novelty is moderate, but the multipartite perspective is a welcome addition if the claims are carefully stated.

major comments (3)
  1. [Section III, eqs. (22)–(30) and claim (2)] The claimed equivalence of I(B+:B−)=0 and I(I:R)→∞ (eq. (2)) and the rewritten island formula (30) rest entirely on the pure-state identity S_vN(I∪R)=S_vN(B+∪B−) used after eq. (22) and again for (23) and (30). This identity is valid inside the eternal pure-state setup, but the ‘conservation of geometric correlation’ is then essentially a rewriting of that identity rather than an independent principle. The purity assumption must be stated explicitly, and the manuscript should clarify that the two conditions share the same critical surface only because of it; otherwise the central interpretive claim overreaches.
  2. [Section IV, eq. (47) and Fig. 3] Eq. (47) for I(I:R+:R−) depends only on ta, a and the outer endpoint e (with te apparently fixed at 0); it is independent of observer time tb and of b. Fig. 3b nevertheless plots the same quantity versus tb (fixed b=10 r+). This is inconsistent. Either the analytic expression must be corrected to include the times of e±, an unstated relation (e.g., te=tb or ta−tb fixed) must be specified for the plot, or the figure must be replaced. In addition, the outer points e± are never defined in the text or shown in Fig. 2, yet they control both the formula and the numerical values in Fig. 3.
  3. [Abstract, Introduction and Conclusion] The language of ‘conservation of geometrical entanglement’ and ‘deep insights about the conservation of geometrical entanglement’ (abstract, Introduction, Conclusion) is not supported by any quantitative conservation law (no relation such as a sum or product of mutual informations remaining constant is derived). The actual observation is that the same geometric locus makes one mutual information vanish and the other diverge. The claims should be rephrased more precisely to ‘complementary behaviour under the scrambling-time condition’ or equivalent.
minor comments (3)
  1. [Throughout] Numerous formatting artifacts appear throughout (e.g., ‘COMPUT A TION’, ‘GRA VITY’, ‘INFORMA TION’) together with minor typos (Conclusion: ‘I(R+:R−) diverges’ should read I(I:R); stray capitalisation such as ‘An important observation’). These should be cleaned for the final version.
  2. [Fig. 2 and Section IV] The outer endpoints e± of the radiation regions R± should be indicated on the Penrose diagram (Fig. 2) and defined in the text at first use, since they enter the tripartite formula and the plots.
  3. [References and Introduction] The paper relies heavily on the authors’ own prior works [86–90] for the Page-curve interpretation. A more balanced discussion of the broader island literature would improve the presentation.

Circularity Check

2 steps flagged

Page-curve interpretation of the shared critical surface rests on self-cited prior works; the new divergence and tripartite formulas are independently computed algebra under the model's purity assumption.

specific steps
  1. self citation load bearing [Abstract; Introduction (paragraphs after eq. (1)); rewritten island formula (30)]
    "In our earlier works, we have shown how the saturation of mutual information between two specific subsystems plays a vital role in obtaining the correct Page curve for the eternal black hole. In those works, we have shown that the mutual information between B+ and B−, that is, I(B+:B−), vanishes at scrambling time, which leads to the correct Page curve. ... = min ext_I {A(∂I)/4GN + S_vN(B+∪B−)|I(B+:B−)=0} = min ext_I {A(∂I)/4GN + S_vN(B+∪B−)|I(I:R)→∞}."

    The physical claim that vanishing I(B+:B−) (or the newly observed divergence of I(I:R)) yields the correct Page curve is justified solely by citation to the authors' own prior papers [86–90]. The present algebra shows only that the two mutual informations share the same critical surface; the Page-curve interpretation of that surface is imported by self-citation and is load-bearing for the strongest claim (eqs. (2) and (30)).

  2. other [Section III, eqs. (21), (28)–(29) and the purity step after (22)]
    "ta−tb=|r∗(a)−r∗(b)|. ... I(B+:B−)|ta−tb=|r∗(a)−r∗(b)|=0. ... I(I:R)|ta−tb=|r∗(a)−r∗(b)|→∞. ... from the pure state property of the von-Neumann entropy, we can write SvN(I∪R)=SvN(B+∪B−)."

