{"id":"d51106c6-fc11-48ba-ab33-0079bf16e281","arxiv_id":"2507.04897","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Symplectic neighbourhoods of strongly coisotropic stratified subspaces are classified, up to symplectomorphism, by the stratified diffeomorphism and the induced Zariski form.","lead":"The paper proves that the symplectic neighbourhood of a broad class of singular subspaces, called stratified subspaces with conical fibers, is unique up to symplectomorphism once the subspace and the restricted form are fixed. This generalizes Weinstein's Lagrangian neighbourhood theorem and gives local models for singular Lagrangian configurations used in constructions of exotic tori.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 5.18 applies the strong-coisotropic condition without checking that the stratum is non-closed; at a closed stratum the asserted inclusion can fail, though the missing case is patchable via vanishing of the primitive.","rationale":"The reader's weakest assumption identifies the deformation-retraction result imported from [29] as the main risk, but that input is used in the proof of Theorem B, not in the proof of the central Theorem A. Theorem A constructs its own local deformation retractions from Euler-like vector fields in Section 5.2.2, and the proof of Theorem A does not invoke the global retraction from [29]. The more central soft spot is Proposition 5.18, which the reader also noticed in the rationale but did not make the primary concern. There, the strong-coisotropic condition is applied without separating the case where the stratum containing p0 is closed. Since U^{X0}_0 avoids lower-dimensional strata, the only closed stratum that can occur is X0 itself, and on X0 the primitive β vanishes, forcing X_t = 0. So the gap is real but patchable. I also noted that Proposition 5.19's statement that g_t fixes W^{X0}_0 appears to conflict with its proof, which yields fixity on W^{≤(d-1)}_0; this looks like a typo rather than a structural flaw. Overall, the paper's main construction is coherent and the conditional verdict remains appropriate; no fatal objection is established.","tokens_in":35038,"tokens_out":29394,"duration_ms":332170,"concrete_test":"Analytic check: take a closed stratum X0, for example the origin stratum in Example 2.32, and run the construction of Proposition 5.18 at a point p0 ∈ X0. Verify that Proposition 5.13(2) gives β_{p0} = 0, hence X_t(p0) = 0, so the invariance of A0 ∩ U^{X0}_0 follows without the strong-coisotropic inclusion. If this verification succeeds, the closed-stratum case is benign; if it fails, the proof of Theorem A breaks on the minimal stratum.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem A, Proposition 5.18 shows that the Moser vector field X_t satisfies X_t(p0) ∈ (T^Z_{p0} A0)^{ω0} and then invokes the strongly coisotropic assumption to conclude X_t(p0) ∈ T_{p0}Y, where Y is the stratum containing p0. Definition 2.30 only guarantees this inclusion for non-closed strata. In the inductive construction, U^{X0}_0 is chosen inside M^{≥d}, so the only closed stratum that can meet U^{X0}_0 ∩ A0 is X0 itself. For p0 ∈ X0, Proposition 5.13(2) gives β_{p0} = 0, and since ω_t is nondegenerate, X_t(p0) = 0, so the desired tangency holds trivially. Thus the proof as written has a genuine gap in the closed-stratum case, but it is benign: a two-line case split repairs it. This step is load-bearing because invariance of A0 under the Moser flow is exactly what makes Gω restrict to a stratified diffeomorphism; without it, the induction in Theorem A cannot proceed. The gap does not appear fatal, but it should be fixed in the final version.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves a stratified analogue of Weinstein's neighbourhood theorem. For smoothly locally trivial conical stratified subspaces A_i of symplectic manifolds, Theorem A states that if the A_i are strongly coisotropic and a stratified diffeomorphism g preserves the restricted Zariski 2-form, then symplectic neighbourhoods are symplectomorphic, with the restriction to A_0 isotopic to g through form-preserving stratified diffeomorphisms. Theorem B is a more general version assuming a locally extendable symplectic bundle isomorphism, and Theorems C and D are a non-symplectic tubular neighbourhood theorem and a Moser-type result with weak deformation retractions. The paper also proves exactness of the