{"id":"b8063190-3ee8-4a9b-9222-0731c2bf930c","arxiv_id":"2411.12114","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In R^{4n+1} with n > 1, there exist pairs of non-isotopic loose Legendrian spheres with Lagrangian concordances in both directions, so Lagrangian concordance is not a partial order.","lead":"This short note proves that in high dimensions, two different Legendrian spheres can be connected by Lagrangian concordances in both directions. This shows the Lagrangian concordance relation is not antisymmetric, so it cannot define a partial order.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The load-bearing step is Remark 5's unproven extension of the Eliashberg–Murphy h-principle from caps to cobordisms with a convex end; without it, Theorem 1 does not follow.","rationale":"I agree with the reader's weakest-assumption analysis. The central construction is otherwise built from established external theorems: Murphy's classification of loose Legendrians [15], Wu's smooth isotopy theorem [18], and the classical Thurston–Bennequin formula of Ekholm–Etnyre–Sullivan [9]. I considered whether the use of Murphy's classification to obtain two non-isotopic loose Legendrians with matching classical invariants could itself be the weak point, but that is a published external result; the paper's Remark 6 also discusses the parity restriction, which is consistent with the literature. I also considered the smooth concordance step, but Wu's theorem gives an isotopy, hence a product concordance, and the author's note about the complexified tangent bundle addresses the formal input. The only step that is both load-bearing and unsupported is the adaptation in Remark 5: the cited h-principle is for caps, and the paper does not supply the relative proof or a reference for the two-ended version. Because this is a missing verification rather than a demonstrated contradiction, the appropriate verdict remains conditional: the paper should be accepted if and only if the h-principle adaptation in Remark 5 is proved or replaced by a precise published statement.","tokens_in":4539,"tokens_out":12691,"duration_ms":147580,"concrete_test":"Re-derive the cobordism version of [11, Theorem 2.2] by following the proof of [11, Theorems 2.2 and 2.3] with a second boundary component at the positive end. Specifically, check whether the h-principle's C^0-small isotopy can be chosen to fix the positive Legendrian boundary pointwise, or at least to preserve its Legendrian isotopy class, and identify the first step in the proof that uses the domain being a disk rather than a general cobordism. If the proof does not carry over, Remark 5 is false as stated and Theorem 1 is unsupported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1 depends on upgrading the smooth concordances supplied by Wu's theorem into exact Lagrangian concordances. The upgrade is made by applying Eliashberg–Murphy [11, Theorem 2.2], but that theorem is explicitly stated for exact Lagrangian caps with a loose concave end. Remark 5 asserts, without proof, that the same h-principle holds for exact Lagrangian cobordisms with a loose concave end and a possibly non-trivial convex end, because the homotopies and isotopies in [11, Theorems 2.2 and 2.3] are compactly supported. This is not a formal consequence as stated: one must show that the relative h-principle with two boundary components is covered by the proof, and that the positive end is fixed as a genuine Legendrian embedding rather than merely a formal one. If this adaptation fails, there is no known mechanism turning the smooth concordances from Λ− to Λ+ and back into the required Lagrangian concordances, so the antisymmetry violation and the partial-order conclusion collapse.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that Lagrangian concordance with connected Legendrian ends is not a partial order on closed, connected Legendrian submanifolds of R^{4n+1}_{st} for n > 1, and that the same holds for the relation given by exact Lagrangian cobordisms in R^{2n+1}_{st}. The proof constructs, for a fixed stably parallelizable simply connected manifold such as S^{2n}, two loose Legendrian embeddings Λ_- and Λ_+ that are not Legendrian isotopic, uses Wu's theorem to obtain smooth concordances between them in both directions, and then invokes an h-principle of Eliashberg–Murphy to upgrade these smooth concordances to exact Lagrangian concordances. The paper also sketches a construction of non-antisymmetric exact Lagrangian cobordisms using previously known examples and the front/spherical spinning construction.","tokens_in":4720,"tokens_out":10844,"duration_ms":115798,"significance":"If the h-principle step is fully justified, the result is a concise and appealing resolution of a natural open question in high-dimensional contact topology, and the auxiliary exact-cobordism statement provides additional context. The proof strategy is coherent and builds on major recent developments (Murphy's classification of loose Legendrians and Eliashberg–Murphy h-principles). However, the central upgrade from smooth to Lagrangian