{"id":"2f727c24-96fd-4de8-97cd-22c1bd4342d5","arxiv_id":"2512.18808","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The k-fold Legendrian lift of a Lagrangian skeleton is a universal (1/k, 1/k)-interlinker in every contact prequantization bundle over a closed integral symplectic manifold.","lead":"Contact prequantization bundles built from any closed symplectic manifold contain explicit Legendrian barriers: removing the k-fold lift of a Lagrangian skeleton forces any Legendrian sweep longer than 1/k to self-intersect or hit the barrier. This extends the authors' S^3 barrier result to all integral symplectic manifolds and yields contact non-embedding corollaries.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 is proved only for skeleta of quasi-holomorphic sections; the quantification over arbitrary polarizations of degree k≥2 is unsupported, and the k=2 construction degenerates.","rationale":"The reader's weakest assumption is exactly the gap between Theorem 1's quantification over all polarizations and the proof's restriction to quasi-holomorphic sections. Section 8 silently narrows the hypothesis, and Section 5 openly treats the needed sections as existing only asymptotically for large k. This is load-bearing because the Hamiltonian-barrier conclusion is obtained from a specific Liouville/skeleton construction tied to s_ε, not from a general polarization. The k=2 degeneracy is a concrete, checkable instance of the same gap: the algebra in Sections 5–6 either has an undefined exponent or forces the critical set onto a level set of |σ|, so the claimed explicit skeleton is not produced. These are internal consistency issues rather than mere disagreement with the consensus, and they are not resolved by the paper's gesture that ε may be taken large. The overall conditional verdict remains appropriate: the dichotomy for k≥3, conditional on the section-reduction, is plausible and the computations are substantial, but the theorem as stated is not established. No change to the reader's conditional verdict is needed; the requested checks should be supplied before the statement is used in its full generality.","tokens_in":30550,"tokens_out":11381,"duration_ms":117653,"concrete_test":"For k=2, take a holomorphic section σ of L² with transverse zeros, choose a generic constant ε, and solve the critical-point equations derived from (5.3) inside SDB(N,τ)\\N_0. If the only interior critical points occur on the level set |σ|=ε and cosψ=1, then the critical set is not the union of stable manifolds used in Corollary 6.3, so Proposition 7.4's skeleton formula fails. Independently, recompute the Hessian in Lemma 5.4 at such a point and check whether the dR² eigenvalue is zero; a zero eigenvalue contradicts the claimed Morse index and shows the k=2 case needs a genuinely different argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 8 begins by taking Γ_k to be 'an isotropic skeleton associated to a quasi-holomorphic section of L^k', but Theorem 1 states the result for any isotropic skeleton associated to a polarization of degree k≥2 (Definition 3.4). The needed bridge — that every such polarization/skeleton is the zero-set skeleton of a κ<1 quasi-holomorphic section σ with ln|σ| Morse — is never proved. Section 5 explicitly acknowledges the limitation: 'Such sections might exist when k≫1 by Donaldson-Giroux theory … or for any other reason.' Donaldson–Giroux gives asymptotic existence for large k, not for all k≥2 and not for a prescribed skeleton. Since the proof of Proposition 7.4 and the subsequent Biran-decomposition step (via [Ops13]) rely on the explicit s_ε construction, the dichotomy of Theorem 1 is established only for section-associated skeleta, not for the polarizations named in the statement.\n\nMoreover, at k=2 the construction itself breaks down: the critical-radius formula contains the undefined exponent 2/(k−2), and Lemma 5.4's Hessian loses its dR² term because the coefficient −k(k−2) vanishes, so −ln|s_ε| is not Morse along the radial direction. The critical-point equation forces |σ(z)|=ε, so for generic ε there are no interior critical points; the claimed equality Skel(s_ε)=Φ^{-1}(Λ_k×(0,1)) cannot be obtained by this argument. Thus the statement as written is not proved even for k=2 quasi-holomorphic sections.