{"id":"7341de01-87e1-41f7-92f9-5c873e67129e","arxiv_id":"2610.24698","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"The authors formulate pseudoentropy in dS/CFT using complex extremal surfaces and show contour independence within homology classes labeled by an integer winding number.","lead":"This paper develops a holomorphic extremal surface prescription for holographic pseudoentropy in the dS/CFT correspondence. The framework resolves limitations of earlier piecewise constructions, extends computations to arbitrary entangling region sizes and all boundary dimensions, and verifies the first law for conical defects.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"The holomorphic extremal surface is only constructed for spherical regions and their harmonic deformations (plus conical defects in dS3); the central claim that the KI=0 prescription yields a unique pseudoentropy for general entangling regions assumes existence and regularity of such surfaces…","rationale":"The reader's verdict of CONDITIONAL already identifies the existence and regularity of turning points as the weakest assumption. The stress-tester's concern directly echoes that: the paper only constructs holomorphic extremal surfaces for spherical regions and their deformations, and for conical defects in dS3. The central claim requires existence for arbitrary entangling regions, yet no general argument is given. The paper explicitly notes that the piecewise construction fails for strips and small spheres, so the holomorphic prescription must succeed there, but no example is provided. This is a genuine gap, but it is fully acknowledged by the authors and by the reader. The concrete test—attempting a strip—would settle whether the gap is merely a matter of demonstration or a deep obstruction. The paper's internal consistency is high: the derivation for spherical regions is rigorous, the deformation analysis uses a well-motivated ansatz, and the first law check for defects is a non-trivial consistency test. The contour classification and n-dependence are clearly argued from the τ→τ+iπ symmetry. The load-bearing concern does not undermine the results for the cases treated; it limits their claimed generality. The reader's recommendation for conditional acceptance (pending clarification of the homology classification in more general geometries) is appropriate. The stress-test finds no reason to alter that verdict.","tokens_in":34153,"tokens_out":1949,"duration_ms":21987,"concrete_test":"Compute the holomorphic extremal surface for a strip entangling region in dS4/CFT3 (e.g., boundary coordinate t_E with x∈[-l/2,l/2] on the equator). Solve the KI=0 equation with boundary condition t_E = T_0 for |x|<l/2 and t_E = 0 elsewhere, using the same complexified geometry (Eq. 2.5). Check whether a regular turning point exists (cf. Eq. 2.27) and whether the area integral can be evaluated without crossing singularities. If no regular turning point exists, the prescription fails for this basic non-spherical region.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim—that the holomorphic RT prescription (KI=0) defines a unique pseudoentropy for each homology class—is verified only for spherical entangling regions, small harmonic deformations thereof (Secs. 2.4, 3.2), and conical defects in dS3 (Sec. 4). For these cases, explicit analytic embeddings are given. However, the general formulation (Sec. 2.3) asserts that for an arbitrary entangling region A, there exists a complex codimension-two extremal surface Σ_A in the complexified dS bulk, anchored on ∂A, with a regular turning point and a well-defined homology classification of integration cycles. No general existence or regularity argument is provided, and the paper acknowledges this limitation: 'the main weakness is the reliance on the unproven assumption that the complexified extremal surface always exists' (reader's weakest_assumption). The paper explicitly states that the piecewise construction fails for certain subregions like the strip (Sec. 1, [b13]), so the holomorphic prescription must succeed where piecewise fails, but no such example is treated. Without a proof or at least a constructive algorithm for generic shapes, the claim that the prescription is a complete replacement for the piecewise construction remains a conjecture supported only by symmetric examples. This is load-bearing because if the extremal surface does not exist for some generic region, the prescription yields no pseudoentropy at all, undermining the claimed universality.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper formulates a holomorphic extremization framework for holographic pseudoentropy in dS/CFT by complexifying the de Sitter bulk and treating the extremal surface as a complex codimension-two surface satisfying KI=0. For spherical entangling regions and their harmonic deformations in any boundary dimension, explicit analytic embeddings are constructed and the universal pseudoentropy expansion is verified through second order in the deformation parameter. The result is independent of the integration contour within each relative homology class, with inequivalent classes (labeled by an integer n) differing by a sign of the universal term in odd boundary dimensions and by finite real shifts in even dimensions. For conical defects in dS3 the first law of pseudoentropy is shown to hold, and the contour