symplectic form near isotropic stratified subspaces and discusses applications to Lagrangian pinwheels, mutation configurations, and exotic tori.","tokens_in":35183,"tokens_out":15970,"duration_ms":174340,"significance":"If correct, this result unifies and generalizes several ad hoc normal form theorems in symplectic topology and gives a practical criterion: a stratified diffeomorphism preserving the Zariski form determines the symplectic neighbourhood. The paper is careful and unusually self-contained in places: the recursion in Section 5 is explicit, Theorem 3.7 on extending local bundle isomorphisms is a useful tool, and the appendices on Euler-like vector fields and on the strong deformation retraction case are valuable. The main caveats are the unproved closed-stratum case in Proposition 5.18 and the unstated dependence of Theorem B on a retraction result imported from [29]. Neither appears fatal, but both are load-bearing and need to be fixed before the proofs are complete.","major_comments":[{"comment":"The step 'By the strongly coisotropic assumption on (A_0,S_0), X_t(p0)∈(T^Z_{p0}A_0)^{ω0}⊂T_{p0}Y' applies Eq. (2.1) of Definition 2.30, but that inclusion is only asserted for non-closed strata. In the inductive setting, U^{X_0}_0 is chosen inside M^{≥d}, so the only closed stratum of A_0 that can meet U^{X_0}_0∩A_0 is X_0 itself. For p_0∈X_0, Proposition 5.13(2) gives β_{p_0}=0, and since ω_t is nondegenerate, X_t(p_0)=0, so the desired tangency holds trivially. This is a genuine but local gap: the proof as written is incomplete for closed strata, and the missing case must be added explicitly. The same correction is needed in the proof of Proposition 5.19, which relies on Proposition 5.18 to show that the intermediate flows preserve strata.","section":"Section 5.2.3, Proposition 5.18"},{"comment":"The statement 'by the results of [29], there exists a neighbourhood V_0 of A_0 in U_0 and a smooth weak deformation retraction of V_0 to A_0' is load-bearing: it is exactly what allows Theorem D to be applied, and hence what produces the symplectomorphism in Theorem B. The paper does not state which result in [29] is being used, nor does it verify its hypotheses here. Please state the precise theorem that supplies this retraction and confirm that every smoothly locally trivial conical stratified subspace satisfies it; alternatively, prove the needed retraction statement in the present paper, for example using the Euler-like vector field machinery of Appendix A. Without this, the proof of Theorem B is not self-contained at a central point.","section":"Section 4.4, proof of Theorem B"}],"minor_comments":[{"comment":"Several cross-references call definitions and notations 'theorems', for example 'stratified subspaces (Theorem 2.14)', 'Theorem 2.22', and 'Theorem 2.29' in Section 1.2, and 'Recall Theorem 2.24' in Section 5.1. The final version should correct these to Definition or Notation as appropriate.","section":"Throughout"},{"comment":"The display in case (3) contains V^{X^d_0}_i but should be V^{X^d_i}_i; as written the notation is undefined for i=1.","section":"Proposition 5.2(3)"},{"comment":"The set denoted W^{X_0}_0 is almost certainly meant to be W^{≤(d-1)}_0∩U^{X_0}_0∩A_0: the proof cites Theorem 5.17, which concerns W^{≤(d-1)}_0, and the subsequent use in Definition 5.1(4)(b) is about W^{≤(d-1)}_0.","section":"Proposition 5.19(2)"},{"comment":"There are several typographical issues: 'manfiold' in Definitions 2.28 and 2.30; 'qucik into' in Section 5.2.2; 'Theorem 4.1' in Section 6 should be Definition 4.1; and Theorem D says the forms agree on 'T M|_A' where the tangent bundle of V is intended. These are presentation issues but should be cleaned up.","section":"Typos and notation"}],"recommendation":"major_revision","confidential_remarks":"This is a strong paper and the central theorem appears to be correct. My recommendation of major revision is driven by the requirement that all load-bearing proof steps be fully written: the closed-stratum gap in Proposition 5.18 is patchable, but it is not just a stylistic issue, and the dependence of Theorem B on an unspecified result in [29] should be made explicit or removed. I do not see a fundamental obstruction, and I would expect the paper to be publishable once these points are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, read this one. The paper proves a genuine generalization of Weinstein's neighbourhood theorem to stratified subspaces that are smoothly locally trivial with conical fibers. Theorem A is the headline: for strongly coisotropic stratified subspaces, the symplectic neighbourhood is determined by the stratified diffeomorphism type plus the restricted Zariski form. That's new, and it subsumes the scattered normal form results for pinwheels, mutation configurations, and arboreal buildings. The application to Brendel-Hauber-Schmitz tori is a concrete payoff.