{"id":"075ad867-1b6a-40a8-8ba3-1a15e0b19b6f","arxiv_id":"2606.27915","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Relative entropy in AdS3/CFT2 equals the linear-order variation of the RT geodesic length divided by 4G_N, obtained from the Bisognano-Wichmann theorem and the holographic dictionary.","lead":"The paper derives a UV-finite version of the Ryu-Takayanagi relation in AdS3/CFT2 by expressing relative entropy between vacuum and coherent states as a Schwarzian functional that matches the variation of bulk geodesic length to linear order. A smart generalist might read it to see how standard QFT theorems and the holographic dictionary can replace divergent entanglement entropy with a finite operator-algebraic quantity.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Order mismatch: relative entropy starts at O(ε²) while equality is claimed to linear order in metric perturbation","rationale":"The reader's weakest assumption targets the 3D rigidity step that promotes boundary data to bulk geometry. The more immediate load-bearing issue is the perturbative-order inconsistency in the central equality itself, which precedes the rigidity argument and is visible already from the abstract's statement of the result. If the derivation actually works at quadratic order, the claim can be repaired by a wording change; the current formulation does not hold.","tokens_in":1676,"tokens_out":396,"duration_ms":58826,"concrete_test":"Expand the Schwarzian functional expression for the relative entropy (obtained after adapting Hollands' result) to first and second order in the coherent-state perturbation parameter around the vacuum; confirm the O(ε) coefficient is zero and check whether the O(ε²) coefficient equals the second variation of the geodesic length computed from the same Bañados data.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The abstract asserts that, to linear order in the metric perturbation, the relative entropy equals the variation of the RT geodesic length divided by 4G_N. By definition, S(ρ||σ) for ρ = σ + ε δρ has vanishing first derivative at ε=0 (since it is nonnegative and minimized at equality), so its expansion begins at O(ε²). The first variation of geodesic length is O(ε). Equating them at linear order is therefore impossible unless both sides are identically zero, which would make the result trivial. The load-bearing step is the adaptation of Hollands' chiral relative entropy result to the diamond (via BW/Borchers) and its identification with the Schwarzian functional: this must produce a quantity whose linear term vanishes and whose quadratic term matches δL/4G_N, but the stated claim does not reflect this structure.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript claims to derive a UV-finite Ryu-Takayanagi relation in AdS₃/CFT₂ by adapting Hollands' exact result for chiral relative entropy to a diamond region via the Bisognano-Wichmann/Borchers theorem. The boundary relative entropy between vacuum and coherent state is expressed as a Schwarzian functional, identified with the asymptotic data of a Bañados geometry through the Fefferman-Graham dictionary, and promoted to the bulk geometry by the rigidity of three-dimensional gravity. To linear order in the metric perturbation, this relative entropy is asserted to equal the variation of the RT geodesic length divided by 4G_N, yielding an operator-algebraic counterpart to the RT formula resting only on the BW/Borchers theorem and the holographic dictionary.","tokens_in":1890,"tokens_out":451,"duration_ms":44137,"significance":"If the central identification is correct, the result would be significant for providing a UV-finite, relative-entropy-based foundation for the RT formula that avoids cutoff dependence and relies solely on standard inputs from the holographic dictionary and the BW/Borchers theorem. This could strengthen the operator-algebraic understanding of holographic entanglement in AdS₃/CFT₂.","major_comments":[{"comment":"Abstract: The assertion that relative entropy equals the variation of the RT geodesic length to linear order in the metric perturbation is inconsistent with the perturbative structure of relative entropy. By definition, S(ρ||σ) for ρ = σ + ε δρ has vanishing first derivative at ε=0 (nonnegative and minimized at equality), so its expansion begins at O(ε²). The first variation of geodesic length is O(ε). Equating the two at linear order is therefore impossible unless both sides vanish identically, which would make the result trivial. The load-bearing step is the adaptation of Hollands' result and its identification with the Schwarzian