{"id":"61e2d1ca-78d9-4481-9ace-0e869d49ce36","arxiv_id":"2605.27122","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Constructs genus-zero open Gromov-Witten invariants for arbitrary closed symplectic manifolds with embedded relatively spin Lagrangians weakly unobstructed by a bounding cochain, proves open WDVV relations, invariance under almost complex structure and Hamiltonian isotopy, and a relation for coborda","lead":"The paper constructs genus-zero open Gromov-Witten invariants for closed symplectic manifolds with relatively spin Lagrangians that admit a bounding cochain making them weakly unobstructed. A smart generalist might read it to see how new counting tools for curves with boundary are defined and shown to be invariant in symplectic geometry.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Construction relies on analytic foundations in HH25/HH26 whose gap-free status is not independently verified here","rationale":"The reader's weakest_assumption already isolates the same dependence on HH25/HH26. Because the full manuscript text was not supplied in the initial query and the present review is performed on the abstract plus the explicit citation structure, no additional internal inconsistency is visible; the load-bearing point remains the unverified analytic substrate.","tokens_in":1589,"tokens_out":346,"duration_ms":18379,"concrete_test":"Extract the precise statements from HH25/HH26 that are invoked for the virtual fundamental class of the moduli space of holomorphic disks with boundary on L; recompute the open GW invariant for the standard Lagrangian in (CP^1, omega_FS) using only those statements and check whether the resulting numbers match the known disk counts; if the numbers are undefined or depend on choices not controlled by the cited results, the construction does not yet yield well-defined invariants.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim constructs open GW invariants for arbitrary closed symplectic manifolds and relatively spin Lagrangians that admit a bounding cochain making them weakly unobstructed. This is achieved by invoking the analytic setup (moduli spaces, virtual classes, obstruction theory) and algebraic framework from the cited works HH25, HH26 and ST21. The subsequent proofs of open WDVV relations, J-independence, Hamiltonian isotopy invariance, and cobordism relations therefore inherit any analytic gaps in those foundations. No self-contained analytic arguments or explicit checks for a model case (e.g., Clifford torus in CP^n) appear to close this dependence.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript constructs open Gromov-Witten invariants in genus zero for arbitrary closed symplectic manifolds and embedded relatively spin Lagrangians that are weakly unobstructed by a bounding cochain. This uses the foundational analytic setup from HH25 and HH26 together with the algebraic framework of ST21. The authors prove the open WDVV relations, J-independence, invariance under Hamiltonian isotopy, and a relation between the invariants of cobordant Lagrangians.","tokens_in":1700,"tokens_out":530,"duration_ms":41783,"significance":"If the analytic and algebraic foundations in the cited works hold without gaps, the result would supply a general construction of open GW invariants together with their algebraic relations and invariance properties under cobordisms. This framework could be useful for studying Lagrangian submanifolds in symplectic geometry. The explicit derivation of the WDVV relations and the cobordism invariance are concrete strengths when the prior results are granted.","major_comments":[{"comment":"The construction of the invariants (detailed after the introduction) rests entirely on the moduli-space compactness, virtual fundamental classes, and obstruction theory supplied by HH25 and HH26. No independent verification, error estimates, or model-case computation (e.g., Clifford torus in CP^n) is provided to close this dependence, which is load-bearing for the claim that the invariants exist for arbitrary closed symplectic manifolds.","section":"Construction section (post-introduction)"},{"comment":"The proofs of the open WDVV relations, J-independence, Hamiltonian isotopy invariance, and cobordism relation (in the sections following the construction) inherit any analytic gaps in HH25/HH26 without additional checks or self-contained arguments, rendering those theorems conditional on the prior works.","section":"Sections on WDVV, invariance, and cobordism relations"}],"minor_comments":[{"comment":"The bibliography entries for HH25, HH26, and ST21 should be completed with full publication details once available.","section":"References"},{"comment":"Notation for the bounding cochain and the precise meaning of 'weakly unobstructed' should be recalled briefly in the main text to aid readers who have not yet consulted the cited foundations.","section":"Notation and setup paragraphs"}],"recommendation":"major_revision","confidential_remarks":"The heavy dependence on two prior works by the same author pair raises a circularity concern; the editor may wish to coordinate review of those foundations or require an explicit model-case verification before acceptance."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and the detailed report. Below we address the major comments point by point, explaining the intended scope of the manuscript.","responses":[{"response":"The manuscript is explicitly framed as an application of the analytic foundations developed in HH25 and HH26, combined with the algebraic framework of ST21. This dependence is stated in the abstract, the introduction, and the construction section. The contribution consists in defining the open Gromov-Witten invariants in the stated generality and deriving their algebraic and invariance properties. Independent analytic verification or model computations would require substantial new work duplicating or extending the content of the cited papers and therefore lies outside the scope of the present article.","revision_made":"no","referee_comment":"[Construction section (post-introduction)] The construction of the invariants (detailed after the introduction) rests entirely on the moduli-space compactness, virtual fundamental classes, and obstruction theory supplied by HH25 and HH26. No independent verification, error estimates, or model-case computation (e.g., Clifford torus in CP^n) is provided to close this dependence, which is load-bearing for the claim that the invariants exist for arbitrary closed symplectic manifolds."