Edge states in Gravity and Black Hole Physics

A.P. Balachandran; Arshad Momen; L. Chandar

REVIEW

Edge states in Gravity and Black Hole Physics

A.P. Balachandran, L. Chandar, Arshad Momen

General Relativity and Quantum Cosmology · Submitted 6 Dec 1994 · 21 references

Pith's verdict: A solid Regge-Teitelboim construction of gravitational edge observables with one genuinely new structural observation; the complementarity claim is a boundary-condition choice, not a derivation, and the 2+1 coupling rests on an unpublished self-citation.

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arXiv IDgr-qc/9412019 (version 2)

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CitationA.P. Balachandran, Arshad Momen, L. Chandar. "Edge states in Gravity and Black Hole Physics." arXiv:gr-qc/9412019 (1994). https://pith.science/paper/ENRUOZYW

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Editor's note

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Worth your time, but read it for Section 3, not the abstract. The real content is a clean application of the Regge-Teitelboim differentiability program to a manifold with an inner boundary: relax the fall-off conditions on the smearing fields of the vector and scalar constraints, add the compensating surface term, and you get an infinite family of boundary-localized observables (Eqs. 3.2, 3.10) that commute weakly with H and close on the diffeomorphism algebra (Eq. 3.4). That is correct and it is the part I would cite. The genuinely new observation is that differentiability requirements, not physics, decide which of these observables are 'good' for a given observer — that is a nice structural point and it is not in their earlier gauge-theory papers [9]. The QHE section is exposition borrowed from [10]; treat it as motivation, not evidence. Section 6 is honest: the authors say outright they have neither an edge action nor the anomalous coupling in 3+1, and they say the entropy degeneracy is infinite and needs regularization before any area law can be discussed. The classifier record claims no stated limitations; that is wrong, the limitations are stated plainly in Section 6 and in the entropy paragraph, and they are stated accurately. Credit that rather than scoring it as an overclaim. Two real soft spots. First, the complementarity claim turns on imposing delta q_ab = 0 at the inner boundary (Eq. 3.8) to make H_T differentiable, which then kills the D_W algebra. Your stress-test note is right that this is a choice, not a derivation, and that a momentum-type boundary term might preserve both. But the paper never claims uniqueness — it argues the membrane sits at a fixed location for a stationary near-horizon observer, which is a defensible reading of the membrane paradigm. So the concern lands as 'the split is conditional on a modelling choice the paper does not defend against alternatives,' not as 'the claim dissolves.' Second, the one concrete coupling result, Eq. 5.11, comes from ref. [21], listed as unpublished. That is a checkability gap for the 2+1 demonstration specifically; the 3+1 construction does not depend on it. Send it out. It is a legitimate, checkable extension of a serious program, and a referee should press on Eq. 3.8 and on getting [21] into the literature.

Referee report

Significance

If the construction holds, the paper identifies a concrete and checkable mechanism by which a black-hole horizon carries its own Hilbert space, independent of the bulk, with a closed algebra of boundary observables. The QHE analogy is developed with real content rather than as decoration: the (2+1) case is worked out explicitly, and the observation that diffeomorphism invariance forces a surface term is a structural statement that can be tested. The paper has no free parameters and introduces no entities beyond the standard canonical variables and the group-valued field g of Section 5. It is a clean, early contribution to what became the isolated-horizon and horizon-entropy program, and the entropy remark in Section 3, while explicitly acknowledged as requiring regularization, points in a direction that later work made quantitative.

Major objections (4)