concordances rests on an unproved extension of the cap theorem, so the current manuscript does not yet establish the main theorem with complete rigor.","major_comments":[{"comment":"The h-principle extension asserted in Remark 5 is the load-bearing step of the proof: it is the only mechanism that turns the smooth concordances LC∞± and LC∞∓ from Wu's theorem into Lagrangian concordances. [11, Theorem 2.2] is stated for exact Lagrangian caps with a loose concave end and no other boundary, whereas the present proof needs the analogous statement for a cobordism with two boundary components, with the convex end fixed as a genuine Legendrian embedding. The compact-supportedness of the homotopies in [11, Theorems 2.2 and 2.3] does not, by itself, imply the relative h-principle with a non-trivial convex end; one must show that the formal solution can be chosen so that its boundary value at the positive end is the prescribed Legendrian embedding, and that the h-principle can be run relative to both ends. As written, the paper gives no proof of this adaptation and no reference for it, so Theorem 1 does not follow from the cited results.","section":"§2, Remark 5"},{"comment":"Even granting the extension in Remark 5, the paper does not verify the formal Lagrangian data required to apply the h-principle to the smooth concordance. The sentence 'the complexified tangent bundle of a concordance over S2n is trivial' does not by itself establish the existence of a Lagrangian monomorphism TL → T(R×R^{4n+1}) covering the embedding, homotopic to the differential, and restricting to the Legendrian differentials at the two ends. The author should either spell out this formal data (for example, by exhibiting a Lagrangian subbundle and a homotopy) or cite the standard h-principle that guarantees it; otherwise the application of the h-principle is incomplete.","section":"§2, paragraph beginning 'For simplicity'"}],"minor_comments":[{"comment":"The sentence 'From [15, Proposition A.4 (c)] it follows that for a fixed rotation class, there is exactly one such couple Λ−, Λ+ up to Legendrian isotopy' is confusingly phrased; Remark 6 clarifies that for even k > 2 there are two Legendrian non-isotopic embeddings with the same classical invariants. Please rephrase to avoid implying uniqueness of the pair.","section":"§2, paragraph after Remark 4"},{"comment":"There is a typo 'Note that the the proof' with a doubled article.","section":"§2, Remark 6"},{"comment":"The arXiv number in reference [16] appears as '22105.02390'; this is likely a typo and should read '2210.02390' or another correct identifier.","section":"References"},{"comment":"The notation 'Si1 × · · ·×Sik × S2' is not defined; please clarify that Si denotes a sphere of dimension i, and explain the role of the indices i1,...,ik.","section":"§3, last paragraph"}],"recommendation":"major_revision","confidential_remarks":"The paper is a short note with a clear and promising idea, and I expect that the main theorem is true. The critical issue is the unproved h-principle extension in Remark 5; the compact-support argument is not a proof of the relative form needed here. I would advise the editor to require the author either to provide a complete proof of the adaptation or to cite a published theorem that covers exact Lagrangian cobordisms with loose concave ends and prescribed convex ends. If that gap cannot be closed, the manuscript should not be accepted in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is the first high-dimensional example of Lagrangian concordances violating antisymmetry, and the argument is probably correct — but the load-bearing step is asserted, not proved. I would send it to a serious referee, and I would expect that referee to demand a proof of Remark 5 before accepting.\n\nWhat's new and good: [8] proved non-antisymmetry for Legendrian knots in R^3; this paper extends the phenomenon to all contact dimensions R^{4n+1} with n>1, using closed connected loose Legendrians such as S^{2n}. The strategy is concise and transparent: Murphy's classification produces two loose, non-isotopic Legendrians with identical rotation class and Thurston-Bennequin number; Wu's embedding theorem gives smooth concordances in both directions; and the Eliashberg-Murphy h-principle is supposed to upgrade those to exact Lagrangian concordances. Section 3 handles exact Lagrangian cobordisms with a separate, simpler topological argument via front spinning; that part reads correctly.\n\nThe soft spot is Remark 5. Eliashberg-Murphy's Theorem 2.2 is stated for exact Lagrangian caps with a loose concave end. The paper applies it to a cobordism with a loose concave end and a possibly non-trivial convex end, and justifies this by saying the homotopies and isotopies in [11, Theorems 2.2 and 2.3] are compactly supported. That is not a proof. The relative h-principle with two boundary components may well hold, but one needs to show the positive end is fixed as an honest Legendrian embedding and that the formal data at both ends is compatible. Since the upgrade from smooth to Lagrangian concordances is the whole content of Theorem 1, this gap is load-bearing. I do not think the theorem is false; I would bet it is true. But as written, the paper is incomplete.