functional; this must be shown to produce a quantity whose linear term vanishes and whose quadratic term matches δL/4G_N, but the stated claim does not reflect this structure.","section":"Abstract"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their detailed reading and for identifying an important issue with the perturbative orders in our abstract. We agree that the current wording is imprecise and will revise the manuscript to correct it.","responses":[{"response":"We agree that the abstract's reference to 'linear order' is incorrect and misleading. Relative entropy between the vacuum and a coherent state is quadratic in the perturbation parameter by construction (vanishing at first order due to the BW theorem and the fact that the coherent state is a unitary excitation). The Schwarzian functional obtained from Hollands' result expands at O(ε²). The first variation of the RT geodesic length is indeed O(ε). We will revise the abstract to state that the relative entropy equals the quadratic term in the expansion that corresponds to the variation of the RT geodesic length divided by 4G_N. In the main text we will add an explicit expansion of the Schwarzian functional demonstrating that the linear term vanishes identically and that the quadratic coefficient matches δL/4G_N via the Fefferman-Graham dictionary and the rigidity of 3d gravity. This revision addresses the load-bearing identification without altering the core result.","revision_made":"yes","referee_comment":"The assertion that relative entropy equals the variation of the RT geodesic length to linear order in the metric perturbation is inconsistent with the perturbative structure of relative entropy. By definition, S(ρ||σ) for ρ = σ + ε δρ has vanishing first derivative at ε=0 (nonnegative and minimized at equality), so its expansion begins at O(ε²). The first variation of geodesic length is O(ε). Equating the two at linear order is therefore impossible unless both sides vanish identically, which would make the result trivial. The load-bearing step is the adaptation of Hollands' result and its identification with the Schwarzian functional; this must be shown to produce a quantity whose linear term vanishes and whose quadratic term matches δL/4G_N, but the stated claim does not reflect this structure."}],"tokens_in":1414,"tokens_out":434,"duration_ms":53723,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper adapts Hollands' chiral relative entropy result to a diamond region via the Bisognano-Wichmann theorem, writes the boundary relative entropy as a Schwarzian functional using the Fefferman-Graham dictionary on Bañados geometries, and invokes 3D gravity rigidity to connect it to the bulk geodesic. This combination for a UV-finite, operator-algebraic RT relation is new.\n\nThe motivation to replace divergent von Neumann entropy with relative entropy is reasonable, and grounding the steps in the BW/Borchers theorem plus the holographic dictionary keeps the inputs standard.\n\nThe load-bearing problem is the order mismatch the stress-test note identifies. Relative entropy between a state and a small perturbation vanishes to first order and begins at quadratic order. The first variation of geodesic length is linear. The abstract states they are equal to linear order in the metric perturbation, which cannot hold unless the result is trivial. If the paper actually derives that the quadratic coefficient in the relative entropy matches the linear variation of the length, that would be a different and potentially interesting statement, but that is not what is claimed. The rigidity step that promotes the boundary Schwarzian to the bulk geodesic also needs explicit checks that it does not assume the target identification.\n\nThis is for specialists in algebraic holography and AdS3/CFT2 entanglement. A reader who wants to see relative entropy used this way might extract something useful once the orders are straightened out. It deserves a serious referee to examine whether the derivation can be repaired or whether the claim must be restated.","headline":"The central claim as written is impossible: relative entropy is O(ε²) while the RT length variation is O(ε), so they cannot match at linear order.","tokens_in":2395,"tokens_out":391,"would_cite":false,"duration_ms":48387,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Relative entropy between vacuum and coherent states equals the first-order variation of the Ryu-Takayanagi geodesic length divided by 4G_N in AdS3/CFT2.","keywords":["relative entropy","Ryu-Takayanagi relation","AdS3/CFT2","Bañados geometries","Fefferman-Graham dictionary","Schwarzian