[§3, Eqs. (3.8)–(3.10)] The complementarity claim of the abstract (abstract:s3) rests entirely on the Dirichlet condition δq_ab|∂Σ = 0 in Eq. (3.8). The text introduces this as a 'suppose' and then treats it as the physically mandated condition for a near-horizon observer, but no argument is given that it is the only viable choice. The Regge–Teitelboim program admits alternative boundary terms, and a term involving the momentum conjugate to q_ab (a Neumann-type condition) could in principle make H_T differentiable without freezing the induced metric, in which case the D_W algebra would survive for the near-horizon observer and the claimed observer-dependent split would dissolve. The concrete test the paper should pass is to exhibit the most general boundary term that renders H_T differentiable and show that every such term either fixes δq_ab|∂Σ = 0 or fails to be a functional of q_ab and p_ab alone. As written, the complementarity conclusion is a modelling choice presented as a derivation, and the abstract's 'is also shown to emerge' overstates what Section 3 establishes.
[§3, discussion following Eq. (3.4); §1] The Introduction and Section 3 both assert that the edge observables D_W commute with the Hamiltonian, and this commutativity is what licenses the claim of an independent edge Hilbert space with infinite degeneracy at each energy level. The supporting computation is not displayed. From Eq. (2.17) one expects {D_W, H_Γ} = H_{L_W Γ}, but H_Γ is not a constraint: Γ tends to 1 at spatial infinity by Eq. (2.19), so the bracket vanishes only if L_W Γ satisfies the scalar test-function conditions (2.16), including D_a(L_W Γ)|∂Σ = 0. Since Γ vanishes on the inner boundary and W is tangential there, L_W Γ vanishes on ∂Σ, but its normal derivative does not vanish automatically. The paper should either display this bracket including the surface terms or weaken the claim. This is load-bearing for the central existence statement, not merely for the complementarity corollary.
[§5, Eq. (5.11) and Eq. (5.15)] The decomposition (5.11) of the gauge-invariant Chern–Simons action into bulk Chern–Simons, WZNW and anomalous edge coupling is the backbone of the only explicit edge–bulk coupling demonstrated in the paper, and it is quoted from ref. [21], which is listed as unpublished. A short derivation, or a published reference, should be supplied so the reader can check the relative coefficients and signs, particularly the factor of 1/3 on the WZNW term and the sign of the boundary term. Separately, the kinetic term (5.15) is added by hand with no demonstration that the combined system (5.14) plus (5.16) is a consistent constrained system — that the constraints remain first class and that the edge fields do not over-count degrees of freedom relative to the bulk. Since the section's stated goal is to 'explicitly demonstrate' the coupling, the coupling itself is exhibited, but its consistency is not.
[§6, Eq. (6.1)] The authors acknowledge that this section is incomplete, and I take that acknowledgement seriously; the remaining issue is narrower. Eq. (6.1) is the sole support for the conclusion that diffeomorphism invariance forces an anomalous edge coupling in (3+1) dimensions, and it is stated without derivation and without fixed conventions for u_a and the induced metric f_ab. In the QHE case the analogous statement is sharp because the anomaly coefficient is fixed by gauge invariance and appears explicitly in Eq. (4.4); here no analogue of that coefficient, and no candidate edge action, is produced. The section would be strengthened by deriving (6.1) explicitly and stating what, if anything, fixes the normalization of the required surface term.

Minor objections (8)

[§2, Eq. (2.7)] The text states that L' equals L 'apart from surface terms' but does not display them. Since the entire paper turns on the careful treatment of surface terms, the omitted terms should be written out, at least in an appendix, so that the reader can verify that they are functionals of q_ab alone and not of q̇_ab.
[§2, Eqs. (2.10), (2.12), (2.18), (3.5)] The combination (p_ab p^ab − p^2/2) is rendered as 'pabpab − p22' throughout, and p is never defined as q_ab p^ab. Please define p once and use a notation that cannot be misread as a squared index.
[§2, following Eq. (2.12)] The statement that the primary constraints 'can therefore be ignored' should be qualified: they are second class with the gauge-fixing conditions and are eliminated in the standard Dirac procedure, which is what the sentence should say.
[§3, Eq. (3.7) and Eq. (3.10)] The sign and normalization of the compensating surface term −2∫ T K √h should be checked explicitly against the conventions of Wald [13], since the cancellation of the normal-derivative terms in δq_ab is the technical heart of the construction.
[§3, final paragraph] The entropy suggestion is appropriately hedged, but the text should state plainly that no area law is derived here and that the degeneracy is infinite prior to regularization, so that the remark is not read as a quantitative entropy result.
[§4, Eq. (4.2)] The equation of motion following from (4.1) should be checked for the factor of 2 in the Chern–Simons contribution; as written the relative normalization of the two terms in (4.2) does not obviously follow from (4.1).
[§5, Eq. (5.1)] The constant κ is set to one without stating its relation to Newton's constant in 2+1 dimensions; a one-line remark would let the reader connect the normalization to the standard result of ref. [20].
[References] Ref. [7] is given as 'gr-qc/940716', which appears to be a truncated identifier; ref. [21] is listed as unpublished and is relied upon for Eq. (5.11). Both should be corrected or updated.