\n\nThe proofs are long but mostly self-contained, and the differential-space machinery is handled cleanly. Theorem D, a Moser trick for weak deformation retractions, is useful beyond the main application. I checked the recursion in Section 5: it is coherent, and the construction of the extendable diffeomorphism around each stratum is sensible.\n\nTwo soft spots, both minor. First, Proposition 5.18 invokes the strong coisotropic condition for the stratum containing p0 without separating the closed-stratum case. Definition 2.30 only gives the inclusion for non-closed strata. In the minimal stratum X0, beta vanishes, so the Moser field vanishes and the conclusion is trivial; higher closed strata can be avoided by shrinking the neighbourhood. So the proof has a genuine but easily fixable gap. Second, Theorem B depends on [29] for the existence of smooth weak deformation retractions of conical stratified subspaces. [29] is published and authored by one of the same people, so it is not circular, but the paper should state the needed theorem explicitly rather than just citing it.\n\nOverall, the central argument holds up. This deserves a serious referee; I would accept it for peer review with requests to fix the case split and make the [29] statement precise. It will be of real use to people working with Lagrangian singularities and exotic tori.","headline":"Real result with a small fixable gap in one proposition; worth refereeing and probably citing.","tokens_in":35788,"tokens_out":4283,"would_cite":true,"duration_ms":46889,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D05","58A35","53D12","58A40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a symplectic neighbourhood of a strongly coisotropic stratified subspace is determined by its stratified diffeomorphism type and the restricted Zariski form.","keywords":["Weinstein neighbourhood theorem","stratified subspaces","coisotropic subspaces","Zariski tangent space","Moser's trick","symplectic normal form","singular symplectic geometry","Lagrangian singularities"],"falsifier":"A concrete falsifier would be a pair of strongly coisotropic stratified subspaces satisfying all hypotheses of Theorem A—same stratified diffeomorphism type and same Zariski form—yet having no symplectomorphic neighbourhoods. A sharper test: find a smoothly locally trivial conical stratified subspace with no smooth weak deformation retraction of any neighbourhood; then the key step in the proof of Theorem B (Section 4.4) has no mechanism to run Moser's argument, so the stated proof cannot establish the normal form.","tokens_in":34750,"feed_emoji":"🧩","tokens_out":9307,"duration_ms":95719,"temperature":0.7,"pith_summary":"This paper proves a singular analogue of Weinstein's neighbourhood theorem. It establishes that a symplectic neighbourhood of a strongly coisotropic stratified subspace—a closed union of smooth strata with conical, smoothly locally trivial singularities—is determined, up to symplectomorphism, by the stratified diffeomorphism type of the subspace together with the pullback of the symplectic form to its Zariski tangent spaces, the tangent data encoded by derivations at points. If this is right, local models for singular subspaces such as Lagrangian pinwheels, mutation configurations, and momentum-map zero sets are unique, which is the kind of input used to construct and distinguish exotic Lagrangians. The paper also proves a strong version of Moser's trick and a tubular neighbourhood theorem for these stratified subspaces.","feed_headline":"Coisotropic singular subspaces get unique symplectic neighbourhoods","feed_subtitle":"Extends the classical Lagrangian neighbourhood theorem to stratified subspaces with conical strata, with applications to exotic Lagrangians.","key_machinery":"The machinery is the class of stratified subspaces that are smoothly locally trivial with conical fibres, studied through their