{"id":"6e3a6017-9e9d-45c3-b9d1-928d237978db","arxiv_id":"2411.17212","paper_version":3,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper asserts canonical Weil-bundle lifts preserve a long list of geometric structures, but several new proofs rely on unproved bracket compatibility and the highlighted non-suspension example is actually a trivial suspension.","lead":"This paper claims that many geometric structures, from Riemannian metrics to contact and Kähler forms, can be lifted from a smooth manifold to its Weil bundle, a higher-order jet-like thickening of the manifold. It gathers known lift results and adds several new claims, but some of the new claims rest on unproved assumptions and one advertised example is incorrect.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Averaged section lifts are not well-defined globally, and the Reeb/Jacobi identities in The proofs of Theorems A(3) and F are assumed rather than proved; the new claims are unsupported as written.","rationale":"The survey parts of the paper are plausibly supported: functorial lifts of Riemannian metrics, Kähler triples, and contact forms via flow prolongation are standard, and Theorems B, C, D, and E rest on material from [6,8,10]. The genuinely new assertions are Theorem A(3) and Theorem F, and both are anchored in the averaged-section construction of §2.3. That construction is where the argument is weakest. First, a section S_j is not a diffeomorphism, so (S_j)_*X cannot be a global vector field unless an extension rule is supplied; the paper supplies none. Second, the proof text for both theorems explicitly postpones the required identities—Theorem A(3) says verification 'requires specific interaction rules' and Theorem F 'Assumes compatibility rules [6]'—so the central new claims are conditional on an unstated lemma. This is not a question of outside consensus; it is an internal proof gap. The reader's weakest assumption is essentially the same compatibility premise, though I would locate the problem one step earlier at the definition of the averaged pushforward. Secondary issues such as the false Lemma 2.12 and the low-rank cosymplectic form in Example 5.2 reinforce the assessment, but the load-bearing defect is the undefined and unproved averaged-lift mechanism. Since the reader already rejected the paper for this reason, my stress test leaves that verdict unchanged.","tokens_in":11528,"tokens_out":39524,"duration_ms":353360,"concrete_test":"Require the missing definition: for A=R[ε]/(ε²), M=R, specify (S_2)_*∂_x as a vector field on all of R×R using only §2.3, i.e., give its value at a point (x,u) not on the image of S_2. If no coordinate formula can be supplied, Theorems A(3) and F are built on an undefined expression. If a formula is supplied, recompute the Schouten identity [Λ^A,Λ^A]=2Ξ^A∧Λ^A for the explicit Jacobi structure Λ=x∂_x∧∂_y+z∂_z∧∂_x, Ξ=1/2∂_y; this will show whether the claimed 'compatibility rules' actually hold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The new mechanism is the averaged lift over sections S_j: X~=(1/l)Σ(S_j)_*X and Λ~=(1/l)Σ(S_j)_*Λ in §2.3. This is not a well-defined global object: each S_j:M→M^A is an embedding whose image has positive codimension, so (S_j)_*X is a vector field only along S_j(M), not on all of M^A, and no extension rule is given. Even if such an extension is intended, the proofs do not supply the required identities. Theorem A(3) says that the averaged field 'matches ξ_{M^A} under the assumption that the canonical lifts ... are compatible with this averaging construction' and that checking η^A(Ξ)=1 and ι_Ξω^A=0 'requires specific interaction rules.' Theorem F says it 'Assumes compatibility rules [6] such that [Λ^A,Λ^A]=...' These are the defining identities of the claimed cosymplectic and Jacobi structures; they are exactly what must be proved. Because the Schouten bracket is bilinear, [Λ^A,Λ^A] is a sum over pairs (S_j)_*Λ,(S_k)_*Λ with a 1/l^2 prefactor; reducing it to (1/l)Σ(S_j)_*[Λ,Λ] drops cross terms and changes normalization unless a non-obvious rule is established. No such rule is stated. This is an internal proof gap, not a disagreement with a different consensus.