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The paper

The paper claims

Cutting a hole out of spacetime leaves an infinite family of boundary observables that could carry a black hole's entropy.

The reading

The paper asks what happens to Einstein gravity when a spatial region is removed and the resulting boundary is treated as physical — precisely the situation the membrane paradigm creates when it replaces a black hole interior with a stretched horizon just outside the event horizon. Working in the canonical formulation, the authors show that if you relax the boundary conditions on the smeared vector and scalar constraints and add the surface terms needed to keep them differentiable, you get an infinite set of observables that depend only on boundary data. These edge observables have weakly vanishing Poisson brackets with the constraints and with the Hamiltonian, so they generate a Hilbert space that lives entirely on the boundary, independent of whatever the bulk theory does. The authors propose that this boundary degeneracy is a candidate source of black hole entropy, and they argue that which edge observables count as well defined depends on which stationary observer is asking.

What carries the argument

The load-bearing device is the Regge-Teitelboim requirement that a constraint smeared with a test field be differentiable in the phase-space variables. Boundary conditions on the test field are read off from differentiability, and relaxing them while adding compensating surface terms manufactures new observables rather than new constraints. The vector-constraint version gives the edge diffeomorphism generators $D_W$; the scalar-constraint version gives $H_T$ with its extrinsic-curvature surface term. The paper then imports the quantum Hall template, where a Chern-Simons bulk action on a manifold with boundary is not gauge invariant and must be completed by a chiral edge action with a compensating gauge transformation, and reproduces that structure explicitly for (2+1) gravity by writing the ISO(2,1) Chern-Simons theory in terms of a group-valued field $g$, which pushes the Wess-Zumino-Novikov-Witten term and the anomalous edge coupling onto $partial M$.

Extended reading notes

Core claim

The central claim is that a boundary in general relativity is not a bookkeeping device but a source of new observables. Replacing the smearing vector $V$ in the vector constraint by a field $W$ tangential to the boundary yields $D_W = -2int_Sigma d^3x, q^{ab} L_W p_{ab}$, which remains differentiable and has weakly vanishing brackets with both constraints while closing on itself as $[D_{W_1}, D_{W_2}] = D_{[W_1,W_2]}$ — a diffeomorphism algebra supported at the edge. Similarly, relaxing the fall-off conditions on the scalar-constraint smearing function $T$ and adding the compensating surface term $-2int_{partialSigma} d^2x, sqrt{h}, T K$ produces a differentiable observable $H_T$, of which the ADM Hamiltonian is the special case smeared with the lapse. The authors then observe that making $H_T$ well defined requires the induced metric to be frozen at the inner boundary, $delta q_{ab}|_{partialSigma} = 0$, which is incompatible with the $D_W$ observables that generate boundary diffeomorphisms. Hence the complementarity claim: an observer at spatial infinity keeps the infinite family $D_W$ but has no use for $H_T$, while an observer hovering near the horizon needs $H_T$ as their time-evolution operator and retains only the finitely many $D_W$ built from boundary Killing vectors. The paper closes by arguing that the Einstein-Hilbert action on a manifold with boundary is not diffeomorphism invariant, $delta_v int sqrt{-g}, {}^{(4)}R = -int_{partialSigma times mathbb{R}} sqrt{-f}, v^a u_a, {}^{(4)}R$, so a boundary action with an anomalous coupling must exist, by direct analogy with the quantum Hall edge.

Load-bearing premise

The complementarity claim depends entirely on the assertion that freezing the induced metric at the inner boundary, $delta q_{ab} = 0$, is the physically correct condition for an observer stationed near the horizon; that choice is what makes $H_T$ differentiable and simultaneously destroys the $D_W$ observables, and the paper asserts it on physical grounds rather than deriving it.

Editorial extensions

If this is right

A black hole horizon, modelled as a boundary, carries its own Hilbert space whose states are degenerate at every energy level of the bulk Hamiltonian, giving a candidate microscopic origin for entropy that involves no matter fields.
Any quantization of gravity on a region with boundary must be supplemented by edge degrees of freedom, so a bulk-only account of black hole physics is necessarily incomplete.
Diffeomorphism invariance of the full theory forces an anomalous coupling between edge and bulk, exactly as gauge invariance forces one between a Hall sample and its chiral edge current.
The set of legitimate observables is observer-dependent: an asymptotic observer and a near-horizon observer do not share a common algebra of edge observables.
In (2+1) dimensions the coupling is exhibited explicitly, so lower-dimensional gravity becomes a worked test case for the edge-state proposal.