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Legendrian barriers in contact prequantization bundles. The main result, Theorem 1, asserts that for a closed symplectic manifold (N,τ) with integral symplectic class, any isotropic skeleton Γ_k of a polarization of degree k≥2 admits a k-fold Legendrian lift Λ_k in the prequantization bundle (P,α), and this lift is a Legendrian barrier: for any closed Legendrian Λ and any smooth contact Hamiltonian H≥1, either there is an H-chord from Λ to Λ_k of length ≤1/k, or the 1/k-sweep of Λ is not embedded. The proof uses Mohnke's trick to convert an embedded long sweep into a Lagrangian submanifold in the negative symplectization, then identifies the negative symplectization with a symplectic disc bundle over N. The technical core constructs, from a quasi-holomorphic section σ of L^k, a Liouville form whose skeleton is exactly the preimage of Λ_k×(0,1), and then applies Biran's decomposition and a ruled-manifold capacity bound. The paper also proves a Lagrangian analogue (Theorem 3) bounding the Cieliebak–Mohnke capacity of complements of polarizations. The proof is largely computational and self-contained, but there is a significant gap between the statement of Theorem 1 and the hypotheses actually used in its proof.","tokens_in":30754,"tokens_out":4055,"duration_ms":39391,"significance":"If the result holds in the stated generality, it is a substantial contribution: it generalizes the Legendrian barrier phenomenon from S^3 (proved in [OS24]) to all prequantization bundles, with the sharp constant 1/k and an explicit, geometrically meaningful barrier Λ_k. The construction via quasi-holomorphic sections and the degenerate compactification of the symplectic disc bundle are useful tools that may find further applications. The paper also gives a clean proof of a Lagrangian barrier theorem (Theorem 3) and provides a detailed account of Mohnke's trick. However, the central dichotomy is proved only for skeleta associated to quasi-holomorphic sections, while the theorem claims all degree-k polarizations. This mismatch, together with an apparent degeneration of the construction at k=2, prevents the main theorem from being established as stated.","major_comments":[{"comment":"Theorem 1 quantifies over all isotropic skeleta associated to a polarization of degree k≥2 (Definition 3.4), but Section 8 begins with 'Let Γ_k ⊂ N be an isotropic skeleton associated to a quasi-holomorphic section of L^k → (N,τ).' No argument is given that every polarization degree k (especially k=2,3) admits a quasi-holomorphic section σ with transverse vanishing and ln|σ| Morse. Section 5 explicitly says such sections 'might exist when k≫1 by Donaldson-Giroux theory ... or for any other reason.' Since Proposition 7.4 and the subsequent Biran-decomposition step rely on the explicit s_ε construction, the proof covers only section-associated skeleta. The theorem should be restated in that restricted class, or a reduction from arbitrary polarizations to this class must be supplied.","section":"Theorem 1 vs. Section 8"},{"comment":"The critical-point analysis in Section 5 produces interior critical points at R = (1 + (ε/|σ(z)|)^{2/(k−2)})^{-1}. For k=2 this exponent is undefined; solving the critical-point equations forces |σ(z)|=ε, so for generic ε there are no interior critical points. Correspondingly, in Lemma 5.4 the Hessian term −k(k−2)ε^2 R^{k−3}/(2(1−R)) dR^2 vanishes, so −ln|s_ε| is not Morse in the radial direction, and Corollary 6.3 cannot identify stable manifolds as claimed. Lemma 6.5's later remark that k=2 can be handled by taking ε≫1 does not restore the missing critical points. Consequently Lemma 6.4 and Proposition 7.4 do not establish the skeleton equality Skel(s_ε)=Φ^{-1}(Λ_k×(0,1)) for k=2, so Theorem 1 is not proved in the stated k≥2 range.","section":"Section 5, k=2 degeneration"}],"minor_comments":[{"comment":"The text 'theomem 1' in the Organization paragraph is a typo for 'Theorem 1'. Also 'Aknowledgements' should be 'Acknowledgements'.","section":"Abstract/Organization"},{"comment":"The notation L^k and L^{⊗k} is used interchangeably across Sections 3.5, 4.4, and 5; this may confuse readers. Consistency is recommended.","section":"Notation"},{"comment":"In the proof of Lemma 3.7, the unexplained symbol κ′(κ) is introduced but never used; either remove it or clarify its role.","section":"Section 3.4"},{"comment":"'a s tuple' should read 'a tuple'. Also 'Poincaré-Dual' should be 'Poincaré dual' in several places.","section":"Definition 3.4"}],"recommendation":"major_revision","confidential_remarks":"The main issue is that Theorem 1 is stated for all degree-k polarizations but proved only for quasi-holomorphic section skeleta; the k=2 degeneration is a genuine obstruction that the current text does not repair. These are load-bearing but probably fixable by narrowing the statement or adding a separate k=2 argument. I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here’s my take. The paper does something real: it upgrades the Legendrian-barrier phenomenon from S^3 [OS24] to every contact prequantization bundle over a closed integral symplectic manifold, with the constant 1/k built into the statement. That is not in the existing literature. The proof is a serious piece of work: a k-fold Legendrian lift construction (Cor 3.6), a degenerate non-symplectic compactification SPB, and a long explicit computation of the Liouville skeleton. The central dichotomy is plausible, and for k≥3 the computations mostly check out.