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REVIEW 3 major objections 4 minor 3 cited by

For slightly deformed spherical entangling surfaces in dS/CFT, the universal part of pseudoentropy is governed by the analytically continued stress-tensor two-point coefficient, with the sphere as a local extremum.

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2026-08-03 19:01 UTC pith:XA3B2A3J

load-bearing objection Einstein gravity result for dS pseudoentropy is solid; the higher-curvature generalization is not on-shell and needs rework. the 3 major comments →

arxiv 2512.02164 v2 pith:XA3B2A3J submitted 2025-12-01 hep-th

Universality of pseudoentropy for deformed spheres in dS/CFT

Giorgos Anastasiou , Ignacio J. Araya , Avijit Das , Javier Moreno This is my paper
classification hep-th MSC 81T4083E30 PACS 04.60.-m11.25.Tq
keywords pseudoentropydS/CFT correspondenceshape deformationsphere extremumstress-tensor two-point coefficienthigher-curvature gravityanalytic continuationholographic entanglement entropy
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 asks what happens to holographic pseudoentropy in the de Sitter/conformal field theory correspondence when the entangling region is a small deformation of a sphere. It claims that the universal part of the pseudoentropy changes at second order in the deformation through a term proportional to the coefficient C_T of the two-point stress-energy tensor correlator of the non-unitary dual CFT, obtained by analytic continuation from AdS/CFT. This makes the sphere a local extremum of pseudoentropy. The same structure is shown to hold for quadratic-curvature gravity, which the authors take as evidence for universality across non-unitary holographic CFTs. A sympathetic reader would care because it extends a known AdS entanglement-entropy formula to a cosmological, non-unitary setting and identifies C_T as the governing data for shape dependence in dS.

Core claim

On the paper's own terms, the discovery is that for an entangling surface given by a unit sphere perturbed as T0 = pi/2 + epsilon sum a_lm Y_lm, the universal piece of dS holographic pseudoentropy takes the form S_u = S^(1)_u + epsilon^2 S^(2)_u. The correction S^(2)_u is C_T times a mode sum with weights (l-1)_d, a rising factorial, and a factor pi/2 (odd d) or 1 (even d). This follows from solving the extremal-surface equations on the timelike and spacelike segments of the bulk and finding that parity-dependent terms cancel between them. The coefficient C_T is identified with the analytic continuation of the AdS/CFT stress-tensor two-point coefficient, via L_* restricted to AdS mapping to

What carries the argument

The central object is the coefficient C_T of the two-point stress-energy tensor correlator, encoded in the OPE as <T T> ~ C_T / x^{2d}, which in dS/CFT is obtained from its AdS/CFT value by the analytic continuation L_*|AdS -> -i L_*|dS. The argument is carried by the holographic extremal-area prescription: the pseudoentropy is the area (in units of 4G) of a codimension-two extremal surface anchored on the deformed sphere, split into a timelike piece in the Lorentzian dS section and a spacelike piece in the Euclidean section. At linear order in the deformation the embedding equations separate and are solved in terms of associated Legendre and hypergeometric functions; the quadratic correctio

Load-bearing premise

The higher-curvature result assumes that the same extremal surface that solves the Einstein gravity problem also extremizes the quadratic-curvature entropy functional, and this is asserted rather than demonstrated; if the true surface differs, the claimed universality for those theories must be revised.

What would settle it

Solve the full variational problem for the quadratic-curvature entropy functional without imposing the vanishing trace of the extrinsic curvature and compare the resulting order-epsilon^2 universal coefficient with the paper's expression; any discrepancy falsifies the higher-curvature universality claim. Alternatively, in an explicit dS/CFT dual (e.g., a known d=3 non-unitary CFT), compute C_T directly from the two-point stress-tensor correlator and check it against the holographic formula.