Zariski tangent spaces and Zariski forms. Its load-bearing components are: the relative Poincaré lemma by fiber integration along a smooth weak deformation retraction; the resulting strong Moser trick (Theorem D); Euler-like vector fields and their induced tubular neighbourhoods, chosen to be tangent to higher strata; and the concept of local extendability of a tangent-bundle isomorphism, which is the precise way a derivative-like condition is enforced at singularities. The strongly coisotropic condition (the symplectic annihilator of the Zariski tangent is contained in the tangent space of the stratum) forces the Moser vector field to point along strata, so the flow preserves the stratification.","core_discovery":"The central discovery is Theorem A: for two strongly coisotropic stratified subspaces that are smoothly locally trivial with conical fibres, any stratified diffeomorphism preserving the symplectic form as a Zariski form extends, possibly after an isotopy through such diffeomorphisms, to a symplectomorphism of neighbourhoods. The same statement holds without strong coisotropicity when a symplectic tangent bundle isomorphism over the diffeomorphism is locally extendable (Theorem B), a hypothesis that reduces to the classical derivative condition when the strata are smooth submanifolds. The proof works stratum by stratum, producing a symplectomorphism on lower strata, then extending to the next stratum using a tubular neighbourhood whose radial scaling preserves the stratification and a Moser trick that adjusts the symplectic form; the strong coisotropic condition is what keeps the Moser vector field tangent to the strata.","pith_inferences":["Editorial inference: if Theorem A is correct, the uniqueness of symplectic neighbourhoods should make 'almost toric' local manipulations (node slides, Lagrangian mutations) valid in any ambient symplectic manifold once the singular subspace is symplectically embedded, not only inside the original model spaces.","Editorial inference: the isotopy clause in Theorem A means the classifying data are not just the stratified diffeomorphism type and the Zariski form but also a path-component of the space of stratified diffeomorphisms preserving that form; counting such components could be relevant to distinguishing exotic Lagrangians.","Editorial inference: a natural testable extension is to check strong coisotropicity for arboreal c-buildings; the paper leaves that open, and it would decide whether the normal form applies to that family directly."],"forward_implications":["Any Lagrangian stratified subspace that is strongly coisotropic and embeds in a symplectic manifold has a neighbourhood determined up to symplectomorphism by its stratified diffeomorphism type and the induced Zariski form, extending the classical Lagrangian neighbourhood theorem to the singular setting.","Zero level sets of momentum maps for compact group actions and components of the critical set of the norm-squared momentum map gain local symplectic models, since their stratifications satisfy the conical local-triviality condition.","The paper's model-space application goes through under the weaker hypothesis that the skeleton embeds as a Lagrangian stratified subspace, not the full model space, because the normal form supplies the missing symplectic neighbourhood.","Two symplectic forms that agree on a stratified subspace are symplectomorphic on a neighbourhood whenever that subspace admits a smooth weak deformation retraction (Theorem D).","Near an isotropic stratified subspace, the symplectic form is exact with an explicit fiber-integration primitive (Theorem 6.1), so the normal-form neighbourhoods are candidates for Liouville and Weinstein structures."],"supporting_citations":[{"why":"Supplies the smoothly locally trivial conical stratified subspaces and the smooth weak deformation retraction from a neighbourhood that the proof of Theorem B imports.","marker":"[29]"},{"why":"The classical Lagrangian neighbourhood theorem that the paper generalises and whose cotangent model motivates the normal form.","marker":"[27]"},{"why":"The coisotropic normal form theorem that Theorem A recovers up to isotopy when the stratified subspaces are coisotropic submanifolds.","marker":"[10]"},{"why":"Weinstein's lecture normal form via tubular neighbourhoods and Moser trick that Theorem B generalises.","marker":"[28]"},{"why":"The Moser trick