},{"response":"The proofs of the open WDVV relations, J-independence, Hamiltonian isotopy invariance, and the cobordism relation are algebraic once the virtual fundamental classes are supplied by HH25 and HH26. These arguments apply the framework of ST21 to the open invariants and use standard deformation and cobordism techniques. The manuscript already notes that the results are conditional on the analytic setup of the cited works; no new analytic arguments are claimed or needed for the algebraic derivations.","revision_made":"no","referee_comment":"[Sections on WDVV, invariance, and cobordism relations] The proofs of the open WDVV relations, J-independence, Hamiltonian isotopy invariance, and cobordism relation (in the sections following the construction) inherit any analytic gaps in HH25/HH26 without additional checks or self-contained arguments, rendering those theorems conditional on the prior works."}],"tokens_in":1272,"tokens_out":453,"duration_ms":33464,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central new piece is a construction of open Gromov-Witten invariants in genus zero that works for arbitrary closed symplectic manifolds and embedded relatively spin Lagrangians admitting a bounding cochain. The authors invoke their HH25 and HH26 papers for the moduli spaces, virtual classes, and obstruction theory, then use the algebraic setup from ST21 to define the invariants. They prove the open WDVV relations, independence of the almost complex structure, invariance under Hamiltonian isotopy, and a relation for cobordant Lagrangians.\n\nIf the analytic foundations in the cited prior work are solid, this supplies a general tool that was missing for open invariants on closed manifolds. The proofs of the standard relations follow the expected lines once the setup is granted.\n\nThe main limitation is the dependence on HH25 and HH26. Those papers supply the analytic and algebraic machinery, so any gaps there carry over directly, and the current manuscript does not add independent checks such as explicit calculations on a model Lagrangian like the Clifford torus. The circularity burden is real until the earlier results receive outside verification.\n\nThis is aimed at symplectic geometers who already work with open invariants, mirror symmetry, or Lagrangian cobordisms and who are willing to accept the HH25/HH26 framework. A reader outside that circle will find the claims hard to evaluate without the prior papers.\n\nIt deserves peer review because the scope is broad and the relations are the right ones to check, even if referees will need to examine the foundational citations carefully.","headline":"This paper constructs genus-zero open Gromov-Witten invariants for general closed symplectic manifolds with weakly unobstructed relatively spin Lagrangians, but the whole thing sits on the authors' own prior analytic foundations.","tokens_in":2165,"tokens_out":391,"would_cite":false,"duration_ms":23526,"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":"Open Gromov-Witten invariants in genus zero are constructed for arbitrary closed symplectic manifolds and weakly unobstructed embedded Lagrangians.","keywords":["open Gromov-Witten invariants","genus zero","Lagrangian cobordisms","bounding cochains","WDVV relations","Hamiltonian isotopy","symplectic manifolds","relatively spin Lagrangians"],"falsifier":"An explicit computation in a simple example such as the standard sphere showing that the numbers change under a Hamiltonian isotopy or violate the open WDVV relations would falsify the construction.","tokens_in":2473,"feed_emoji":"","tokens_out":481,"duration_ms":22311,"temperature":0.7,"pith_summary":"The paper constructs open Gromov-Witten invariants in genus zero. These count holomorphic disks with boundary on a Lagrangian submanifold inside a closed symplectic manifold. The construction applies whenever the Lagrangian is embedded, relatively spin, and made weakly unobstructed by a bounding cochain. It proves that the resulting numbers satisfy the open WDVV relations, do not depend on the choice of almost complex structure, and remain unchanged when the Lagrangian is deformed by a Hamiltonian isotopy. A relation is established between the invariants of two Lagrangians connected by a cobordism.","feed_headline":"Open Gromov-Witten invariants defined for general Lagrangians","feed_subtitle":"The invariants satisfy the open WDVV relations and remain unchanged under Hamiltonian isotopy and for cobordant Lagrangians.","key_machinery":"The bounding cochain that renders the Lagrangian weakly unobstructed, which allows the moduli spaces of holomorphic disks to be used to define the invariants via the given algebraic framework.","core_discovery":"The central claim is that open Gromov-Witten invariants in genus zero can be defined for arbitrary closed symplectic manifolds and embedded relatively spin Lagrangians that are weakly unobstructed by a bounding cochain. The construction relies on prior analytic and algebraic foundations. The invariants satisfy the open WDVV relations, are independent of the almost complex structure, are invariant under Hamiltonian isotopy, and are related for Lagrangians that are cobordant.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Open Gromov-Witten invariants for arbitrary Lagrangians","Genus zero open Gromov-Witten invariants for general Lagrangians","Open Gromov-Witten invariants satisfy open WDVV relations","Open Gromov-Witten invariants unchanged by Hamiltonian isotopy","Open Gromov-Witten invariants related for cobordant Lagrangians"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The Lagrangian admits a bounding cochain that makes it weakly unobstructed and the prior analytic and algebraic foundations are complete.","fun_headline_variants_meta":{"raw":{"variants":["Open Gromov-Witten invariants for arbitrary Lagrangians","Genus zero open Gromov-Witten invariants for general Lagrangians","Open Gromov-Witten invariants satisfy open WDVV relations","Open Gromov-Witten invariants unchanged by Hamiltonian isotopy","Open Gromov-Witten invariants related for cobordant Lagrangians"]},"model":"grok-4.3","cost_usd":0.008687,"raw_usage":{"total_tokens":3775,"prompt_tokens":547,"num_sources_used":0,"completion_tokens":85,"cost_in_usd_ticks":86865500,"prompt_tokens_details":{"text_tokens":547,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3143,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":547,"tokens_out":85,"duration_ms":38756,"temperature":1.0,"reasoning_tokens":3143,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T14:29:41.523787+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit computation in a simple example such as the standard sphere showing that the numbers change under a Hamiltonian isotopy or violate the open WDVV relations would falsify the construction.","supporting_citations":[],"review_version":1}