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

If this is right

  • The sphere is a local extremum of universal pseudoentropy in dS/CFT, since the epsilon^2 correction keeps the sign of the unperturbed value.
  • The shape dependence of dS pseudoentropy is fixed by the analytically continued stress-tensor two-point coefficient, extending the AdS/CFT shape-deformation formula for entanglement entropy to non-unitary holographic CFTs.
  • In even dimensions the universal finite part is free of UV-regulator contamination, making it a clean observable tied to timelike entanglement entropy.
  • For quadratic-curvature gravity the same structure survives with C_T rescaled by coupling-dependent coefficients, suggesting universality across higher-curvature theories.
  • The analytic continuation L_*|AdS -> -i L_*|dS maps the AdS C_T to the dS C_T, reinforcing the view that some dS/CFT data is obtainable by continuation from AdS/CFT.

Where Pith is reading between the lines

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

  • If C_T is the controlling coefficient, it gives a practical way to extract C_T for non-unitary CFTs from dS holography; testing it in explicit dS/CFT models (such as known d=3 constructions) would be a direct check.
  • The paper only establishes a local extremum; following the unitary-CFT story, one might conjecture the sphere is a global extremum in d=3 for the holographic universality class, but that requires additional argument.
  • Because C_T can vanish or become complex in non-unitary theories, the shape-deformation formula cannot hold for all non-unitary CFTs; the paper's own discussion already notes this and restricts to the holographic universality class.
  • The higher-curvature extrapolation relies on an unproven extremal-surface condition; a full variational treatment of the quadratic-curvature functional could either confirm or correct the reported C_T rescaling.

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 / 4 minor

Summary. The paper studies holographic pseudoentropy in dS/CFT for a slightly deformed spherical entangling surface B^{d-1}_ε. In Einstein-dS gravity it claims the universal part admits the expansion S_u = S^(1) + ε² S^(2), with S^(2) given by Eq. (7), controlled by the analytically continued stress-tensor two-point coefficient C_T of Eq. (18); parity-dependent terms from timelike and spacelike parts of the RT surface are said to cancel. The same structure is claimed for quadratic curvature gravity, with C_T modified as in Eq. (21). The authors interpret the result as an extension of Mezei's formula to the non-unitary dS/CFT setting and as evidence that the sphere is a local extremum of pseudoentropy.

Significance. If correct, this is a natural and useful extension of the AdS/CFT shape-dependence formula to the non-unitary dS/CFT setting. The explicit separation into timelike/spacelike segments and the claimed parity cancellations are nontrivial, and the quadratic-curvature generalization is a reasonable test of universality. The manuscript is also honest about limitations, notably the C_T=0 caveat for certain non-unitary CFTs. However, the central Einstein-gravity computation is partly presented by assertion: the key intermediate expressions are not derived, and the higher-curvature extension rests on an unjustified identification of the extremal surface. These issues prevent acceptance in the paper's current form.