\n\nOther observations: the use of Murphy's Proposition A.4(c) is terse, though Remark 6 helps. The citation pattern is reasonable; the author's own prior work is used for exactly the nuggets it proves. The exposition is clear and the note is short, which is a virtue.\n\nWho this is for: symplectic topologists working on Lagrangian cobordisms and Legendrian classification. The statement is significant enough that the gap is worth trying to close. My recommendation: send it to peer review, and make the referee check Remark 5 carefully. If the adaptation is correct, the paper is publishable essentially as is; if not, it should be rewritten as a conditional result or a conjecture.","headline":"A plausible and important high-dimensional counterexample to the partial-order question, but the key h-principle step is asserted rather than proved.","tokens_in":5268,"tokens_out":2994,"would_cite":false,"duration_ms":32321,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D12","53D42"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that Lagrangian concordance is not a partial order in high dimensions: non-isotopic Legendrian submanifolds can be Lagrangian concordant in both directions.","keywords":["Lagrangian concordance","Legendrian submanifold","partial order","loose Legendrian","h-principle","exact Lagrangian cobordism","front spinning","anti-symmetry"],"falsifier":"Construct a smooth concordance with loose Legendrian ends that satisfies the formal conditions of the flexibility theorem but provably cannot be deformed to an exact Lagrangian concordance; such an example would invalidate the adaptation in Remark 5 and collapse the proof of Theorem 1.","tokens_in":4311,"feed_emoji":"🔄","tokens_out":9909,"duration_ms":90755,"temperature":0.7,"pith_summary":"This paper aims to establish that, in high dimensions, the relation \"there is a Lagrangian concordance from one closed Legendrian submanifold to another\" is not a partial order. The author constructs, for each $n>1$, a pair of closed, connected Legendrian submanifolds $\\Lambda_-$ and $\\Lambda_+$ of the standard contact vector space $\\mathbb{R}^{4n+1}_{st}$ that are not Legendrian isotopic, yet admit Lagrangian concordances in both directions. Mutual comparability without isotopy violates antisymmetry, so the concordance relation cannot be a partial order in those dimensions. The same conclusion is drawn for exact Lagrangian cobordisms with connected Legendrian ends in $\\mathbb{R}^{2n+1}_{st}$, using examples whose ends are not even diffeomorphic. The result matters because Lagrangian concordance has been a candidate geometric notion of ordering Legendrian submanifolds by complexity.","feed_headline":"In high dimensions, Lagrangian concordance is not a partial order","feed_subtitle":"Non-isotopic Legendrian submanifolds can be Lagrangian concordant in both directions, so anti-symmetry fails.","key_machinery":"The machinery has three parts. First, a classification of loose Legendrian embeddings says that in $\\mathbb{R}^{4n+1}$ with $n>1$, for a fixed manifold and rotation class there are exactly two non-isotopic loose Legendrian embeddings up to Legendrian isotopy; this produces the two ends. Second, the Thurston--Bennequin formula shows the two ends share this classical invariant, so they cannot be distinguished by it. Third, the h-principle for exact Lagrangian embeddings with loose concave ends, extended from caps to cobordisms with possibly non-trivial convex ends, converts the smooth concordances supplied by the smooth embedding theorem into exact Lagrangian concordances. Loose ends are what make the h-principle applicable: the formal data can be genuinely realized because the concave ends are loose.","core_discovery":"The central claim is Theorem 1: for every $n>1$, there exists a pair of closed, connected Legendrian submanifolds $\\Lambda_-$ and $\\Lambda_+$ of the standard contact vector space $\\mathbb{R}^{4n+1}_{st}$ that are not Legendrian isotopic, but for which there are Lagrangian concordances $L_\\pm$ from $\\Lambda_-$ to $\\Lambda_+$ and $L_\\mp$ from $\\Lambda_+$ to $\\Lambda_-$. Since antisymmetry would force the two ends to be isotopic, the existence of such a pair implies that the Lagrangian concordance relation is not a partial order on closed, connected Legendrian submanifolds (Corollary 2). The construction takes $\\Lambda$ to be any closed, stably parallelizable, simply connected manifold, for instance $S^{2n}$, chooses two loose Legendrian embeddings of $\\Lambda$ with the same rotation class that are not Legendrian isotopic, and then uses a classical smooth embedding theorem to obtain smooth concordances in both directions. The h-principle for exact Lagrangian embeddings with loose concave ends is invoked to upgrade these smooth concordances to genuine Lagrangian concordances. The final section repeats the same conclusion for exact Lagrangian cobordisms with connected ends, using a pair of