functional","holographic dictionary","Bisognano-Wichmann theorem"],"falsifier":"An explicit calculation of the relative entropy for a chosen coherent state in the boundary CFT2 that fails to match the first-order change in the corresponding RT geodesic length in the Bañados bulk geometry.","tokens_in":2587,"feed_emoji":"","tokens_out":785,"duration_ms":39185,"temperature":0.7,"pith_summary":"The paper derives a Ryu-Takayanagi relation in AdS3/CFT2 that replaces the ultraviolet-divergent von Neumann entanglement entropy with relative entropy, which stays finite. It adapts an exact result for chiral relative entropy to diamond regions, writes the boundary quantity as a Schwarzian functional, and uses the Fefferman-Graham dictionary to match this to the asymptotic data of a Bañados geometry. Three-dimensional gravity rigidity then lifts the identification into the bulk, so that relative entropy equals the linear change in RT length over 4G_N. The argument uses only the Bisognano-Wichmann/Borchers theorem and the holographic dictionary, supplying an operator-algebraic and ultraviolet-finite version of the relation.","feed_headline":"Relative entropy equals first-order Ryu-Takayanagi length change in AdS3","feed_subtitle":"A finite operator-algebraic version of the holographic entanglement formula follows from the Bisognano-Wichmann theorem and dictionary alone","key_machinery":"The Schwarzian functional for boundary relative entropy, identified with Bañados asymptotic data via the Fefferman-Graham dictionary and promoted to bulk geometry by three-dimensional gravity rigidity.","core_discovery":"Adapting Hollands' exact result for the chiral relative entropy to a diamond region, we express the boundary relative entropy between the vacuum and a coherent state as a Schwarzian functional, which the Fefferman-Graham dictionary identifies with the asymptotic data of a Bañados geometry; the rigidity of three-dimensional gravity promotes this boundary identification to the bulk. To linear order in the metric perturbation, the relative entropy then equals the variation of the RT geodesic length divided by 4G_N. The construction rests only on the Bisognano-Wichmann/Borchers theorem and the holographic dictionary, giving a UV-finite, operator-algebraic counterpart to the RT relation.","pith_inferences":["The same boundary-to-bulk promotion step may not hold in higher dimensions where gravity is less rigid.","Relative entropy could serve as a regularized replacement for other holographic quantities that currently rely on divergent entropies.","The method isolates the role of the Bisognano-Wichmann theorem, suggesting it might be tested in non-holographic models with modular flow."],"forward_implications":["The equality holds to linear order in metric perturbations.","The relation is ultraviolet-finite by construction.","The derivation applies to diamond regions and uses only the Bisognano-Wichmann/Borchers theorem plus the holographic dictionary.","It supplies an operator-algebraic version of the Ryu-Takayanagi formula without reference to divergent entanglement entropy."],"fun_headline_variants":["Relative entropy matches RT length variation in AdS3","UV-finite RT relation from relative entropy in AdS3/CFT2","Relative entropy equals first-order RT geodesic change in AdS3","Operator algebra RT formula via Bisognano-Wichmann in AdS3"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The rigidity of three-dimensional gravity allows boundary data from the Schwarzian functional and Fefferman-Graham dictionary to fix the bulk geometry.","fun_headline_variants_meta":{"raw":{"variants":["Relative entropy matches RT length variation in AdS3","UV-finite RT relation from relative entropy in AdS3/CFT2","Relative entropy equals first-order RT geodesic change in AdS3","Operator algebra RT formula via Bisognano-Wichmann in AdS3"]},"model":"grok-4.3","cost_usd":0.003764,"raw_usage":{"total_tokens":1957,"prompt_tokens":688,"num_sources_used":0,"completion_tokens":73,"cost_in_usd_ticks":37637000,"prompt_tokens_details":{"text_tokens":688,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1196,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":688,"tokens_out":73,"duration_ms":16528,"temperature":1.0,"reasoning_tokens":1196,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T04:02:03.481152+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit calculation of the relative entropy for a chosen coherent state in the boundary CFT2 that fails to match the first-order change in the corresponding RT geodesic length in the Bañados bulk geometry.","supporting_citations":[],"review_version":1}