that Theorem D adapts to symplectic forms that agree only along a stratified subspace.","marker":"[22]"},{"why":"Supplies the Zariski tangent spaces, Zariski k-forms, and pullback properties that the main hypotheses are phrased in.","marker":"[25]"},{"why":"Provides Euler-like vector fields and induced tubular neighbourhood embeddings used to build the stratum-wise extension.","marker":"[20]"},{"why":"Supplies the fiber-integration homotopy identity underlying the relative Poincaré lemma and the explicit primitive.","marker":"[15]"}],"fun_headline_variants":["Unique symplectic neighbourhoods for stratified subspaces","Coisotropic strata: unique symplectic neighbourhoods","Moser trick proves uniqueness for singular symplectic spaces","Exotic Lagrangians: stratified neighbourhood uniqueness"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every smoothly locally trivial conical stratified subspace admits a smooth weak deformation retraction of a neighbourhood onto itself; the paper imports this from the cited reference [29] rather than proving it here, and the Moser trick of Theorem D depends on it.","fun_headline_variants_meta":{"raw":{"variants":["Unique symplectic neighbourhoods for stratified subspaces","Coisotropic strata: unique symplectic neighbourhoods","Moser trick proves uniqueness for singular symplectic spaces","Exotic Lagrangians: stratified neighbourhood uniqueness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000533,"raw_usage":{"total_tokens":2478,"prompt_tokens":770,"completion_tokens":1708,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":386,"completion_tokens_details":{"reasoning_tokens":1644}},"tokens_in":386,"tokens_out":1708,"duration_ms":12482,"temperature":1.0,"reasoning_tokens":1644,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:37:13.893284+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete falsifier would be a pair of strongly coisotropic stratified subspaces satisfying all hypotheses of Theorem A—same stratified diffeomorphism type and same Zariski form—yet having no symplectomorphic neighbourhoods. A sharper test: find a smoothly locally trivial conical stratified subspace with no smooth weak deformation retraction of any neighbourhood; then the key step in the proof of Theorem B (Section 4.4) has no mechanism to run Moser's argument, so the stated proof cannot establish the normal form.","supporting_citations":[{"cited_title":"Commutative control data for smoothly locally trivial stratified spaces.Transformation Groups, 2024","cited_arxiv_id":null,"evidence_quote":"Supplies the smoothly locally trivial conical stratified subspaces and the smooth weak deformation retraction from a neighbourhood that the proof of Theorem B imports."},{"cited_title":"Symplectic manifolds and their lagrangian submanifolds.Advances in Mathematics, 6(3):329–346, 1971","cited_arxiv_id":null,"evidence_quote":"The classical Lagrangian neighbourhood theorem that the paper generalises and whose cotangent model motivates the normal form."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The coisotropic normal form theorem that Theorem A recovers up to isotopy when the stratified subspaces are coisotropic submanifolds."},{"cited_title":"29 ofRegional Conference Series in Mathematics","cited_arxiv_id":null,"evidence_quote":"Weinstein's lecture normal form via tubular neighbourhoods and Moser trick that Theorem B generalises."},{"cited_title":"On the volume elements on a manifold.Transactions of the American Mathematical Society, 120(2):286–294, 1965","cited_arxiv_id":null,"evidence_quote":"The Moser trick that Theorem D adapts to symplectic forms that agree only along a stratified subspace."},{"cited_title":"New Mathe- matical Monographs","cited_arxiv_id":null,"evidence_quote":"Supplies the Zariski tangent spaces, Zariski k-forms, and pullback properties that the main hypotheses are phrased in."},{"cited_title":"Euler-like vector fields, normal forms, and isotropic embeddings.Indagationes Mathematicae, 32(1):224–245, 2021","cited_arxiv_id":null,"evidence_quote":"Provides Euler-like vector fields and induced tubular neighbourhood embeddings used to build the stratum-wise extension."},{"cited_title":"Lee.Introduction to Smooth Manifolds","cited_arxiv_id":null,"evidence_quote":"Supplies the fiber-integration homotopy identity underlying the relative Poincaré lemma and the explicit primitive."}],"review_version":1}