{"id":"1db42294-65a4-464a-86e4-b78721332524","arxiv_id":"2605.26556","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":1,"one_line_summary":"A β-deformed version of motivic Segre classes of Schubert cells is constructed via the connective formal group law, with rational representatives via lattice models and structure constants via Knutson-Tao puzzles proven using quantum group intertwiners for d=1.","lead":"The paper defines a β-deformation of motivic Segre classes of Schubert cells in d-step flag varieties using the connective formal group law. This interpolates between motivic classes at β=1 and Segre-Schwartz-MacPherson classes at β=0, with explicit formulas and proofs given for the d=1 case.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"The connective formal group law deformation must preserve geometric/algebraic properties for correct β=0/1 specializations and for the GKM condition to hold in the cobordism ring quotient.","rationale":"The reader's weakest_assumption is precisely the load-bearing compatibility claim identified above. Because the original review had access only to the abstract, the full manuscript would still need to be examined for the explicit verification of the specializations and the GKM argument; no other technical gap is apparent from the strongest_claim.","tokens_in":1782,"tokens_out":416,"duration_ms":23789,"concrete_test":"For the d=1 case, compute the lattice-model rational functions explicitly for the smallest Schubert cells, substitute β=0 and β=1, and compare the resulting classes against the known motivic Segre and Segre-Schwartz-MacPherson representatives; separately, check whether the canonical elements satisfy the stated GKM localization conditions at all fixed points in the cobordism ring.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The central construction defines the β-deformed motivic Segre classes via the connective formal group law, asserts that these specialize to the known motivic Segre classes (β=1) and Segre-Schwartz-MacPherson classes (β=0), supplies rational representatives for d=1 via a lattice model, derives structure constants via Knutson-Tao puzzles and â2 quantum-group intertwiners, and finally realizes the classes as quotients of canonical elements in a quotient of the equivariant algebraic cobordism ring of the cotangent bundle after verifying a GKM-type condition on those elements. The single load-bearing step is the claim that the chosen deformation automatically preserves the geometric and algebraic features required for both the specializations and the GKM condition to be satisfiable; if that compatibility fails, the lattice-model representatives, the puzzle formula, and the cobordism interpretation all lose their claimed meaning. No other internal inconsistency is visible in the stated claims.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript defines a one-parameter β-deformation of the motivic Segre classes of Schubert cells in the d-step flag variety via the connective formal group law. This deformation is asserted to specialize to the motivic Segre classes at β=1 and to the Segre-Schwartz-MacPherson classes at β=0. For the d=1 case, rational-function representatives are constructed via a solvable lattice model, and a combinatorial formula for the structure constants in the deformed basis is proved using Knutson-Tao puzzles together with intertwiners for the multi-parameter quantum group of type â2. The deformed classes are realized as quotients of canonical elements in a quotient of the equivariant algebraic cobordism ring of the cotangent bundle of the flag variety, after verifying a GKM-type condition on those elements.","tokens_in":2038,"tokens_out":500,"duration_ms":14176,"significance":"If the compatibility of the deformation with the required geometric and algebraic properties holds, the work supplies an explicit bridge between motivic and classical Segre classes together with combinatorial tools (lattice models, puzzle formulas) and a cobordism-ring interpretation. The explicit use of solvable lattice models and the proof of the puzzle formula via quantum-group intertwiners constitute concrete, verifiable contributions for the d=1 case.","major_comments":[{"comment":"The central construction (abstract and opening sections) defines the β-deformed classes directly from the connective formal group law and asserts that the resulting objects specialize correctly at β=0,1 while satisfying the GKM condition needed for the cobordism interpretation. Because this compatibility is invoked both for the specializations and for the final realization as quotients of canonical elements, an explicit verification that the lattice-model representatives and the GKM condition remain valid for generic β (rather than only at the endpoints) is required; without it, the combinatorial formula and the cobordism claim lose their claimed geometric meaning.","section":"construction of the β-deformed classes and the GKM verification paragraph"}],"minor_comments":[{"comment":"Notation for the