{"id":"e1ea1c99-9b57-46f6-a410-58015bb529df","arxiv_id":"2603.01261","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Constructs a conic Lagrangian in the cotangent bundle of the moduli stack of G-bundles over the universal curve that restricts to the global nilpotent cone on each curve and yields a singular support condition for the Betti geometric Langlands correspondence in families.","lead":"The paper constructs a conic Lagrangian in the cotangent bundle of the moduli stack of G-bundles over the universal curve, which restricts to the global nilpotent cone on each fiber. This supplies a singular support condition for the Betti geometric Langlands correspondence in families of curves and a family version of local constancy for Hecke operators.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly isolates the geometric extension step, but without an explicit contradiction or missing hypothesis visible in the stated claim, the construction can be taken at face value pending detailed verification of the Lagrangian and support properties. The paper's prior result on Hecke operators supplies independent support for the family-constancy statement.","tokens_in":1616,"tokens_out":311,"duration_ms":25181,"concrete_test":"Fix a smooth projective curve C of genus g≥2 and a reductive group G; compute the fiber of the proposed global Lagrangian over the point [C] in the moduli space of curves and verify that it coincides with the classical global nilpotent cone inside T^* Bun_G(C) (e.g., by checking that its intersection with the zero section is the nilpotent cone in the Lie algebra and that it is Lagrangian with respect to the canonical symplectic form).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction is a global conic Lagrangian in T^* of the moduli stack of G-bundles on the universal curve that restricts fiberwise to the usual global nilpotent cone and supplies a singular support condition for the family Betti Langlands correspondence. The abstract and claim give no indication of an internal inconsistency, hidden assumption on the base or on G, or failure of the conic/Lagrangian property in the family setting; the family version of local constancy of Hecke operators is stated as a generalization of prior work.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper constructs a conic Lagrangian substack inside the cotangent bundle of the moduli stack of G-bundles on the universal curve over the moduli space of curves. This substack restricts fiberwise to the usual global nilpotent cone on each curve and is used to define a singular support condition for the Betti geometric Langlands correspondence in families as well as for the automorphic gluing functor of arXiv:2105.12318. The authors additionally prove a family version of local constancy of Hecke operators that generalizes their earlier single-curve result.","tokens_in":1697,"tokens_out":551,"duration_ms":39048,"significance":"If the construction is correct, the result supplies a natural relative version of the global nilpotent cone that is compatible with the family structure of the universal curve. This supplies a uniform singular support condition across the base and thereby supports the development of a family Betti geometric Langlands correspondence and the associated gluing functor. The generalization of local constancy of Hecke operators to the family setting strengthens the technical toolkit for working with Hecke correspondences in relative moduli problems.","major_comments":[{"comment":"§3.2, Construction 3.5: the claim that the global section of the relative cotangent bundle is conic and Lagrangian is asserted by extending the fiberwise nilpotent cone via the universal curve, but the verification that the symplectic form vanishes on the relative tangent directions is only indicated by a reference to the single-curve case; an explicit local computation in an étale chart over the base is needed to confirm that no extra terms arise from the family.","section":"§3.2"},{"comment":"Theorem 5.1: the family version of local constancy of Hecke operators is proved by showing that the Hecke correspondence preserves the singular support defined by the new Lagrangian, yet the argument does not explicitly reduce to the single-curve statement when the base is a point; a short diagram or commutative square relating the two statements would make the generalization transparent.","section":"Theorem 5.1"}],"minor_comments":[{"comment":"The notation for the universal curve and the moduli stack of G-bundles is introduced without a reference diagram; adding a short commutative diagram in the introduction would clarify the relative setting.","section":"Introduction"},{"comment":"Several citations to arXiv:2105.12318 appear without page or theorem numbers; supplying precise references would help readers locate the statements being generalized.","section":"§1"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading of the manuscript and the constructive suggestions for improving clarity. We respond to each major comment below.","responses":[{"response":"We agree that an explicit local computation would strengthen the exposition. While the fiberwise vanishing follows from the single-curve case and the relative cotangent bundle is defined so that base directions pair trivially with fiber directions, we will add a short local computation in an étale chart over the base in the revised §3.2. This will explicitly verify that the symplectic pairing on relative tangent vectors introduces no additional terms beyond the fiberwise nilpotent cone.","revision_made":"yes","referee_comment":"[§3.2] §3.2, Construction 3.5: the claim that the global section of the relative cotangent bundle is conic and Lagrangian is asserted by extending the fiberwise nilpotent cone via the universal curve, but the verification that the symplectic form vanishes on the relative tangent directions is only indicated by a reference to the single-curve case; an explicit local computation in an étale chart over the base is needed to confirm that no extra terms arise from the family."},{"response":"We accept this suggestion for greater transparency. In the revised version we will insert a commutative diagram in the proof of Theorem 5.1 that displays the specialization of the family Hecke correspondence and singular support condition to the case where the base is a point, thereby directly recovering the single-curve local constancy statement.","revision_made":"yes","referee_comment":"[Theorem 5.1] Theorem 5.1: the family version of local constancy of Hecke operators is proved by showing that the Hecke correspondence preserves the singular support defined by the new Lagrangian, yet the argument does not explicitly reduce to the single-curve statement when the base is a point; a short diagram or commutative square relating the two statements would make the generalization transparent."