degeneracy persists. The work extends the piecewise Hartle–Hawking construction to arbitrary spherical region size and to the full n-saddle family, providing a unified complex-geometric description.","tokens_in":34602,"tokens_out":3176,"duration_ms":36482,"significance":"If the results hold, the holomorphic prescription offers a systematic way to compute pseudoentropy in dS/CFT without the limitations of the piecewise construction. The closed-form solutions for spherical regions and their deformations confirm universality and provide a testing ground for more complex backgrounds (e.g., Schwarzschild-dS). The use of Cauchy's theorem to organize integration cycles into homology classes elegantly explains the n-dependence and unifies the different saddles. The paper is mathematically careful and contains reproducible analytic derivations.","major_comments":[{"comment":"The holomorphic RT prescription is stated as a general formalism, but its applicability relies on the existence of a suitable complex extremal surface for any entangling region. The authors acknowledge this as an unproven assumption in Sec. 5, but the language in Sec. 2.3 (e.g., 'The area therefore depends on Γ only through its relative homology class') may give the impression that the existence is already guaranteed. I recommend adding a brief sentence at the end of Sec. 2.3 making the existence assumption explicit, so that readers are not misled into believing the prescription has been established for arbitrary regions.","section":"Sec. 2.3","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The turning point expression τ^{(n)}_⋆ = ±log(cot(T0/2)) − iπ(1/2−n) is presented with a choice of sign corresponding to t_E=0 or π. The text states that the choice Re(τ)≥0 is taken, but it would be helpful to explicitly state which t_E value this sign selects, to avoid ambiguity.","section":"Sec. 2.4.1, Eq. (2.28)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The caption of Fig. 2 describes the right panel as 'the same disk-like entangling region of unit radius perturbed harmonically'. Since the undeformed radius is T0 (not necessarily π/2), the phrase 'unit radius' may confuse; consider rewording to 'the same entangling region perturbed harmonically'.","section":"Sec. 2.4.2, Fig. 2","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The universal coefficient Su(B^{d-1}) is written proportional to (-i)^{d-1}. It would improve clarity to note explicitly that this factor is the same for all n, and that the n-dependent sign/flip already appears in Eq. (2.21).","section":"Sec. 3.1, Eq. (3.9)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The quantity c_{dS} is defined as −3iL/(2GN), which is complex. Since the CFT central charge in dS/CFT is complex, it would be helpful to add a brief comment that this matches the non-unitary central charge expected from the dS/CFT dictionary.","section":"Sec. 4, Eq. (4.15)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The uniformization analysis near the turning point is clear and shows that local winding does not produce a finite contribution. The text already notes (p. 34, last paragraph) that genuine contour dependence arises from global monodromies. Adding a one-sentence summary connecting this local result to the global n-dependence of Appendix E would improve readability.","section":"Appendix C","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is well-written and technically sound. Its central weakness—the assumption that a complex extremal surface exists for a general entangling region—is acknowledged by the authors themselves and is not overclaimed. The paper's contribution is in providing a practical holomorphic framework and explicit solutions for a non-trivial class of regions, which goes beyond earlier piecewise constructions. The results for the first law in dS3 are a nice consistency check. I recommend minor revision; the authors should add the clarifying statement suggested in the major comment and address the minor points. The citation pattern and fit to the journal are appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"This is a solid paper. The central idea—replacing the piecewise Hartle–Hawking construction with a single holomorphic extremal surface in the complexified de Sitter bulk—is clean and well motivated. The authors work out explicit analytic solutions for spherical entangling regions in all dimensions, handle harmonic deformations to second order, and check the first law for conical defects in dS3. The contour homology classes labeled by an integer n are a nice touch, and the dependence of the universal term on n is worked out. The math is careful: they derive the extremality condition from area variation, solve the radial equation by separation of variables, and address branch-cut issues in even dimensions via uniformization (Appendix C). The results for the cases treated are reliable.\n\nThe main soft spot is the one the reader and the stress-test note both flag: the existence of a holomorphic extremal surface for arbitrary entangling regions is assumed, not proved. The paper is honest about this—it states in the discussion that the contour independence is a property of the geometric vacuum and that degeneracy may break for black holes or quantum corrections. Still, the claim that the holomorphic prescription is a replacement for the piecewise construction would be stronger with at least a constructive existence argument for generic shapes, or a counterexample where it fails. Without that, the universality claim is supported only by symmetric examples (spheres and their harmonic deformations). This is not a fatal flaw—the paper does what it sets out to do for those examples—but it is a real limitation that potential users should keep in mind.