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that a wide range of geometric structures on a smooth manifold M, including lcs, lcc, cosymplectic, contact, Jacobi, Sasakian, Walker, sub-Riemannian, orientation, Riemannian, and Kähler structures, admit canonical lifts to the Weil bundle M^A. The proofs rely on the Weil functor formalism, on known canonical lifts from the literature, and, for the Reeb field of a cosymplectic structure and for Jacobi structures, on averaged lifts defined by (1/l)Σ_j (S_j)_* over sections S_j. The paper also gives coordinate examples, discusses non-preserved Einstein and Calabi-Yau conditions, and includes a short appendix of definitions.","tokens_in":11877,"tokens_out":8776,"duration_ms":83024,"significance":"Several of the claimed lifts are already standard: contact, symplectic, Riemannian, and Kähler lifts appear in the cited literature by Kolář–Michor–Slovák, Morimoto, and Okayama. The genuinely new mechanism in this manuscript is the averaged section lift in §2.3, used for the Reeb field in Theorem A(3) and for the Jacobi structure in Theorem F. That mechanism is not rigorously defined, and the proofs explicitly assume the very identities that must be proved. The paper also contains a plainly false lemma about smooth functions on Weil bundles. If the averaging construction were replaced by a well-defined global construction with genuine proofs of the bracket and contraction identities, the cosymplectic and Jacobi results would be a useful contribution; as written, the central claims are not established. The paper does have useful expository parts: Section 2.2's distinction between pullbacks and canonical lifts is clear, and the coordinate examples in Section 5 are concrete.","major_comments":[{"comment":"The averaged vector field X~ = (1/l)Σ_j (S_j)_*X is not shown to be a well-defined global vector field on M^A. Each S_j:M→M^A is an embedding whose image has positive codimension when l>1, so (S_j)_*X is defined only along S_j(M); no extension to all of M^A is given. The proof of Theorem A(3) states only that Ξ 'matches' ξ_{M^A} under an assumption of compatibility with the averaged construction, and that verifying η^A(Ξ)=1 and ι_Ξω^A=0 'requires specific interaction rules.' These two equations are exactly the defining Reeb identities of the asserted cosymplectic structure, so the theorem is not proved.","section":"§2.3 / Theorem A(3)"},{"comment":"The proof of Theorem F assumes [Λ^A,Λ^A]_{SN} = (1/l)Σ_j (S_j)_*[Λ,Λ]_{SN} and [Ξ^A,Λ^A]_{SN}=0. The Schouten bracket is bilinear, so the left-hand side of the first identity contains l^2 cross terms with a 1/l^2 prefactor; no rule is given that eliminates or combines those cross terms. The second identity is also assumed. These are precisely the Jacobi identities that characterize a Jacobi structure, so the argument is circular rather than a proof.","section":"Theorem F / §3"},{"comment":"Lemma 2.12 asserts that every smooth real-valued function on M^A is constant along the fibers of π~_M. This is false: for A = R[ε]/(ε^2), M^A is the tangent bundle TM, and fiber coordinates such as the components of a tangent vector are smooth, nonconstant functions on the fibers. Consequently the identification C^∞(M^A) ≅ C^∞(M) in Remark 2.13 is incorrect. This lemma is not directly used in the main proofs, but it is a stated foundational result and should either be corrected or removed.","section":"Lemma 2.12 / Remark 2.13"},{"comment":"Example 5.2 defines a cosymplectic lift using η^A = dz_1, whose Reeb field is ∂/∂z_1, a canonical section-type lift of ∂/∂z. This is not the averaged field (1/l)Σ_j (S_j)_*(∂/∂z) featured in Theorem A(3). The relation between the two constructions is never explained, so the example does not illustrate the theorem's Reeb-field construction and leaves the status of the averaged Reeb field unclear.","section":"Example 5.2 / Theorem A(3)"}],"minor_comments":[{"comment":"In the paragraph after the pullback discussion, the phrase 'lifts the full algebraic structure of A and the functorial nature of T^A' is grammatically incomplete; 'using' or 'exploiting' appears to be missing.","section":"Introduction"},{"comment":"The notation D^A and (g_D)^A for the lifted distribution and metric is not defined; the proof should specify how the Weil functor is applied to a subbundle of TM and why the lifted distribution is a smooth subbundle of TM^A.","section":"Proposition H"},{"comment":"Corollary 3.1 states that the map [θ]↦[θ^A] is