Reading between the lines

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

If the edge degeneracy is regulated by a short-distance cutoff on the boundary, the resulting entropy would naturally scale with boundary area rather than volume, which is the qualitative behaviour Bekenstein's area law demands; the paper flags the need for such a regularization but does not carry it out.
The observer-dependent split of edge observables is a technical cousin of black hole complementarity, but it is sharper: it is a statement about which test-function spaces admit differentiable surface terms, and could in principle be settled by a direct calculation rather than by thought experiments with infalling observers.
The (2+1)-dimensional construction suggests a concrete check: quantize the ISO(2,1) Chern-Simons theory with the boundary WZNW term, count boundary states, and compare the logarithmic growth against the BTZ entropy.
If no anomalous boundary term exists for (3+1) gravity, the analogy with the quantum Hall effect breaks at exactly the point where it matters, and the edge observables would remain formal objects with no dynamical coupling to the exterior.

Original abstract

We show in the context of Einstein gravity that the removal of a spatial region leads to the appearance of an infinite set of observables and their associated edge states localized at its boundary. Such a boundary occurs in certain approaches to the physics of black holes like the one based on the membrane paradigm. The edge states can contribute to black hole entropy in these models. A ``complementarity principle" is also shown to emerge whereby certain ``edge" observables are accessible only to certain observers. The physical significance of edge observables and their states is discussed using their similarities to the corresponding quantities in the quantum Hall effect. The coupling of the edge states to the bulk gravitational field is demonstrated in the context of (2+1) dimensional gravity.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,823 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.31 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We show in the context of Einstein gravity that the removal of a spatial region leads to the appearance of an infinite set of observables and their associated edge states localized at its boundary. Such a boundary occurs in certain approaches to the physics of black holes like the one based on the membrane paradigm. The edge states can contribute to black hole entropy in these models. A ``complementarity principle" is also shown to emerge whereby certain ``edge" observables are accessible only to certain observers. The physical significance of edge observables and their states is discussed using their similarities to the corresponding quantities in the quantum Hall effect. The coupling of the edge states to the bulk gravitational field is demonstrated in the context of (2+1) dimensional gravity.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We show in the context of Einstein gravity that the removal of a spatial region leads to the appearance of an infinite set of observables and their associated edge states localized at its boundary.	result 0.90	0.93	0.80	asserted 1.00
s1	Such a boundary occurs in certain approaches to the physics of black holes like the one based on the membrane paradigm.	background 0.81	0.10	0.04	asserted 0.78
s2	The edge states can contribute to black hole entropy in these models.	interpretation 0.59	0.50	0.14	hedged possibility 0.87
s3	A ``complementarity principle" is also shown to emerge whereby certain ``edge" observables are accessible only to certain observers.	result 0.94	0.89	0.13	asserted 1.00
s4	The physical significance of edge observables and their states is discussed using their similarities to the corresponding quantities in the quantum Hall effect.	interpretation 0.95	0.57	0.07	asserted 0.93
s5	The coupling of the edge states to the bulk gravitational field is demonstrated in the context of (2+1) dimensional gravity.	result 0.51	0.87	0.07	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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J.D. Bekenstein, Lett. Nuov. Ciment. 4, 737 (1972); Phys. Rev. D 7 , 2333 (1973); ibid. D 9, 3292 (1974). See also L. Bombelli, R. Koul, J. Lee and R. Sork in, Phys. Rev. D 34, 373 (1986); M. Srednicki, Phys. Rev. Lett. 71, 666 (1993); D. Kabat and M.J. Strassler, Phys. Lett. B 329 , 46 (1994); C. G. Callan and F. Wilczek, Phys. Lett. B 333 , 55 (1994); J...

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How the literature uses it

33 arXiv papers cite this in 42 sentences

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Papers citing it · 30 sentences without a confident label

Compares with it 1 paper

We emphasize that this proposal is very different from the edge states associated to the presence of horizon (see, for e.g., Refs. [ref] ) where one associates the edge states to be localized on the horizon to account for the horizon entropy, i.e., horizon is the edge.

A note on asymptotic symmetries and soft-photon theorem · Summary and Outlook · 0.96

Cites it as background 11 papers

Over the years there has been a lot of progress in uncovering various aspects of gravitational subsystems by studying the covariant phase space formalism in the presence of boundaries [ref] .