\n\nThe soft spots are concentrated in the statement-versus-proof gap. Theorem 1 quantifies over all polarizations of degree k≥2. The proof, however, starts with a κ-quasi-holomorphic section σ of L^k, with κ<1 and ln|σ| Morse. The paper itself says such sections 'might exist when k≫1 by Donaldson–Giroux theory.' No bridge is given from an arbitrary polarization to such a section, and for small k (especially k=2) existence is not covered. So the theorem as stated is not proved.\n\nThe k=2 case is not a minor technicality. The critical-point radius formula contains the exponent 2/(k−2), which is undefined at k=2, and Lemma 5.4's Hessian coefficient −k(k−2) vanishes, so the Morse-index argument collapses. The paper gestures at taking ε large when k=2, but that does not repair the critical-point analysis. Thus the statement is not proved for k=2 even for quasi-holomorphic sections. This is a real gap, not a nitpick.\n\nThere are also several load-bearing citations to the author's own prior work ([Ops13], [Ops15]) and to SFT compactness without verification in the degenerate non-symplectic setting. That may be acceptable for a research article, but the reader should push for clarity.\n\nCredit where due: the paper is honest about its assumptions — Section 5 explicitly flags the quasi-holomorphic existence issue. The algebraic core is written out in enough detail to be checked. The non-symplectic compactification is a clever move. If the polarization-to-section reduction is filled in (or the theorem statement narrowed) and the k=2 case is either repaired or excluded, the paper will be a strong contribution.\n\nWho is this for? Anyone working on contact rigidity, Legendrian barriers, or quantitative contact topology. It deserves a serious referee. I would send it to review, with the expectation of major revision.","headline":"Generalizes the S^3 Legendrian barrier to prequantization bundles, but the main theorem is proved only for quasi-holomorphic-section skeleta, not for all polarizations, and k=2 degenerates.","tokens_in":132,"tokens_out":2397,"would_cite":true,"duration_ms":53865,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D35","53D12","53D10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Prequantization bundles carry explicit Legendrian barriers: removing one makes long embedded contact flows impossible.","keywords":["Legendrian barrier","prequantization bundle","isotropic skeleton","contact Hamiltonian","Lagrangian barrier","symplectic disc bundle","quasi-holomorphic section","contact rigidity"],"falsifier":"A direct test: in the standard prequantization $S^{2n+1} \\to \\mathbb{CP}^n$ with the degree-$2$ skeleton made of half great circles, look for a closed Legendrian $\\Lambda$ and a contact Hamiltonian $H\\ge 1$ such that $\\Phi^t_H(\\Lambda)$ stays embedded for $t\\in [0,1/2]$ and avoids the two-point Legendrian lift $\\Lambda_2$. Finding one would disprove the $1/k$ bound. Alternatively, exhibiting a degree-$2$ polarization whose skeleton is not realizable by any $\\kappa < 1$ quasi-holomorphic section with non-degenerate norm logarithm would show the proof does not cover the theorem's stated hypotheses.","tokens_in":30247,"feed_emoji":"🛡️","tokens_out":9024,"duration_ms":84870,"temperature":0.7,"texified_at":"2026-08-05T20:44:58.914197+00:00","pith_summary":"This paper proves that every prequantization bundle—the contact circle bundle over a closed symplectic manifold with integral symplectic class—contains explicit Legendrian barriers. For a polarization of degree $k\\ge 2$, the barrier is the minimal Legendrian lift of the polarization's isotropic skeleton: it meets each fiber over the skeleton in $k$ equally spaced points. The main theorem says that any closed Legendrian submanifold moved by any contact Hamiltonian $H\\ge 1$ either hits the barrier within time $1/k$, or its $1/k$-time sweep fails to be embedded. In other words, deleting the barrier destroys the possibility of long embedded Reeb or Hamiltonian cylinders. This gives a quantitative contact rigidity phenomenon in all dimensions and recovers the earlier three-dimensional sphere result as a special case.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":8560,"prompt_tokens":859,"completion_tokens":7701,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":859,"completion_tokens_details":{"reasoning_tokens":6818}},"feed_headline":"Legendrian lift of a Lagrangian barrier is a barrier","feed_subtitle":"Removing it makes long embedded contact flows impossible; every sweep of time 1/k either chords or self-intersects.","key_machinery":"The proof's load-bearing object is a Liouville polarization of the negative symplectization, identified with a symplectic disc bundle over $N$. Starting from a quasi-holomorphic section $\\sigma$ of the $k$th power of the tautological line bundle whose norm has non-degenerate critical points, the paper extends $\\sigma$ to a compactifying projective sphere bundle via $s_\\varepsilon=(1-R)^{k/2}P_R\\sigma+\\varepsilon R^{k/2}P_R\\sigma_\\infty$. The zero set of $s_\\varepsilon$ is a smooth divisor, and the associated Liouville form has a skeleton equal to $\\Phi^{-1}(\\Lambda_k \\times (0,1))$—the radial product of