major comments (3)
  1. [Holographic pseudoentropy for perturbed spheres, Eqs. (12)–(17)] Eqs. (12)–(13) are the solutions of the linearized shape equations (10)–(11), and Eqs. (14)–(17) are the O(ε²) area contributions. These are the computational core of the main result, yet no derivation is given; in particular the claimed parity cancellations that reduce the sums to Eq. (7) are only asserted. The absence of even a representative intermediate step (for d=3 or d=4) makes the central claim hard to verify. A revised version should include the derivation, at least in a supplementary file or appendix, and ideally an independent numerical or CFT-side cross-check of the final coefficient.
  2. [Higher-curvature gravity, Eqs. (19)–(21)] Eq. (20) is obtained by evaluating Eq. (19) on a surface with K_i=0. But K_i=h^{ab}K^i_{ab} is only the trace; the full λ3 term is quadratic in the extrinsic curvature tensor. The stationarity condition of the full Dong–Camps functional contains δ[K^i_{ab}K_i^{ab}] = 2K_i^{ab}δK_{iab}+..., which does not vanish when only the trace K_i vanishes, since the traceless part of K_i^{ab} is generically nonzero on the deformed area-extremal surface. Thus K_i=0 extremizes the area functional but not the entropy functional (19); substituting it into (19) is off-shell. The linearized shape equations for T_ℓ(τ) and T_E,ℓ(τ_E) acquire λ3-dependent terms, so the coefficient in Eq. (21) is not established. The authors should solve the corrected variational equations at O(ε) and recompute S^(2) for quadratic gravity.
  3. [Eq. (18) and Discussion, C_T identification] The identification of the coefficient in Eq. (7) with the two-point stress-tensor coefficient C_T of the dual non-unitary CFT is not tested by an independent calculation. Eq. (18) is the standard AdS/CFT C_T with the analytic continuation L→−iL inserted; no dS/CFT computation of ⟨T T⟩ is shown. Since one of the main claims is that C_T controls shape dependence of pseudoentropy, an independent check, or at least a direct citation of an explicit dS/CFT two-point function computation, is needed. I also note the Discussion already concedes that non-unitary CFTs with C_T=0 escape the formula, which qualifies the word 'universality' and should be reflected in the abstract.
minor comments (4)
  1. [Eq. (5) and Eq. (4)] The notation T0 is used both as the Euclidean time coordinate in the deformation profile (5) and as a characteristic length of the subregion in Eq. (4). This is confusing; use a separate symbol such as t_E^* for the profile.
  2. [Abstract and Discussion] The statements that the correction 'retains the sign' of the unperturbed result and that the sphere is a local extremum assume C_T is real and positive. Since C_T is allowed to be complex or negative in non-unitary CFTs, the sign statement should be qualified (e.g., by specifying a phase convention or a restricted class of theories).
  3. [Figure 2] Figure 2 is schematic but no deformation parameters or ℓ values are given. It should either be labelled as illustrative or accompanied by the actual values used.
  4. [Page 4, around Eqs. (14)–(17)] The text says 'in both even and odd dimensions, we find: i) real, finite contributions...'. Since the pseudoentropy is generally complex and the prefactors contain (-i)^{d-1}, the word 'real' is misleading; this presumably refers to real coefficients after extracting the overall phases. Please rephrase.

Circularity Check

1 steps flagged

The O(ε²) coefficient is computed from the extremal area, then Eq. (18) renames that coefficient C_T; the central 'C_T-controlled' claim is therefore partly a relabeling, though the shape dependence and extremum sign are genuinely derived.

specific steps
  1. self definitional [Eq. (7) and Eq. (18), section 'Holographic pseudoentropy for perturbed spheres']
    "In the latter expressions, we identify the coefficient of the two-point stress-energy tensor for CFTs dual to Einstein-dS gravity as CT = (−i)^{d−1} Γ(d+ 2) / [8π^{(d+2)/2}(d−1)Γ(d/2)] L^{d−1}_⋆/G , (18)"

    The prefactor multiplying the ℓ,m sum in the computed O(ε²) universal pseudoentropy — Eqs. (14)–(17) — is exactly the L^{d−1}_⋆/G combination that Eq. (18) renames C_T. No independent evaluation of the dual non-unitary CFT's ⟨TT⟩ two-point coefficient is performed; the paper itself says 'we conjecture this coefficient to be the C_T'. Thus the claim that S_u^(2) is controlled by C_T is true by definition of C_T: the coefficient was computed from the area functional and then labelled C_T. The nontrivial shape-dependent sum, the cancellations between timelike and spacelike sectors, and the sign of the extremum are independent content, but the advertised C_T-universality statement reduces to a relabeling unless C_T is obtained from the CFT side.

full rationale

The Einstein-dS computation itself is self-contained: the extremal profile equations (10)–(11) are solved with the stated boundary and junction conditions, and the resulting area integrals (14)–(17) are genuine first-principles calculations. Summing them does reproduce the Pochhammer-weighted ℓ,m sum in Eq. (7) without any input from a dual CFT two-point function. That part is not circular. The circularity is confined to the interpretive step: Eq. (18) defines C_T to be precisely the coefficient that already appears in the computed O(ε²) entropy, and Eq. (7) is then presented as a C_T-controlled universality statement. Since the non-unitary CFT's C_T is not independently computed or measured, this is a definition/conjecture rather than a derivation. The same relabeling recurs in Eq. (21) for quadratic-curvature gravity. Separately, the higher-curvature section imposes K_i=0 as the extremal-surface condition on the Dong–Camps functional (19) without extremizing the λ3 K^i_ab K_i^ab term; I treat that as a derivation gap or correctness concern, not as circularity. Overall, because the central shape-dependence and extremum results are derived, but the C_T identification is a constructed label, the circularity score is moderate: 4.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The main Einstein result introduces no fitted constants: L_* and G are standards inputs and the deformation amplitudes a_ℓm parametrize the boundary profile. The quadratic-gravity part brings in undetermined couplings λ1-λ3. The main conjectural input is the identification of Eq. (18) with the CFT stress-tensor coefficient C_T, supported by analytic continuation rather than an independent non-unitary CFT computation.