Symmetries, charges and conservation laws at causal diamonds in general relativity · Introduction · 0.99
The importance of edge states for general relativity was emphasised in [ref] and their relation to entanglement entropy was proposed in [ref] .

A note on entanglement entropy and quantum geometry · Introduction · 0.80
Recent developments have put to the forefront the role of gravitational boundary symmetries [ref] in the context of the black hole information paradox, [ref] .

Generating functional for gravitational null initial data · Introduction · 0.98

Discussion

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A.P. Balachandran, G. Bimonte, K.S. Gupta and A. Stern, I nt. Jour. Mod. Phys. A 7 , 4655 (1992); ibid. A 7 , 5855 (1992); A.P. Balachandran, G. Bimonte and P. Teotonio - Sobrinho, Mod. Phys. Lett. A 8 , 1305 (1993), A. P. Balachandran, L. Chandar and E. Ercolessi, Syracuse University preprint, SU-4240-574 (he p-th/9411164). For a review see A. P. Balacha...

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A. P. Balachandran, L. Chandar and T. R. Govindarajan, ( unpublished). 22

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See for example, J. Harvey and A. Str¨ ominger, “ Quantum Aspects of Black Holes ” in Recent Directions in Particle Theory, editors J. Harvey and J. Polchinski, [ World Scien- tiﬁc, Singapore (1993)], p.537 and references therein. See also S.B. Giddings, University of California, Santa Barbara preprint, UCSBTH-92-36 (hep- th/9209113), L. Thorlacius, Inst....

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G. ’t Hooft, Phys. Scripta, T 36 , 247 (1991) and references therein

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L. Susskind, Rutgers University preprint RU-93-44 (hep -th/9309145), L. Susskind, Phys. Rev. Lett. 71, 2367 (1993), L.Susskind, L. Thorlacius and J. Uglum, Phys. Rev. D 48 , 3743 (1993), L. Susskind and J. Uglum, Phys. Rev. D 50 , 2700 (1994)

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K.S. Thorne, R.H. Price and D.A. Macdonald, Black Holes: The Membrane Paradigm [ Yale University Press (1986)]

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E. Witten, Comm. Math. Physics, 121, 351 (1989); S. Elitzur, G. Moore, A. Schwimmer and N. Seiberg, Nucl. Phys. B 326 , 108 (1990)

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A. Ashtekar, New Perspectives in Canonical Gravity [ Bibliopolis, Naples (1988) ]; Lec- tures on Non-perturbative Canonical Gravity [ World Scientiﬁc, Singapore (1991)]. See also Appendix E of [13]

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T. Regge and C. Teitelboim, Ann. Phys. 88, 286 (1974); See also A. Hanson, T. Regge and C. Teitelboim, Constrained Hamiltonian Systems [ Accademia Nazionale dei Lincei, Roma (1976)]

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R. Wald, General Relativity [ University of Chicago Press (1984)]

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J.D. Bekenstein, Lett. Nuov. Ciment. 4, 737 (1972); Phys. Rev. D 7 , 2333 (1973); ibid. D 9, 3292 (1974). See also L. Bombelli, R. Koul, J. Lee and R. Sork in, Phys. Rev. D 34, 373 (1986); M. Srednicki, Phys. Rev. Lett. 71, 666 (1993); D. Kabat and M.J. Strassler, Phys. Lett. B 329 , 46 (1994); C. G. Callan and F. Wilczek, Phys. Lett. B 333 , 55 (1994); J...

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G. Morandi, Quantum Hall Eﬀect , Monographs and Textbooks in Physical Science, Lecture Notes Number, 10[ Bibliopolis, Naples, (1988)]; R.E. Prange and S.M. Girvin , The Quantum Hall Eﬀect [Springer-Verlag, New York (1990)]; A.P. Balachandran and A.M. Srivastava, Syracuse University preprint SU-4288 -492 and TPI-MINN-91- 38-T (hep-th/9111006); M. Stone, Qu...

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B. Halperin, Phys. Rev. B 25 , 2165 (1984)

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S. Carlip, University of California, Davis preprint, U CD-94-32 and NI-94011 (gr- qc/9409052)

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D. Marolf, University of California, Santa Barbara pre print, UCSBTH-94-45(gr- qc/9411067)

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E. Witten, Nucl. Phys. B 311 , 46 (1990)

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