the Legendrian barrier, where $\\Phi$ is the identification between the disc bundle and the negative symplectization. A decomposition theorem for symplectic disc bundles then identi","core_discovery":"The central discovery is that the Legendrian lift of a Lagrangian barrier is itself a Legendrian barrier. Let $(N,\\tau)$ be a closed symplectic manifold with integral symplectic class, $\\Gamma_k$ an isotropic skeleton of a polarization of degree $k\\ge 2$, and $\\Lambda_k$ its minimal Legendrian lift to the prequantization bundle. For every closed Legendrian $\\Lambda$ and every smooth contact Hamiltonian $H\\ge 1$, either there is an $H$-chord from $\\Lambda$ to $\\Lambda_k$ of length at most $1/k$, or the union of $\\Phi^t_H(\\Lambda)$ over $t\\in [0,1/k]$ is not embedded. When $H$ is autonomous this becomes a chord from $\\Lambda$ to itself or to $\\Lambda_k$ of length $\\le 1/k$. A direct consequence is that no contact form $\\alpha' \\le \\alpha$ admits a strong contact embedding of $D(1/k)^n \\times S^1$ into $P\\setminus \\Lambda_k$.","pith_inferences":["The proof requires the skeleton to be realizable by a quasi-holomorphic section with control constant κ<1; known existence results only guarantee such sections for large k. If some degree-2 polarization fails this condition, the theorem as stated would need a separate treatment for small k.","The same strategy, using a degenerate symplectic form on a projective compactification, may produce barriers in other open symplectic manifolds with rational symplectic class, not only negative symplectizations of prequantization bundles.","The paper's (δ,δ')-universal interlinker notion suggests that the constants 1/k could be sharp for contact capacities; testing whether δ and δ' can be tuned independently would give finer quantitative invariants.","The k=1 case appears to be genuinely exceptional—the statement already fails for complex projective space with the Fubini–Study form—so the rational-area mechanism underlying the proof seems to be the decisive feature."],"forward_implications":["For any contact form α'≤α, there is no strong contact embedding of D(1/k)^n×S^1 into P\\Λ_k; the largest round cylinder that fits contactomorphically has radius at most 1/k.","Any Legendrian isotopy in P\\Λ_k generated by a contact Hamiltonian H≥1 has discriminant length at least k, so the barrier forces every long isotopy to pass through non-embedded stages.","The area class of an isotropic skeleton of degree k is discrete with values in (1/k)Z; this rationality is exactly what makes the k-fold Legendrian lift well-defined.","For any smooth polarization of degree k, every Lagrangian immersion in M\\L_k bounds a symplectic disc of area <1/k, so the Lagrangian capacity of the complement is at most 1/k.","For autonomous Hamiltonians the statement reduces to a chord from Λ to itself or to Λ_k of length ≤1/k, giving a concrete realization of the paper's notion of universal interlinking."],"fun_headline_variants":["Legendrian lift blocks long cylinders","Barrier lifts to Legendrian, stops embeddings","Obstructing cylinders via lifted barriers","Prequantization barriers: no long contact flows","Lifted Lagrangian barrier limits contact chords"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The theorem is stated for every polarization of degree $k\\ge 2$, but the proof needs the skeleton to be the skeleton of a quasi-holomorphic section of the $k$th power line bundle with control constant $\\kappa < 1$, transverse zeros, and non-degenerate critical points of the norm's logarithm; such sections are only known to exist for large $k$.","fun_headline_variants_meta":{"raw":{"variants":["Legendrian lift blocks long cylinders","Barrier lifts to Legendrian, stops embeddings","Obstructing cylinders via lifted barriers","Prequantization barriers: no long contact flows","Lifted Lagrangian barrier limits contact chords"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000108,"raw_usage":{"total_tokens":794,"prompt_tokens":566,"completion_tokens":228,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":310,"completion_tokens_details":{"reasoning_tokens":163}},"tokens_in":310,"tokens_out":228,"duration_ms":3405,"temperature":1.0,"reasoning_tokens":163,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T14:54:29.507599+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct test: in the standard prequantization $S^{2n+1} \\to \\mathbb{CP}^n$ with the degree-$2$ skeleton made of half great circles, look for a closed Legendrian $\\Lambda$ and a contact Hamiltonian $H\\ge 1$ such that $\\Phi^t_H(\\Lambda)$ stays embedded for $t\\in [0,1/2]$ and avoids the two-point Legendrian lift $\\Lambda_2$. Finding one would disprove the $1/k$ bound. Alternatively, exhibiting a degree-$2$ polarization whose skeleton is not realizable by any $\\kappa < 1$ quasi-holomorphic section with non-degenerate norm logarithm would show the proof does not cover the theorem's stated hypotheses.","supporting_citations":[],"review_version":1}