free parameters (1)
  • λ1, λ2, λ3 = undetermined
    Quadratic gravity couplings in L_QG = R − 2Λ + λ1R² + λ2R_μνR^μν + λ3R_μνρσR^μνρσ; the higher-curvature result C_T|QG depends on them through a_d and the λ3 term.
axioms (5)
  • domain assumption dS/CFT correspondence and the RT prescription for pseudoentropy, Eq. (8)
    The whole computation interprets the area of extremal codimension-two surfaces in dS as pseudoentropy; this is the central holographic premise.
  • domain assumption Mixed Lorentzian/Euclidean Hartle-Hawking geometry with junction conditions at τ=τ_E=0
    The extremal surface is glued across the τ=0 hypersurface using T_ℓ(0)=T_{E,ℓ}(0) and ∂_τ T_ℓ(0)=i ∂_{τ_E}T_{E,ℓ}(0); the factor i is essential for the parity cancellations in Eqs. (14)-(17).
  • domain assumption Analytic continuation L_*|AdS → −i L_*|dS transfers CFT data
    Used to identify the coefficient in Eqs. (18) and (21) with C_T; the authors acknowledge that scalar primary two-point coefficients do not continue this way [50], so the assumption is not automatic.
  • domain assumption Dong-Camps entropy functional applies to dS via the replica trick, Eq. (19)
    Extends the entropy functional to quadratic gravity in dS; relies on the cited result [44].
  • ad hoc to paper Extremal surface for S_QG is given by K_i=0
    Asserted before Eq. (20); the full functional contains λ3∫K^i_ab K_i^ab, so this should be derived from the full variational problem rather than assumed.

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read the original abstract

We determine the universal part of pseudoentropy for small shape deformations of spherical entangling surfaces in the context of de Sitter/conformal field theory (dS/CFT) correspondence. The leading correction at quadratic order in the deformation parameter is controlled by the analytic continuation of the coefficient of the two-point stress-energy tensor correlator in AdS/CFT (i.e., $\left. L_{*} \right|_{\text{AdS}}\rightarrow -i \left. L_{*} \right|_{\text{dS}}$), thereby establishing the sphere as a local extremum. The same structure holds in higher-curvature theories, as we check explicitly for quadratic curvature gravity, suggesting a universal behavior across non-unitary holographic CFTs. Our findings extend the Mezei formula to the dS/CFT setting and indicate that the shape dependence of pseudoentropy in dS holography resembles that of entanglement entropy in AdS space. Thus, we conjecture this coefficient to be the $C_T$ for the non-unitary CFT dual.

Figures

Figures reproduced from arXiv: 2512.02164 by Avijit Das, Giorgos Anastasiou, Ignacio J. Araya, Javier Moreno.

Figure 1
Figure 1. Figure 1: In green, the entangling region B d−1 ϵ is the per￾turbation around the unit-ball B d−1 , stereographically projected on S d−1 . global dSd+1 coordinates, the metric ds 2 = gµνdx µdx ν reads ds 2 = L 2 ⋆ [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (Left) Front view of the RT surface for small perturbations around the ball-shaped entangling region at [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

discussion (0)

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Forward citations

Cited by 3 Pith papers

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  2. Universal relation between $C_{T}$ and the CFT Weyl anomaly

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    In any even-dimensional CFT, C_T = [d/(d−1)]·[(d+1)!/(d/2−1)!]·c, where c is the coefficient of the quadratic-in-Weyl term in the trace anomaly.

  3. Renormalized pseudoentropy in dS/CFT

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