    Once purity is used to replace S(I∪R) by S(B+∪B−), the identical geometric locus that nullifies the numerator/denominator combination in I(B+:B−) automatically sends the denominator of I(I:R) to zero. The claimed 'equivalence' and 'conservation of geometric correlation' therefore hold by construction inside the pure-state model; they are not an independent dynamical prediction.

full rationale

The paper's new algebraic results (explicit formula for I(I:R) in (26)–(27), its divergence under the geometric locus (29), and the tripartite expression (44)–(47)) are derived from standard 2d CFT distance formulas and the pure-state identity S(I∪R)=S(B+∪B−) that is native to the eternal JT+bath setup. These steps do not reduce to fitted parameters or to a definitional tautology. The interpretive claim that either I(B+:B−)=0 or I(I:R)→∞ is necessary and sufficient for the correct post-Page Page curve (eqs. (2), (30)) does, however, import its physical meaning wholesale from the authors' earlier series [86–90]. That is ordinary self-citation of prior results rather than a uniqueness theorem or an ansatz smuggled by citation, so the circularity burden remains modest. No fitted-input-as-prediction or renaming-of-known-result patterns appear. Score 3 reflects one load-bearing self-citation chain that is not the sole content of the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper works entirely within the standard island-formula and free-2d-CFT toolkit. No free parameters are fitted; the only inputs are the JT metric, the free-CFT entropy formulas, and the pure-state property of a Cauchy slice. No new entities are postulated.

axioms (4)
  • domain assumption Von Neumann entropy of free 2d CFT intervals is given by the Calabrese–Cardy logarithmic distance formulas (eqs. (13), (24), (25), (36)–(39)).
    Invoked throughout §§III–IV; standard but essential for every explicit expression.
  • domain assumption The full Cauchy slice is in a pure state, so S(I∪R)=S(B+∪B−).
    Used immediately after eq. (22) and again to obtain eq. (43); without it the mutual-information identities collapse.
  • domain assumption The island formula (1) correctly computes the fine-grained entropy of Hawking radiation.
    Background assumption of the entire program; not re-derived here.
  • ad hoc to paper The geometric condition ta−tb=|r∗(a)−r∗(b)| is the scrambling time at which I(B+:B−) vanishes.
    Taken from the authors’ earlier series and re-used as the trigger for the new divergence claim.

pith-pipeline@v1.1.0-grok45 · 23385 in / 2484 out tokens · 23722 ms · 2026-07-11T14:49:53.692873+00:00 · methodology

0 comments
read the original abstract

In this article, we have presented one of the very important observations, regarding the behavior of mutual information of two different sets of subsystems in the after Page time scenario. This provides us with some deep insights about the conservation of geometrical entanglement. In our earlier works, we have shown how the saturation of mutual information between two specific subsystems plays a vital role in obtaining the correct Page curve for the eternal black hole. In those works, we have shown that the mutual information between $B_+$ and $B_-$, that is, $I(B_{+}: B_{-})$, vanishes at scrambling time, which leads to the correct Page curve. That means that at scrambling time, there is no correlation between $B_+$ and $B_-$. Remarkably, it is observed that at this particular value of observer's time, the mutual information between $\mathcal{I}$ and $R$, that is, $I(\mathcal{I}:R)$, becomes infinity. This indicates that the regions $\mathcal{I}$ and $R$ become maximally entangled. This provides us with a notion of conservation of geometric correlation between different regions on the Cauchy slice. In this work, we have also provided a way to calculate the tripartite mutual information of regions $\mathcal{I}$,$R_+$ and $R_-$, that is, $I(\mathcal{I}:R_+:R_-)$ on the Cauchy slice using the earlier results involving the bipartite regions. This is a new result which was missing in the earlier literature.

Figures

Figures reproduced from arXiv: 2607.04706 by Anirban Roy Chowdhury, Souvik Paul, Sunandan Gangopadhyay.

Figure 1
Figure 1. Figure 1: FIG. 1: Penrose diagram of an eternal black hole in JT [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Penrose diagram of an eternal black hole in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The left panel shows the variation of the tripartite information between [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

discussion (0)

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Reference graph

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