multi-parameter quantum group of type â2 and the precise definition of the lattice model should be introduced with a short self-contained paragraph or diagram to aid readers unfamiliar with the â2 intertwiners.","section":"d=1 combinatorial section"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and the detailed comment on the central construction. We address the point below and will revise the manuscript to incorporate an explicit verification as requested.","responses":[{"response":"We agree that an explicit verification for generic β is necessary to fully support the geometric claims. The β-deformed classes are constructed algebraically via the connective formal group law over the ring ℤ[β], and the lattice-model representatives for d=1 are defined with weights that are polynomials in β. The GKM-type condition on the canonical elements in the quotient of the equivariant algebraic cobordism ring can therefore be verified by direct (albeit tedious) computation that holds identically as an identity in β, after which the specializations at β=0 and β=1 are obtained by substitution. The quantum-group intertwiners used for the puzzle formula are likewise defined over the multi-parameter ring that includes β. To make this verification fully explicit rather than implicit in the algebraic setup, we will add a dedicated subsection (new Section 4.3 or equivalent) that carries out the GKM check for a generic β and records the resulting identities. This revision will be included in the next version of the manuscript.","revision_made":"yes","referee_comment":"The central construction (abstract and opening sections) defines the β-deformed classes directly from the connective formal group law and asserts that the resulting objects specialize correctly at β=0,1 while satisfying the GKM condition needed for the cobordism interpretation. Because this compatibility is invoked both for the specializations and for the final realization as quotients of canonical elements, an explicit verification that the lattice-model representatives and the GKM condition remain valid for generic β (rather than only at the endpoints) is required; without it, the combinatorial formula and the cobordism claim lose their claimed geometric meaning."}],"tokens_in":1432,"tokens_out":402,"duration_ms":22194,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The key thing to know is that this paper defines a beta-deformation of motivic Segre classes of Schubert cells using the connective formal group law. The deformation specializes to the usual motivic Segre classes at beta=1 and to Segre-Schwartz-MacPherson classes at beta=0. For the d=1 case it gives explicit rational function representatives from a solvable lattice model and proves a combinatorial formula for the structure constants via Knutson-Tao puzzles and intertwiners for the multi-parameter quantum group of type a2 hat. It also shows the classes arise as quotients of canonical elements in a quotient of the equivariant algebraic cobordism ring of the cotangent bundle after verifying a GKM-type condition.\n\nWhat is new is the specific use of the connective formal group law to produce this interpolation together with the d=1 combinatorial tools and the cobordism-ring interpretation. The paper does well in spelling out concrete representatives for d=1 and in connecting the structure constants to existing puzzle combinatorics and quantum-group methods.\n\nThe main place to check is whether the deformation preserves the geometric and algebraic properties needed for the specializations to hold and for the GKM condition to be satisfiable in the cobordism ring. The abstract states that the canonical elements satisfy the GKM-type condition, so that verification is the load-bearing step. If it is carried out correctly and without hidden choices, the rest follows. No other internal gaps are visible from the claims.\n\nThis is narrow work aimed at people working on motivic cohomology, Schubert calculus, or equivariant cobordism. A reader in that subfield would get value from the d=1 formulas and the deformation construction. It deserves a serious referee because the results are specific and the methods are standard enough to evaluate in detail. I would send it to peer review.","headline":"The paper defines a beta-deformation of motivic Segre classes via the connective formal group law, supplies d=1 lattice-model representatives and a puzzle formula proved with quantum-group intertwiners, and realizes them in a cobordism ring quotient after a GKM check.","tokens_in":2504,"tokens_out":473,"would_cite":false,"duration_ms":27735,"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":"The connective formal group law yields a β-deformation of motivic Segre classes of Schubert cells that specializes to known classes and admits explicit combinatorial descriptions for d=1.","keywords":["motivic Segre classes","Schubert cells","connective formal group law","β-deformation","Knutson-Tao puzzles","algebraic cobordism","flag varieties","quantum groups"],"falsifier":"A