\n\nThe reliance on the authors' own previous work (Refs. [56,57]) for the universal expansion is fine: the new holomorphic derivation is self-contained and extends those results to arbitrary T0 and all n. The citation pattern is appropriate.\n\nVerdict: This paper is a genuine advance for the dS/CFT subfield. It deserves a serious referee and should be published after minor revision. The revision should add a brief discussion of the existence assumption: even just a paragraph sketching why one might expect a complex extremal surface to exist for a given region (e.g., by analytic continuation of the real solution), and what obstructions could arise. That would address the main concern without changing the paper’s core results.","headline":"A careful, well-motivated paper that gives a consistent holomorphic extremal-surface prescription for pseudoentropy in dS/CFT, solving a known limitation of the piecewise construction for the cases it treats.","tokens_in":35148,"tokens_out":1281,"would_cite":true,"duration_ms":15432,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"Holomorphic extremal surfaces compute pseudoentropy for any spherical region in dS/CFT.","keywords":["pseudoentropy","dS/CFT correspondence","holomorphic extremal surfaces","complexification","Ryu–Takayanagi prescription","conical defects","first law of pseudoentropy","contour homology"],"falsifier":"Compute the pseudoentropy for a strip subregion in dS$_4$/CFT$_3$ using the holomorphic prescription; if the complex contour yields a finite, consistent answer while the piecewise construction does not exist for that region, the framework is validated. Alternatively, compute the second-order shape coefficient for a non-harmonic deformation (e.g., a bump with finite support) and verify whether the universal term still matches the Mezei-like formula derived here.","tokens_in":35583,"feed_emoji":"🌌","tokens_out":5604,"duration_ms":57425,"temperature":0.7,"pith_summary":"This paper formulates the extremization problem for codimension-two surfaces in de Sitter space directly in holomorphic coordinates, treating the bulk metric as the analytic continuation of the Lorentzian dS metric. It shows that the resulting equation, $K^I=0$, defines a unique complex extremal surface for a given homology class of integration contours, and that the area (hence the holographic pseudoentropy) is independent of the contour within each class. For spherical entangling regions and their small harmonic deformations in any boundary dimension, the universal terms of the pseudoentropy are computed and shown to match and extend the earlier piecewise Euclidean–Lorentzian construction. The construction also reproduces the first law of pseudoentropy for conical defects in dS$_3$. The work provides a unified framework that overcomes the limitations of the piecewise prescription and opens the way to study more general backgrounds and subregions.","feed_headline":"Complex surfaces unify pseudoentropy in de Sitter space","feed_subtitle":"A single holomorphic extremal-surface equation replaces the piecewise construction and works for any region size and dimension.","key_machinery":"The central machinery is the holomorphic extremality equation $K^I=0$, the vanishing of the complex mean curvature vector of a codimension-two surface in the complexified de Sitter manifold $M_\\mathbb{C}$ (obtained by promoting all global coordinates to complex numbers). The surface is anchored on a real entangling region at future infinity, and the integration contour is chosen from a homology class labelled by an integer $n$ that counts winding around the $\\tau$-cylinder. The invariance under continuous contour deformations (Cauchy's theorem) and the discrete isometry $\\tau\\to\\tau+i\\pi$ organize the pseudoentropy into classes.","core_discovery":"The holomorphic Ryu–Takayanagi prescription (the condition $K^I=0$) yields a unique pseudoentropy for a given homology class of integration contours on the complexified extremal surface. For spherical regions and their harmonic deformations, the result matches and extends the piecewise Hartle–Hawking construction to arbitrary region size and arbitrary boundary dimension. Inequivalent homology classes, labelled by an integer $n$ associated with the shift $\\tau \\to \\tau + i\\pi$, differ by a sign of the universal term in odd boundary dimensions and by real finite shifts in even dimensions. For conical defects in dS$_3$, the first law of pseudoentropy holds and the contour degeneracy within each homology class persists.","pith_inferences":["If the holomorphic prescription is correct, it should apply to non-spherical subregions (e.g., strips) and reproduce finite pseudoentropies where the piecewise construction fails; this prediction can be tested numerically.","The integer $n$ labeling homology classes may correspond to different sheets in the analytic continuation of the boundary replica twist correlator; a direct CFT derivation of the $n$-dependence would strengthen the duality.","For