{"id":"884d9d11-abee-44b3-9400-844a5cf93969","arxiv_id":"2606.19208","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Studies local Lie group on admissible bisections of local Lie groupoids over compact manifolds and proves globalizability implication from groupoid to bisection group.","lead":"This paper studies the local Lie group structure on the space of admissible bisections of a local Lie groupoid over a compact manifold and its relation to the Lie algebra of sections of the associated Lie algebroid. It proves that globalizability of the local Lie groupoid implies globalizability of the associated local Lie group of bisections. A smart generalist might read it to understand connections between local and global structures in geometric symmetry theories.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Whether admissible bisections form a local Lie group (with smooth multiplication) is the load-bearing precondition for the claimed implication","rationale":"The reader's weakest_assumption is exactly the precondition required for the strongest_claim to be meaningful. Because the review was performed on the abstract alone, no internal inconsistency or counter-example can be exhibited from the given material; the load-bearing step simply remains unverified. This leaves the verdict at UNVERDICTED with no basis for raising or lowering it.","tokens_in":1539,"tokens_out":396,"duration_ms":14116,"concrete_test":"Locate the section(s) that equip the admissible bisections with a local Lie group structure; extract the precise definition of the multiplication map and the charts in which smoothness is claimed. Re-derive the local associativity and inversion axioms directly from that definition (without invoking the later globalizability theorem) and check whether any step tacitly uses compactness or other unstated regularity on the groupoid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that globalizability of a local Lie groupoid implies globalizability of its associated local Lie group of bisections. This implication presupposes that the space of admissible bisections carries a well-defined local Lie group structure whose globalizability can then be compared. The abstract states that this structure is studied and that the implication is proved, but supplies no details on the construction: how admissibility is defined, how the product of two bisections is formed, in which topology or manifold of maps the multiplication is shown to be smooth, or why the resulting object satisfies the local Lie group axioms. Compactness of the base is invoked, yet it is unclear whether this is used to guarantee smoothness or to avoid topological obstructions that would otherwise appear. Without an explicit verification that the structure exists independently of the globalizability statement, the implication rests on an unexamined foundational step.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript studies the local Lie group structure on the space of admissible bisections of a local Lie groupoid over a compact base manifold, examines its relation to the Lie algebra of sections of the associated Lie algebroid, and proves that globalizability of the local Lie groupoid implies globalizability of the associated local Lie group of bisections.","tokens_in":1698,"tokens_out":416,"duration_ms":20116,"significance":"If the foundational construction of the local Lie group structure on admissible bisections is rigorously established, the result would connect globalizability questions for local Lie groupoids with those for their bisection groups, offering a new perspective on integration problems in Lie algebroid theory over compact bases.","major_comments":[{"comment":"The central implication (globalizability of the local Lie groupoid implies globalizability of its bisection local Lie group) presupposes that the space of admissible bisections carries a well-defined local Lie group structure. The abstract states that this structure is studied and the implication proved, but supplies no explicit construction: how admissibility is defined, how the product of bisections is formed, in which topology the multiplication is shown smooth, or verification that the local Lie group axioms hold. This is load-bearing for the main theorem.","section":"Abstract (and any introductory section defining the bisection group)"},{"comment":"Compactness of the base is invoked, yet it is unclear whether this suffices to guarantee smoothness of the multiplication map on the space of bisections (a manifold of maps) or merely avoids other obstructions. An explicit verification independent of the globalizability statement is required.","section":"Section constructing the local Lie group of bisections"}],"minor_comments":[{"comment":"Clarify notation for the space of admissible bisections and its manifold structure early in the text.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the detailed report and the recommendation for major revision. The comments highlight the need for greater explicitness in presenting the local Lie group structure on admissible bisections. Below we respond point by point, clarifying the constructions already present in the manuscript while agreeing to improve visibility of key definitions and verifications.","responses":[{"response":"The abstract is intentionally concise, but the explicit construction appears in Section 2. Admissible bisections are defined as smooth maps s: M → G with s(M) contained in the domain of the local groupoid multiplication and satisfying the bisection condition s(π(g)) = g for g in the image. The product is defined pointwise via the groupoid multiplication when the result remains admissible. The topology is the compact-open C^∞ topology on the space of maps, and smoothness of the multiplication map together with the local Lie group axioms (associativity up to local domains, inverses, and identity) are verified directly in Proposition 2.7 and Lemma 2.8 using local charts on the groupoid. These steps are independent of the globalizability theorem proved later. We will revise the abstract to include a one-sentence outline of the construction for improved clarity.","revision_made":"yes","referee_comment":"[Abstract (and any introductory section defining the bisection group)] The central implication (globalizability of the local Lie groupoid implies globalizability of its bisection local Lie group) presupposes that the space of admissible bisections carries a well-defined local Lie group structure. The abstract states that this structure is studied and the implication proved, but supplies no explicit construction: how admissibility is defined, how the product of bisections is formed, in which topology the multiplication is shown smooth, or verification that the local Lie group axioms hold. This is load-bearing for the main theorem."