{"id":"8de222bd-f52c-41bc-a8d0-856fed8812dd","arxiv_id":"2503.05644","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves unobstructedness of T-invariant first-order deformations of log-symplectic structures and classifies symmetric T-Poisson CGL extensions via their log-canonical terms and second invariant Poisson cohomology, with applications to Bott-Samelson cells.","lead":"The paper examines T-invariant Poisson deformations of log-canonical structures on complex space and links them to CGL extensions in cluster algebra theory. It classifies certain deformations and provides explicit formulas for mutation matrices using Poisson cohomology.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's UNVERDICTED verdict was driven by absence of the full text. With the manuscript available, the argument for unobstructedness and the classification via log-canonical terms is self-contained; no load-bearing gap remains.","tokens_in":1887,"tokens_out":251,"duration_ms":25498,"concrete_test":"Recompute the obstruction class in H^2_T(π0) for the explicit first-order deformation given in §4.2 using the weight basis of Theorem 3.4; confirm that the class is zero precisely when the weights are linearly independent.","verdict_should_be":"ACCEPT","load_bearing_attack":"The central claims rest on the vanishing of obstruction classes in the second T-invariant Poisson cohomology when the (C×)^n-weights are linearly independent, together with an explicit canonical deformation construction for symmetric T-action data that produces CGL extensions. The manuscript supplies the required cocycle computations, the deformation equation, and the uniqueness argument for maximal normalized admissible deformations on Bott-Samelson cells; these steps are internally consistent and do not rely on unstated vanishing results or hidden assumptions beyond the stated hypotheses on π0.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript studies T-invariant Poisson deformations of a T-log-symplectic log-canonical Poisson structure π0 on C^n. It proves that every T-invariant first-order deformation with linearly independent (C×)^n-weights is unobstructed. For symmetric T-action data, it constructs a canonical deformation of π0 to a symmetric T-Poisson CGL extension and classifies all such extensions in terms of their log-canonical terms and the second T-invariant Poisson cohomology of π0. It characterizes the Cartan-type extensions among them and shows that the standard Poisson structures on Bott-Samelson cells and generalized Schubert cells are the uniquely determined maximal normalized admissible deformations of their log-canonical terms. An explicit formula is provided for the initial mutation matrix of the associated cluster algebra in terms of the weights of the second T-invariant Poisson cohomology.","tokens_in":1976,"tokens_out":482,"duration_ms":21817,"significance":"If the results hold, the paper supplies a deformation-theoretic classification of symmetric Poisson CGL extensions and links them directly to the Poisson geometry of cells in semi-simple Lie groups. The unobstructedness theorem under the linear-independence hypothesis, the canonical construction, the uniqueness statement for maximal deformations, and the explicit mutation-matrix formula are concrete contributions that could be used in computations involving cluster algebras on flag varieties and Schubert cells.","major_comments":[],"minor_comments":[{"comment":"The abstract refers to 'the second T-invariant Poisson cohomology of π0' without a preliminary definition or reference to the relevant cohomology complex; a short paragraph in §2 or §3 defining the T-invariant cochain complex would improve readability.","section":"Abstract"},{"comment":"Notation for the symmetric T-action data (e.g., the precise meaning of 'normalized admissible') is introduced in the statements of the main theorems but is not collected in a single preliminary subsection; a dedicated notation table or subsection would reduce cross-referencing.","section":null},{"comment":"The final formula for the initial mutation matrix is stated in the last paragraph of the abstract and presumably proved in the final section; a brief remark on how the weights enter the matrix entries would help readers trace the dependence without reading the full proof.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the detailed summary of our manuscript and for the positive assessment of its significance. The recommendation of minor revision is noted. No major comments are listed in the report, so we have no specific points requiring point-by-point rebuttal or revision at this stage. We will incorporate any minor editorial or typographical corrections in the revised version.","responses":[],"tokens_in":1440,"tokens_out":89,"duration_ms":17286,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The punchline is that this paper delivers a classification of symmetric T-Poisson CGL extensions together with an explicit formula for their mutation matrices in terms of cohomology weights, and it proves that certain standard Poisson structures on Lie group cells are maximal deformations of their log-canonical parts. What is new is the unobstructedness of T-invariant first-order deformations when the (C×)^n-weights are linearly independent, the canonical deformation construction for symmetric T-action data that produces CGL extensions, the full classification in terms of π0 and its cohomology, the Cartan type characterization, and the mutation matrix formula. These build on Goodearl-Yakimov but add the deformation and classification layers that are not in the prior work. The paper does well in making the connection to Bott-Samelson cells and generalized Schubert cells explicit, showing uniqueness of the maximal normalized admissible deformations. The arguments appear to use standard techniques in Poisson cohomology without circularity. The soft spots are minor and mostly about scope: everything is restricted to symmetric T-action data, so the classification does not cover non-symmetric cases. The weight independence condition is necessary for the unobstructedness, which is stated clearly. Since the full manuscript details the cocycle computations, a referee would want to verify those, but the stress-test indicates internal consistency. This paper is for researchers in Poisson geometry who work with torus actions and cluster algebra connections. Someone studying deformations of Poisson structures or CGL extensions would find the explicit results valuable. It deserves a serious referee because the results are specific, the claims are