cobordisms between a Legendrian $S^2$ and a Legendrian $T^2$, extended to higher dimensions by front spinning; here the ends are not even diffeomorphic.","pith_inferences":["A natural extension, not stated in the paper, is that among loose Legendrians in even dimensions the Lagrangian concordance relation may be as flexible as smooth concordance: any smooth concordance with loose ends might be upgradeable to a Lagrangian one.","The paper's method does not directly cover odd-dimensional contact vector spaces, since the classification it relies on gives a unique loose embedding per formal class there; whether two-way Lagrangian concordances exist in those dimensions remains open.","For exact Lagrangian cobordisms, the two-way examples connect non-isotopic and even non-diffeomorphic ends, suggesting that any partial-order structure in that setting would need to compare different topologies rather than merely Legendrian isotopy classes."],"forward_implications":["Corollary 2 follows directly: Lagrangian concordances with connected Legendrian ends do not define a partial order on closed, connected Legendrian submanifolds of $\\mathbb{R}^{4n+1}_{st}$ for $n>1$.","In the exact Lagrangian cobordism setting, anti-symmetry fails in $\\mathbb{R}^{2n+1}_{st}$ for all $n>1$; the constructed cobordisms connect Legendrian ends with different diffeomorphism types, such as $S^2$ and $T^2$ after spinning.","The two ends $\\Lambda_-$ and $\\Lambda_+$ have the same Thurston--Bennequin number and rotation class, and both have acyclic Legendrian contact homology, so those invariants cannot prevent two-way Lagrangian concordance.","The non-partial-order phenomenon occurs even among closed, connected, loose Legendrian submanifolds, not merely in non-compact or disconnected settings."],"supporting_citations":[{"why":"This classification supplies the two non-isotopic loose Legendrian embeddings with the same rotation class that are used as the ends $\\Lambda_-$ and $\\Lambda_+$.","marker":"[15, Proposition A.4]"},{"why":"This h-principle is the tool that upgrades the smooth concordances to exact Lagrangian concordances, after the adaptation stated in Remark 5.","marker":"[11, Theorem 2.2]"},{"why":"This classical result gives smooth isotopy of embeddings of a connected $2n$-manifold into $\\mathbb{R}^{4n+1}$, producing the smooth concordances needed as input.","marker":"[18]"},{"why":"This formula shows the Thurston--Bennequin number is a topological invariant, so the two ends have matching classical invariants.","marker":"[9, Proposition 3.2]"},{"why":"This supplies the exact Lagrangian endocobordisms between $S^2$ and $T^2$ that are used to violate anti-symmetry for exact cobordisms.","marker":"[7, Section 2.3]"},{"why":"This provides the front spinning construction that extends the $S^2/T^2$ example to all higher dimensions.","marker":"[14]"}],"fun_headline_variants":["Two-way Lagrangian concordances break partial order in high dimensions","High-dimensional Lagrangians: concordance both ways despite non-isotopy","Concordance without isotopy: anti-symmetry fails in high dimensions","Two-way concordance between non-isotopic Legendrians: anti-symmetry fails"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on an unproven adaptation of a flexibility theorem: the result for exact Lagrangian caps with loose concave ends is assumed to hold for cobordisms with a possibly non-trivial convex end, on the grounds that the relevant homotopies and isotopies are compactly supported.","fun_headline_variants_meta":{"raw":{"variants":["Two-way Lagrangian concordances break partial order in high dimensions","High-dimensional Lagrangians: concordance both ways despite non-isotopy","Concordance without isotopy: anti-symmetry fails in high dimensions","Two-way concordance between non-isotopic Legendrians: anti-symmetry fails"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00121,"raw_usage":{"total_tokens":5004,"prompt_tokens":986,"completion_tokens":4018,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":602,"completion_tokens_details":{"reasoning_tokens":3937}},"tokens_in":602,"tokens_out":4018,"duration_ms":29658,"temperature":1.0,"reasoning_tokens":3937,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:53:37.648489+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a smooth concordance with loose Legendrian ends that satisfies the formal conditions of the flexibility theorem but provably cannot be deformed to an exact Lagrangian concordance; such an example would invalidate the adaptation in Remark 5 and collapse the proof of Theorem 1.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This classical result gives smooth isotopy of embeddings of a connected $2n$-manifold into $\\mathbb{R}^{4n+1}$, producing the smooth concordances needed as input."},{"cited_title":"Golovko, A note on the front spinning construction , Bulletin of the London Mathematical Society, 46 (2014), no","cited_arxiv_id":null,"evidence_quote":"This provides the front spinning construction that extends the $S^2/T^2$ example to all higher dimensions."}],"review_version":1}