an isomorphism on H^1 because M^A deformation retracts onto M, but it does not prove that the canonical lift of an exact form is exact; this naturality should be justified explicitly.","section":"Corollary 3.1"},{"comment":"The displayed comparison of transition Jacobians, det(J(ψ^A_{αβ})) ≈ (det J(ψ_{αβ}))^l, is only heuristic; the orientation form used and the convention for the lifted volume element should be specified.","section":"Theorem E proof"},{"comment":"References [7] and [12] do not appear to be cited in the text, while several citations to [6] and [8] would benefit from precise proposition or chapter numbers.","section":"References"}],"recommendation":"reject","confidential_remarks":"The manuscript's genuinely new contribution is the averaged section lift in §2.3, and that is precisely the part that is not rigorously defined and whose key identities are assumed rather than proved. In addition, Lemma 2.12 is plainly false. I do not see a local fix within the current scope; the averaging construction would need to be replaced by a well-defined global construction with new proofs of the Reeb and Schouten bracket identities."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my read of arXiv:2411.17212. The paper is best treated as a survey of known canonical lifts to Weil bundles, plus a set of new claims that are not proved. The standard material—Riemannian, symplectic, contact, Kähler, orientation lifts—is correctly assembled from Kolář–Michor–Slovák and Okayama, and the functorial proofs for Kähler integrability and the Lagrangian submanifold lift are clean. The lcs and lcc lift statements also follow from functoriality and are likely fine.\n\nThe trouble is the new mechanism. The averaged lifts in §2.3 define (S_j)_*X only along the image of each section, not on all of M^A, and no extension rule is given. Even if that were fixed, the proofs of Theorem A(3) and Theorem F assume compatibility rules that are exactly the defining identities of the claimed structures: η^A(Ξ)=1, ι_Ξ ω^A=0, and [Λ^A,Λ^A]=2Ξ^A∧Λ^A. The authors admit as much in the text (\"requires specific interaction rules,\" \"Assumes compatibility rules [6]\"). That is circular, not a proof. The Schouten bracket of averaged bivectors also has cross terms and a 1/l^2 prefactor; the paper gives no reason they collapse to the claimed (1/l)Σ(S_j)_*[Λ,Λ].\n\nThere's also a clear false statement: Lemma 2.12 says any smooth function on M^A is constant along fibers. This contradicts the local coordinates the paper itself uses (x_{i,k}, k≥2, are nonconstant along fibers). So the lemma is simply wrong.\n\nOne place where the reader's critique misses: Example 5.2. The construction gives M^A ≅ (R^{2n})^A × R^l with l>1. That is correctly not a suspension in the standard sense (P × R). So don't repeat that as a flaw.\n\nWho gets value from this? Someone looking for a single place that lists which structures survive Weil prolongation will find the survey useful. Researchers should not rely on the new lift theorems as stated.\n\nFor peer review: I'd send it out, because the standard part is a legitimate survey and a referee can pin down the exact gaps in the new claims. But my own verdict is that the new results are unsupported as written, and the paper would need heavy revision to become a valid research contribution.","headline":"A survey of standard Weil lifts with new lift claims that are assumed rather than proved; the averaged-section mechanism is not well-defined globally.","tokens_in":11,"tokens_out":4708,"would_cite":false,"duration_ms":102411,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["58A32","53D05","53D10","53C15","53C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Weil-bundle functor canonically lifts eleven classical geometric structures — lcs, lcc, cosymplectic, contact, Jacobi, Sasakian, Walker, sub-Riemannian, orientation, Riemannian, and Kähler — preserving the equations that define them.","keywords":["Weil bundle","canonical lift","Weil prolongation","cosymplectic structure","Jacobi structure","Sasakian structure","contact structure","locally conformal symplectic"],"falsifier":"Compute, in local coordinates on $(\\mathbb R^3)^{\\mathbf A}$ for a 3-dimensional Weil algebra such as $\\mathbb R[u]/(u^3)$, the averaged Reeb field $\\Xi=\\frac13\\sum_j(S_j)_*\\partial_z$ for the standard cosymplectic structure $\\omega=dx\\wedge dy$, $\\eta=dz$, then check whether $\\eta^{\\mathbf A}(\\Xi)=1$ and $\\iota_\\Xi\\omega^{\\mathbf A}=0$ for the lifted forms used in the proof. A single choice of sections or trace where either identity fails would refute the Reeb-field construction in Theorem A(3).","tokens_in":43,"feed_emoji":"📐","tokens_out":7505,"duration_ms":123907,"temperature":0.7,"pith_summary":"This paper claims that a single functorial construction, the Weil bundle (or manifold of infinitely near points) $M^{\\mathbf A}$ of a smooth manifold $M$, canonically inherits a wide family of geometric structures from $M$. The list includes locally conformal symplectic and cosymplectic structures, contact and Jacobi structures, Sasakian, Walker, sub-Riemannian, Riemannian, oriented, and Kähler structures. If the claims are right, tools from one manifold transfer to its 'thickened' bundle while preserving the defining equations, so symplectic, contact, and complex geometry can be studied on Weil bundles without ad hoc constructions. The paper also produces an explicit cosymplectic Weil bundle that is not a trivial suspension of a symplectic manifold, showing the lifted geometry is genuinely new.","feed_headline":"Cosymplectic, contact, Jacobi structures all lift to Weil bundles","feed_subtitle":"A single functorial thickening preserves the defining equations of eleven classical geometries.","key_machinery":"The load-bearing object is the Weil functor $T^{\\mathbf A}:M\\mapsto M^{\\mathbf A}$, which sends a manifold to the bundle of algebra homomorphisms $C^\\infty(M)\\to\\mathbf A$ over its points, together with the canonical lift $T\\mapsto T^{\\mathbf A}$ of functions, vector fields, forms, tensor fields, and connections. The lift is functorial: it commutes with $d$, $L_X$, $\\iota_X$, brackets, and algebraic operations, and real-valued results are obtained by applying a normalized trace $\\operatorname{Tr}:\\mathbf A\\to\\mathbb R$. For characteristic vector fields the paper uses a second mechanism, the averaged lift $\\frac1l\\sum_{j=1}^l (S_j)_*$ built from sections $S_j$ associated to a basis of $\\mathbf A$, which projects to the base vector field but is not a Lie-algebra homomorphism. These two lift types, canonical and averaged, are what carry every theorem in the paper.","core_discovery":"The central discovery is that canonical lifts—denoted $\\omega^{\\mathbf A}$, $g^{\\mathbf A}$, $\\beta^{\\mathbf A}$, $J^{\\mathbf A}$—preserve the algebraic identities characterizing each structure, provided the Weil algebra has odd dimension for contact, cosymplectic, lcc, and Sasakian cases. For example, $d(\\omega^{\\mathbf A})=(d\\omega)^{\\mathbf A}$ turns $d\\omega=-\\theta\\wedge\\omega$ into $d\\omega^{\\mathbf A}=-\\theta^{\\mathbf A}\\wedge\\omega^{\\mathbf A}$; the lift of a Killing field is Killing; the lift of the Levi-Civita connection is the Levi-Civita connection of the lifted metric; and the lifted Nijenhuis tensor vanishes when the base one does. For Reeb fields, the paper distinguishes two lift types: the flow prolongation $X^{\\mathbf A}$ for contact structures, and the averaged section lift $\\frac1l\\sum_j (S_j)_*X$ for cosymplectic and Jacobi structures. A further result shows that a Lagrangian submanifold lifts to a Lagrangian submanifold of the symplectic Weil bundle.","pith_inferences":["Beyond the paper, the same averaged-lift recipe could be iterated, lifting a Jacobi structure on $M^{\\mathbf A}$ to $M^{\\mathbf A\\otimes\\mathbf B}$, which would give a hierarchy of nilpotent thickenings of Jacobi geometry.","Since Einstein and Ricci-flat conditions are not preserved, a natural next check is whether curvature invariants of $g^{\\mathbf A}$ decompose into base curvature plus $\\mathbf A$-algebraic terms; computing scalar curvature for the flat-base examples in Section 5 would be a short test.","The non-suspension cosymplectic example suggests that the cosymplectic topology of Weil bundles differs from the base, so the lifted Reeb flows are worth studying even when the base flow is trivial."],"forward_implications":["A symplectic manifold lifts to a symplectic Weil bundle, and a Kähler manifold lifts to a Kähler Weil bundle with integrable complex structure; Lagrangian submanifolds lift to Lagrangian submanifolds.","For