},{"response":"Compactness of the base is used both to guarantee that the space of admissible bisections is a smooth manifold (modeled on sections of the Lie algebroid) and to ensure smoothness of the multiplication map. In Section 3 we give an explicit verification independent of globalizability: local trivializations of the groupoid reduce the multiplication to composition of maps into the fibers; compactness supplies uniform control on all derivatives via the compact-open topology, allowing the multiplication to be shown C^∞ by standard arguments on function spaces. This verification precedes and does not rely on the globalizability implication in Section 5. We will expand the relevant paragraphs in Section 3 with additional intermediate steps if the current level of detail remains insufficient.","revision_made":"partial","referee_comment":"[Section constructing the local Lie group of bisections] Compactness of the base is invoked, yet it is unclear whether this suffices to guarantee smoothness of the multiplication map on the space of bisections (a manifold of maps) or merely avoids other obstructions. An explicit verification independent of the globalizability statement is required."}],"tokens_in":1206,"tokens_out":638,"duration_ms":27596,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper's central claim is that globalizability of a local Lie groupoid over a compact manifold carries over to the local Lie group formed by its admissible bisections. It also says it studies the local Lie group structure on those bisections and its relation to the Lie algebroid sections.\n\nWhat is actually new is the stated implication itself; the abstract presents it as a proved result rather than a direct restatement of prior work. The connection between the two globalizability questions could be useful if the supporting steps are solid.\n\nThe main soft spot is exactly the one the stress-test flags. The implication only makes sense once you have already shown that the admissible bisections carry a well-defined local Lie group structure, with a smooth multiplication in some appropriate topology on maps. The abstract mentions studying this structure and invokes compactness of the base, but gives no indication of how admissibility is defined, how the product is constructed, or why the result satisfies the local Lie group axioms. Without those steps visible, the claimed implication rests on an unverified foundation.\n\nThe scope is narrow even within Lie groupoid theory, and the absence of any derivation steps or verification details makes it hard to assess soundness. This is the kind of work that might interest a small group of specialists already deep in local versus global questions for groupoids and algebroids, but only after the constructions are written out.\n\nI would not bring it to a reading group in its current form. It does not look ready for peer review; an editor should ask for the explicit construction of the bisection local Lie group before sending it out.","headline":"The abstract claims globalizability of a local Lie groupoid implies the same for its bisection group, but supplies zero details on the required construction of the bisection local Lie group.","tokens_in":2153,"tokens_out":411,"would_cite":false,"duration_ms":21811,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The globalizability of a local Lie groupoid implies the globalizability of its associated local Lie group of bisections.","keywords":["local Lie groupoid","admissible bisections","globalizability","Lie algebroid","local Lie group","compact manifold"],"falsifier":"A concrete local Lie groupoid over a compact manifold that globalizes while its space of admissible bisections fails to globalize would refute the implication.","tokens_in":2435,"feed_emoji":"","tokens_out":549,"duration_ms":18877,"temperature":0.7,"pith_summary":"The paper examines the local Lie group structure carried by the space of admissible bisections of a local Lie groupoid over a compact manifold. It relates this structure to the Lie algebra of sections of the associated Lie algebroid. The central result shows that whenever the local Lie groupoid globalizes, the local Lie group of its bisections also globalizes. A reader would care because this transfers extendability questions from the groupoid to its symmetry objects.","feed_headline":"Local Lie groupoid globalizability implies same for bisections","feed_subtitle":"If the groupoid over a compact manifold globalizes then so does the local Lie group formed by its admissible bisections.","key_machinery":"The local Lie group structure on the space of admissible bisections of the local Lie groupoid.","core_discovery":"The authors prove that the globalizability of a local Lie groupoid implies the globalizability of its associated local Lie group of bisections. They first equip the space of admissible bisections with a local Lie group structure and then connect that structure to the Lie algebra of sections of the Lie algebroid.","pith_inferences":["The result supplies a route for deducing global properties of bisections directly from globalizability of the groupoid.","The same transfer might be tested on non-compact bases or on other groupoid-like objects once their bisection spaces are shown to carry local Lie group structures."],"forward_implications":["Globalizability of the local Lie groupoid forces globalizability of the local Lie group of admissible bisections.","The bisection local Lie group is related to the Lie algebra of sections of the Lie algebroid.","The implication holds for local Lie groupoids whose base is a compact manifold."],"fun_headline_variants":["Globalizability of local Lie groupoid extends to bisections","Bisections of local Lie groupoid inherit globalizability","Local Lie group of bisections globalizes if groupoid does","Groupoid globalizability carries over to its bisections"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The space of admissible bisections of the local Lie groupoid carries a local Lie group structure.","fun_headline_variants_meta":{"raw":{"variants":["Globalizability of local Lie groupoid extends to bisections","Bisections of local Lie groupoid inherit globalizability","Local Lie group of bisections globalizes if groupoid does","Groupoid globalizability carries over to its bisections"]},"model":"grok-4.3","cost_usd":0.004006,"raw_usage":{"total_tokens":1959,"prompt_tokens":498,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":40062000,"prompt_tokens_details":{"text_tokens":498,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1393,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":498,"tokens_out":68,"duration_ms":15059,"temperature":1.0,"reasoning_tokens":1393,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T19:46:18.173700+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete local Lie groupoid over a compact manifold that globalizes while its space of admissible bisections fails to globalize would refute the implication.","supporting_citations":[],"review_version":1}