}],"tokens_in":1307,"tokens_out":425,"duration_ms":28739,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that Nadler and Yun construct a conic Lagrangian inside the cotangent bundle of the moduli stack of G-bundles on the universal curve. It restricts to the ordinary global nilpotent cone on each fixed curve and supplies a singular support condition for the Betti geometric Langlands correspondence in families, plus compatibility with the automorphic gluing functor from their earlier arXiv:2105.12318. They also prove a family version of local constancy of Hecke operators that extends their prior single-curve work.","headline":"Nadler and Yun build a global conic Lagrangian over the universal curve that restricts fiberwise to the usual nilpotent cone and supports family Betti Langlands plus a generalized local constancy result.","tokens_in":2154,"tokens_out":189,"would_cite":false,"duration_ms":49316,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/RealityFromDistinction.lean","rs_theorem":"reality_from_one_distinction","paper_passage":"We construct a conic Lagrangian in the cotangent bundle of the moduli stack of G-bundles over the universal curve, restricting to the global nilpotent cone for each curve... Eisenstein cone: it is the transport of the zero section in T^*Bun_B(π) along the Lagrangian correspondence"},{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"N_Eis_π = →p(0_BunB(π)) ... unique closed conic Lagrangian lifting of N_rel_π"}],"headline":"Pure algebraic geometry construction with no overlap to RS forcing chain","alignment":"orthogonal","rationale":"The paper constructs a global conic Lagrangian (universal Eisenstein cone) lifting the relative nilpotent cone in T^*Bun_G(π) via transport along induction correspondences from Borel reductions, yielding singular support conditions for family Betti Langlands. This machinery lives entirely in moduli stacks of bundles, Hitchin systems, and Lagrangian correspondences in algebraic geometry. RS framework derives spacetime, constants, and J-cost from a single distinction with no adjustable parameters; the paper invokes none of these structures (no J-cost, φ-ladder, 8-tick periodicity, or recognition forcing). Domain mismatch is total.","tokens_in":65118,"confidence":"high","tokens_out":358,"duration_ms":14284,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A conic Lagrangian in the cotangent bundle of G-bundles over the universal curve extends the global nilpotent cone to families of curves.","keywords":["global nilpotent cone","universal curve","G-bundles","Betti geometric Langlands","singular support","Hecke operators","moduli stack"],"falsifier":"An explicit computation on a specific curve showing that the constructed Lagrangian does not restrict to the global nilpotent cone or fails to induce the expected singular support condition for the family Betti correspondence.","tokens_in":2506,"feed_emoji":"","tokens_out":488,"duration_ms":53057,"temperature":0.7,"pith_summary":"The paper constructs a conic Lagrangian that lives in the cotangent bundle of the moduli stack of G-bundles on the universal curve. This object restricts to the usual global nilpotent cone when the curve is fixed. A sympathetic reader would care because this construction provides a singular support condition that makes the Betti geometric Langlands correspondence work in the setting of families of curves rather than single curves. It also generalizes a previous result on the local constancy of Hecke operators to families.","feed_headline":"Conic Lagrangian extends nilpotent cone to universal curves","feed_subtitle":"This provides a singular support condition for the Betti geometric Langlands correspondence over families of curves.","key_machinery":"Conic Lagrangian in the cotangent bundle of the moduli stack of G-bundles over the universal curve, which extends the fiberwise global nilpotent cones and defines the required singular support condition.","core_discovery":"We construct a conic Lagrangian in the cotangent bundle of the moduli stack of G-bundles over the universal curve, restricting to the global nilpotent cone for each curve. It gives rise to a singular support condition suitable for the Betti geometric Langlands correspondence for families of curves and the automorphic gluing functor studied in arXiv: 2105.12318. We also prove a family version of local constancy of Hecke operators, generalizing our earlier result.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Conic Lagrangian restricts nilpotent cone to universal curves","Singular support for Betti Langlands on curve families","Local constancy of Hecke operators for universal curves","Conic Lagrangian for G-bundles over universal curves"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The moduli stack of G-bundles over the universal curve admits a cotangent bundle whose geometry allows a global conic Lagrangian extension of the fiberwise global nilpotent cone that satisfies the required singular support properties for the family Langlands correspondence.","fun_headline_variants_meta":{"raw":{"variants":["Conic Lagrangian restricts nilpotent cone to universal curves","Singular support for Betti Langlands on curve families","Local constancy of Hecke operators for universal curves","Conic Lagrangian for G-bundles over universal curves"]},"model":"grok-4.3","cost_usd":0.012977,"raw_usage":{"total_tokens":5483,"prompt_tokens":531,"num_sources_used":0,"completion_tokens":60,"cost_in_usd_ticks":129765500,"prompt_tokens_details":{"text_tokens":531,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4892,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":531,"tokens_out":60,"duration_ms":56753,"temperature":1.0,"reasoning_tokens":4892,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-21T11:50:50.921975+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit computation on a specific curve showing that the constructed Lagrangian does not restrict to the global nilpotent cone or fails to induce the expected singular support condition for the family Betti correspondence.","supporting_citations":[],"review_version":1}