even boundary dimensions, the pseudoentropy contains a logarithmic term whose coefficient is universal; this coefficient should match the A-type anomaly of the nonunitary CFT, providing a cross-check.","In excited states or black-hole backgrounds, the contour degeneracy is expected to be lifted; the present work sets a baseline for future studies of complex saddles in Schwarzschild–dS."],"forward_implications":["The universal expansion of pseudoentropy through second order in shape deformations is confirmed, establishing a Mezei-like formula for dS/CFT that constrains the CFT central charges.","The contour independence within each homology class supports the interpretation of the bulk geometry as emergent, since different geometric realizations (Lorentzian, Euclidean, or genuinely complex) yield the same pseudoentropy for the vacuum.","The integer label $n$ provides a new quantum number for complex saddles; its effect on the pseudoentropy (sign flip in odd dimensions, shift in even dimensions) is a testable signature in holographic computations.","The first law of pseudoentropy is verified for conical defects, indicating that the holomorphic framework can be extended to excited states and may capture the linearized bulk dynamics."],"supporting_citations":[{"why":"Introduces the piecewise Hartle–Hawking construction for holographic pseudoentropy and defines the reduced transition matrix.","marker":"[37,38]"},{"why":"Computes pseudoentropy for deformed spheres in dS/CFT using the piecewise construction, which the present work extends to arbitrary size and dimension.","marker":"[56,57]"},{"why":"Demonstrates the first law of pseudoentropy in dS/CFT for an interval; the present work verifies it for conical defects.","marker":"[59]"},{"why":"Proposes complex codimension-two extremal surfaces for timelike entanglement entropy in AdS/CFT, providing the template for the holomorphic approach.","marker":"[62]"},{"why":"Establishes that extremal surfaces in dS space naturally become complex, motivating the need for a holomorphic formulation.","marker":"[7,8]"},{"why":"Discusses complex extremal surfaces in AdS and Lifshitz black holes, showing that different analytic branches contribute to holographic entropy.","marker":"[68]"}],"fun_headline_variants":["Holomorphic surfaces yield unique pseudoentropy in dS/CFT","One equation for pseudoentropy across all de Sitter dimensions","Pseudoentropy first law holds for conical defects in dS3","Complex extremal surfaces label pseudoentropy by homology class","Harmonic deformations confirm pseudoentropy's second-order expansion"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The complexified de Sitter metric (Eq. 2.5) is taken as the holomorphic extension of the Lorentzian metric, and the existence of a smooth turning point (Eq. 2.27) for the extremal surface in the complex domain is assumed; in even boundary dimensions, the branch points of the area integrand are resolved by a local uniformization that may not hold for more general backgrounds or higher-order deformations.","fun_headline_variants_meta":{"raw":{"variants":["Holomorphic surfaces yield unique pseudoentropy in dS/CFT","One equation for pseudoentropy across all de Sitter dimensions","Pseudoentropy first law holds for conical defects in dS3","Complex extremal surfaces label pseudoentropy by homology class","Harmonic deformations confirm pseudoentropy's second-order expansion"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000765,"raw_usage":{"cost":0.00076485948,"is_byok":false,"cost_details":{"upstream_inference_cost":0.00076485948,"upstream_inference_prompt_cost":7.777616e-05,"upstream_inference_completions_cost":0.00068708332},"total_tokens":4637,"prompt_tokens":856,"completion_tokens":3781,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":0,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":3691}},"tokens_in":856,"tokens_out":3781,"cost_source":"openrouter_usage","duration_ms":38864,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":3691,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-20T18:50:36.754937+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Compute the pseudoentropy for a strip subregion in dS$_4$/CFT$_3$ using the holomorphic prescription; if the complex contour yields a finite, consistent answer while the piecewise construction does not exist for that region, the framework is validated. Alternatively, compute the second-order shape coefficient for a non-harmonic deformation (e.g., a bump with finite support) and verify whether the universal term still matches the Mezei-like formula derived here.","supporting_citations":[{"cited_title":"Fujiki, M","cited_arxiv_id":null,"evidence_quote":"Demonstrates the first law of pseudoentropy in dS/CFT for an interval; the present work verifies it for conical defects."},{"cited_title":"Geometric Interpretation of Timelike Entanglement Entropy","cited_arxiv_id":"2408.15752","evidence_quote":"Proposes complex codimension-two extremal surfaces for timelike entanglement entropy in AdS/CFT, providing the template for the holomorphic approach."},{"cited_title":"Complex Entangling Surfaces for AdS and Lifshitz Black Holes?","cited_arxiv_id":"1407.2900","evidence_quote":"Discusses complex extremal surfaces in AdS and Lifshitz black holes, showing that different analytic branches contribute to holographic entropy."}],"review_version":1}