{"id":"9345dedd-72bc-4b41-b493-9c461f942ed2","arxiv_id":"2512.02164","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The quadratic shape-deformation correction to pseudoentropy in dS/CFT is controlled by the analytically continued stress-tensor coefficient C_T, making the sphere a local extremum across Einstein and quadratic-curvature gravity.","lead":"This paper derives a universal formula for how holographic pseudoentropy in de Sitter/CFT changes when a spherical entangling surface is slightly deformed, with the correction set by the stress-tensor two-point coefficient C_T. It extends the known AdS/CFT Mezei formula to non-unitary dS duals and checks the same structure in quadratic-curvature gravity.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Higher-curvature section asserts K_i=0 extremizes the Dong–Camps functional; the λ3 K^2 term's second variation likely shifts the first-order surface, leaving Eq. (21) unproven.","rationale":"The reader's weakest assumption correctly identifies the soft spot: the higher-curvature section imposes K_i=0 as the extremal condition for the full Dong–Camps functional, but the variation of the λ3 K^2 term is not captured by that condition. This is the single most load-bearing concern because Eq. (21) is the only evidence for the claimed universality across non-Einstein theories; if the extremal surface shifts, the central coefficient changes. The Einstein-gravity calculation, by contrast, is a direct area computation and less exposed. I agree with the reader that the paper should not be accepted without deriving the true extremal condition or showing that the λ3 term's second variation does not affect the linearized surface equations. The obstacle is computational, not conceptual, so a conditional verdict (or unchanged conditional) is appropriate rather than rejection.","tokens_in":10817,"tokens_out":5779,"duration_ms":61980,"concrete_test":"Compute the second variation of S_QG in Eq. (19) around the undeformed sphere for a shape perturbation t_E = π/2 + ε a Y_ℓm f(τ). Derive the linearized Euler–Lagrange equation for f including the δ²(−2λ3∫ K^i_ab K_i^ab) term. Solve to first order in λ3 with the same boundary/junction conditions, evaluate S_QG to O(ε²), and compare the coefficient of Σ a²_ℓm (ℓ−1)_d with Eq. (21). If any additional λ3-dependent term appears, Eq. (21) is incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central universality claim for quadratic-curvature gravity rests on Eq. (21), obtained by substituting the area-extremal surface (K_i=0) into the Dong–Camps functional (19). However, the full functional contains −2λ3 ∫_Σ K^i_ab K_i^ab. Imposing K_i=0 removes the λ2 K_i K^i term but does not extremize the λ3 term. Around the undeformed sphere K_i=0, the first variation of the λ3 term vanishes, yet its second variation does not. Therefore the linearized shape equation for Tℓ(τ) (Eqs. (10)–(11)) acquires λ3-dependent terms that the paper does not solve. If the corrected surface differs from the area-extremal one at order ε, the O(ε²) entropy receives extra contributions, altering the coefficient in Eq. (21). The paper asserts, without showing, that K_i=0 extremizes the full functional; this is a gap in the derivation of the higher-curvature universality claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11071,"tokens_out":8994,"duration_ms":88736,"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":[{"comment":"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.","section":"Holographic pseudoentropy for perturbed spheres, Eqs. (12)–(17)"},{"comment":"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.","section":"Higher-curvature gravity, Eqs. (19)–(21)"},{"comment":"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.","section":"Eq. (18) and Discussion, C_T identification"}],"minor_comments":[{"comment":"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.","section":"Eq. (5) and Eq. (4)"},{"comment":"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).","section":"Abstract and Discussion"},{"comment":"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.","section":"Figure 2"},{"comment":"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.","section":"Page 4, around Eqs. (14)–(17)"}],"recommendation":"major_revision","confidential_remarks":"The Einstein-gravity part may well be sound, but the manuscript is too terse in the central computation and the higher-curvature section contains a genuine variational gap. The authors should be given the opportunity to fix the higher-curvature derivation and to provide more of the Einstein-gravity calculation. If the quadratic-gravity issue cannot be resolved, the higher-curvature universality claim should be removed or substantially weakened."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the Einstein part of this paper is a genuine new result and probably right; the quadratic-curvature part is not. The authors extend Mezei's formula to dS/CFT for deformed spheres, and the calculation hangs together: they solve the linearized extremal-surface equations, combine the timelike and spacelike segments, and find that the parity-dependent terms cancel, leaving a compact universal coefficient. Identifying that coefficient with C_T via analytic continuation from AdS is a conjecture—they say so themselves—but it is a useful conjecture, and the paper flags the C_T=0 caveat for broader non-unitary CFTs.\n\nThe soft spot is the higher-curvature section. They evaluate the Dong-Camps functional on the area-extremal surface (K_i=0) and call it on-shell. It isn't. The λ3 K^2 term is quadratic in the extrinsic curvature, so its first variation at the undeformed sphere vanishes, but its second variation does not. That means the linearized shape equation for T_ℓ(τ) gets λ3-dependent terms, which the paper never solves. So Eq. (21) is unproven, and the claimed universality for quadratic curvature gravity is not established. The stress-test note is correct on this point.\n\nMinor soft spots: the key intermediate results (12)-(17) are quoted without derivation, and there is no independent CFT check. For a Letter that is acceptable if a supplement is provided. The discussion is honest about the limitations of the C_T identification, which I credit.\n\nThis paper is for the dS/CFT and holographic-entropy community. The Einstein result is a solid starting point and deserves referee time. The quadratic-curvature claim should not survive as is; it needs either a proper treatment of the full extremal condition or a clear restriction of the claim to whatever surface actually solves the full functional. I would send it to peer review with the expectation of a major revision.","headline":"Einstein gravity result for dS pseudoentropy is solid; the higher-curvature generalization is not on-shell and needs rework.","tokens_in":11559,"tokens_out":7484,"would_cite":true,"duration_ms":78408,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81T40","83E30"],"pacs":["04.60.-m","11.25.Tq"],"model":"deepseek-v4-flash","headline":"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.","keywords":["pseudoentropy","dS/CFT correspondence","shape deformation","sphere extremum","stress-tensor two-point coefficient","higher-curvature gravity","analytic continuation","holographic entanglement entropy"],"falsifier":"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.","tokens_in":10703,"feed_emoji":"🌀","tokens_out":7397,"duration_ms":65571,"temperature":0.7,"pith_summary":"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.","feed_headline":"Sphere is a local extremum for dS pseudoentropy","feed_subtitle":"The second-order shape correction is fixed by the stress-tensor two-point coefficient, mirroring the AdS entanglement-entropy formula.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Universal shape correction for dS pseudoentropy","Pseudoentropy in dS/CFT: sphere sets the extremum","dS pseudoentropy mirrors AdS entanglement formula","Mezei formula extends to de Sitter holography","Non-unitary CFTs share universal pseudoentropy law"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Universal shape correction for dS pseudoentropy","Pseudoentropy in dS/CFT: sphere sets the extremum","dS pseudoentropy mirrors AdS entanglement formula","Mezei formula extends to de Sitter holography","Non-unitary CFTs share universal pseudoentropy law"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000251,"raw_usage":{"total_tokens":1402,"prompt_tokens":762,"completion_tokens":640,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":506,"completion_tokens_details":{"reasoning_tokens":565}},"tokens_in":506,"tokens_out":640,"duration_ms":6046,"temperature":1.0,"reasoning_tokens":565,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T19:01:58.796630+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}