direct check for small d=1 cases in which the rational function representatives from the lattice model fail to match the expected specialization at β=1 would falsify the deformation construction.","tokens_in":2673,"feed_emoji":"","tokens_out":906,"duration_ms":32672,"temperature":0.7,"pith_summary":"The paper defines a one-parameter β-deformation of the motivic Segre classes of Schubert cells in d-step flag varieties using the connective formal group law. This deformation specializes to the original motivic classes when β equals 1 and to the Segre-Schwartz-MacPherson classes when β equals 0. In the d equals 1 case the deformed classes have explicit rational function representatives coming from a solvable lattice model. The structure constants in this deformed basis are given by a combinatorial formula proved with Knutson-Tao puzzles and intertwiners from a multi-parameter quantum group. The construction is realized as quotients of canonical elements in a quotient of the equivariant algebraic cobordism ring of the cotangent bundle of the flag variety after verifying a GKM-type condition.","feed_headline":"β-deformation links motivic Segre classes to cobordism via puzzles","feed_subtitle":"The construction specializes correctly at β=0 and 1 while giving lattice-model representatives and puzzle formulas for d=1.","key_machinery":"The β-deformation of motivic Segre classes defined via the connective formal group law, with rational representatives from the solvable lattice model and structure constants from Knutson-Tao puzzles in the d=1 case.","core_discovery":"We use the connective formal group law to define a one-parameter (β-)deformation of the motivic Segre classes of Schubert cells in the d-step flag variety. This β-deformation specializes to the motivic Segre classes of Schubert cells when β=1 and to the Segre-Schwartz-MacPherson classes of Schubert cells when β=0. We define rational function representatives for the β-deformed classes in the d=1 case in terms of a solvable lattice model, and we prove a combinatorial formula for the structure constants in the β-deformed basis in the d=1 case using Knutson-Tao puzzles. The proof of the puzzle formula involves intertwiners for representations of the multi-parameter quantum group of type â2. We s","pith_inferences":["The lattice model solvability could be checked in higher d to test whether explicit representatives extend beyond the one-step case.","The quantum group intertwiners used for the puzzle proof may connect the construction to other representation-theoretic approaches in Schubert calculus.","The GKM verification step offers a method that might apply to deformations of similar classes on other homogeneous spaces."],"forward_implications":["The β-deformed classes specialize to the motivic Segre classes when β=1.","The β-deformed classes specialize to the Segre-Schwartz-MacPherson classes when β=0.","The β-deformed classes in the d=1 case have rational function representatives from the solvable lattice model.","The structure constants of the β-deformed basis are given by the combinatorial puzzle formula.","The β-deformations correspond to quotients of canonical elements in the equivariant algebraic cobordism ring that satisfy a GKM condition."],"fun_headline_variants":["Connective formal group law deforms motivic Segre classes","β-deformation of motivic Segre classes in d-step flag varieties","Lattice model yields reps for β-deformed Segre classes d=1","Knutson-Tao puzzles give formula for β-deformed structure constants","β-deformed classes as cobordism ring quotients satisfying GKM"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The connective formal group law supplies a deformation of the motivic Segre classes that preserves the geometric and algebraic properties needed for the specializations at β=0 and β=1 to hold and for the GKM condition to be satisfiable in the cobordism ring.","fun_headline_variants_meta":{"raw":{"variants":["Connective formal group law deforms motivic Segre classes","β-deformation of motivic Segre classes in d-step flag varieties","Lattice model yields reps for β-deformed Segre classes d=1","Knutson-Tao puzzles give formula for β-deformed structure constants","β-deformed classes as cobordism ring quotients satisfying GKM"]},"model":"grok-4.3","cost_usd":0.005579,"raw_usage":{"total_tokens":2703,"prompt_tokens":728,"num_sources_used":0,"completion_tokens":84,"cost_in_usd_ticks":55787000,"prompt_tokens_details":{"text_tokens":728,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1891,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":728,"tokens_out":84,"duration_ms":14037,"temperature":1.0,"reasoning_tokens":1891,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T17:52:29.321905+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A direct check for small d=1 cases in which the rational function representatives from the lattice model fail to match the expected specialization at β=1 would falsify the deformation construction.","supporting_citations":[],"review_version":1}