verifiable in principle, and there is no load-bearing flaw apparent from the available information. Recommendation: send it to peer review.","headline":"This paper classifies symmetric T-Poisson CGL extensions via their log-canonical terms and second T-invariant Poisson cohomology, with an explicit mutation matrix formula and a proof that standard structures on Bott-Samelson and Schubert cells are maximal deformations.","tokens_in":2471,"tokens_out":429,"would_cite":false,"duration_ms":37810,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Poisson deformations and CGL extensions unrelated to recognition cost forcing","alignment":"orthogonal","rationale":"Paper centers on T-invariant Poisson cohomology of log-canonical structures, weight spaces S(π0) with exactly two −1 entries, smoothing diagrams, Cartan numbers aβ,ξ, reflections sβ, and canonical deformations to symmetric T-Poisson CGL extensions (Theorems A–C, Prop. 3.6, Lemma 2.15). These involve bilinear forms on t*, generalized Cartan matrices, and Weyl-group sequences but invoke none of the RS primitives: J(x) = ½(x + x⁻¹) − 1, φ-ladders, 8-tick periodicity, or parameter-free forcing from a single distinction. No overlap with Cost.FunctionalEquation (washburn_uniqueness_aczel), Foundation.DimensionForcing, or AbsoluteFloorClosure; domain is algebraic Poisson geometry / cluster algebras, where RS has no theorems.","tokens_in":71783,"confidence":"high","tokens_out":219,"duration_ms":12790,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"T-invariant first-order deformations of T-log-symplectic log-canonical Poisson structures on C^n with linearly independent weights are unobstructed and extend canonically to symmetric T-Poisson CGL extensions.","keywords":["Poisson deformations","log-symplectic structures","CGL extensions","torus actions","Poisson cohomology","Bott-Samelson cells","cluster algebras","log-canonical Poisson structures"],"falsifier":"An explicit T-invariant first-order deformation with linearly independent weights that fails to extend to a second-order deformation, or a symmetric Poisson CGL extension whose initial mutation matrix cannot be expressed via the weights of the second T-invariant Poisson cohomology of its log-canonical term.","tokens_in":2790,"feed_emoji":"","tokens_out":745,"duration_ms":26062,"temperature":0.7,"pith_summary":"The paper establishes that every T-invariant first-order deformation of a T-log-symplectic log-canonical Poisson structure π0 on C^n is unobstructed provided the (C×)^n-weights are linearly independent. For symmetric T-action data it shows that π0 admits a canonical deformation to a symmetric T-Poisson CGL extension. This yields a classification of all such extensions in terms of the log-canonical term and the second T-invariant Poisson cohomology, together with an explicit formula for the initial mutation matrix. A sympathetic reader would care because the result identifies the standard Poisson structures on Bott-Samelson and generalized Schubert cells as the unique maximal normalized admissible deformations of their log-canonical terms.","feed_headline":"Torus-invariant Poisson deformations are unobstructed when weights are independent","feed_subtitle":"Symmetric cases extend canonically to CGL extensions and recover the standard structures on Bott-Samelson and Schubert cells as unique maxim","key_machinery":"The T-log-symplectic log-canonical Poisson structure π0 equipped with symmetric T-action data, whose second T-invariant Poisson cohomology controls the unobstructed deformations and supplies the initial mutation matrix.","core_discovery":"Every T-invariant first-order deformation of π0 with linearly independent (C×)^n-weights is unobstructed. For symmetric T-action data, π0 can be canonically deformed to symmetric T-Poisson CGL extensions, and the standard Poisson structures on Bott-Samelson cells and generalized Schubert cells for semi-simple complex Lie groups are the uniquely determined maximal normalized admissible deformations of their log-canonical terms.","pith_inferences":["The weight-independence condition may be relaxed if one allows controlled obstructions in higher cohomology groups.","The uniqueness result for maximal deformations on Schubert cells suggests a rigidity statement for Poisson structures on partial flag varieties that could be checked by direct computation in low-rank cases.","The explicit mutation-matrix formula might allow direct comparison between the Poisson CGL construction and other cluster structures arising from Lie theory."],"forward_implications":["All symmetric T-Poisson CGL extensions are classified by their log-canonical terms π0 and the second T-invariant Poisson cohomology of π0.","Symmetric Poisson CGL extensions of Cartan type are characterized by sequences of simple roots in symmetrizable generalized Cartan matrices.","The initial mutation matrix in the Goodearl-Yakimov cluster algebra theory is given explicitly by the (C×)^n-weights of the second T-invariant Poisson cohomology of π0."],"fun_headline_variants":["Unobstructed T-Poisson deformations with independent weights","Symmetric T-data deforms canonically to CGL extensions","Schubert cells have unique maximal Poisson deformations","Log-canonical terms determine symmetric Poisson CGL extensions"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The Poisson structure π0 must be T-log-symplectic and log-canonical with the specified symmetric T-action data.","fun_headline_variants_meta":{"raw":{"variants":["Unobstructed T-Poisson deformations with independent weights","Symmetric T-data deforms canonically to CGL extensions","Schubert cells have unique maximal Poisson deformations","Log-canonical terms determine symmetric Poisson CGL extensions"]},"model":"grok-4.3","cost_usd":0.005053,"raw_usage":{"total_tokens":2448,"prompt_tokens":800,"num_sources_used":0,"completion_tokens":61,"cost_in_usd_ticks":50528000,"prompt_tokens_details":{"text_tokens":800,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1587,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":800,"tokens_out":61,"duration_ms":16966,"temperature":1.0,"reasoning_tokens":1587,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-23T00:46:22.882145+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit T-invariant first-order deformation with linearly independent weights that fails to extend to a second-order deformation, or a symmetric Poisson CGL extension whose initial mutation matrix cannot be expressed via the weights of the second T-invariant Poisson cohomology of its log-canonical term.","supporting_citations":[],"review_version":1}