odd-dimensional Weil algebras, contact, cosymplectic, lcc, and Sasakian structures lift with explicit Reeb fields: flow prolongation for contact, section-averaged sums for cosymplectic.","The canonical connection on $M^{\\mathbf A}$ is the Levi-Civita connection of the lifted metric, geodesics lift through canonical sections, and Killing fields lift to Killing fields.","The example on $(\\mathbb R^{2n+1})^{\\mathbf A}$ with $l>1$ is cosymplectic but not a product $P\\times\\mathbb R$, so the lifted structure is not a trivial suspension.","Bracket-generating sub-Riemannian distributions lift to bracket-generating distributions, and parallel null distributions lift to parallel null distributions, so Walker and sub-Riemannian geometries pass to Weil bundles."],"supporting_citations":[{"why":"introduces the Weil algebra and the bundle of infinitely near points that defines $M^{\\mathbf A}$.","marker":"[13]"},{"why":"supplies the canonical lift formalism, functoriality of $T^{\\mathbf A}$, commutativity with $d$, $L_X$, $\\iota_X$, and the trace convention used throughout.","marker":"[6]"},{"why":"provides the prolongation of connections to bundles of infinitely near points, used for Levi-Civita, geodesic, and parallelism claims.","marker":"[8]"},{"why":"establishes the canonical contact lift on Weil bundles, used for Theorem D.","marker":"[10]"},{"why":"shows symplectic structures induce symplectic structures on Weil bundles, the starting case extended by the paper.","marker":"[1]"},{"why":"characterizes cosymplectic manifolds as suspensions, the reference against which the non-suspension example is measured.","marker":"[4]"},{"why":"supplies the submersion/section lemmas used to relate forms and vector fields along the Weil projection.","marker":"[11]"}],"fun_headline_variants":["Weil lift: one construction, all classical geometries preserved","New cosymplectic manifolds from canonical Weil lifts","Weil bundles inherit contact, Jacobi, Sasakian, and more","Canonical lifts preserve defining equations on Weil bundles"],"cache_read_input_tokens":14464,"weakest_assumption_plain":"The load-bearing premise is that averaging vector fields by pushing them forward along sections interacts with lifted differential forms exactly as the canonical lift does; the paper asserts this compatibility rather than proving it, and the cosymplectic and Jacobi theorems collapse if it fails.","fun_headline_variants_meta":{"raw":{"variants":["Weil lift: one construction, all classical geometries preserved","New cosymplectic manifolds from canonical Weil lifts","Weil bundles inherit contact, Jacobi, Sasakian, and more","Canonical lifts preserve defining equations on Weil bundles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001133,"raw_usage":{"total_tokens":4723,"prompt_tokens":976,"completion_tokens":3747,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":3677}},"tokens_in":592,"tokens_out":3747,"duration_ms":24679,"temperature":1.0,"reasoning_tokens":3677,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:24:34.571732+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, in local coordinates on $(\\mathbb R^3)^{\\mathbf A}$ for a 3-dimensional Weil algebra such as $\\mathbb R[u]/(u^3)$, the averaged Reeb field $\\Xi=\\frac13\\sum_j(S_j)_*\\partial_z$ for the standard cosymplectic structure $\\omega=dx\\wedge dy$, $\\eta=dz$, then check whether $\\eta^{\\mathbf A}(\\Xi)=1$ and $\\iota_\\Xi\\omega^{\\mathbf A}=0$ for the lifted forms used in the proof. A single choice of sections or trace where either identity fails would refute the Reeb-field construction in Theorem A(3).","supporting_citations":[{"cited_title":"Li, Topology of cosymplectic/cokaehler manifolds , Asian J","cited_arxiv_id":null,"evidence_quote":"characterizes cosymplectic manifolds as suspensions, the reference against which the non-suspension example is measured."},{"cited_title":"Weil, Th´ eorie des points proches sur les vari´ et´ es diff´ erentiables, Colloque de Topologie et G´ eom´ etrie Diff´ erentielle, Strasbourg, 1953, C.N.R.S.,Paris, 1953, pp","cited_arxiv_id":null,"evidence_quote":"introduces the Weil algebra and the bundle of infinitely near points that defines $M^{\\mathbf A}$."},{"cited_title":"Kol´ aˇ r, P","cited_arxiv_id":null,"evidence_quote":"supplies the canonical lift formalism, functoriality of $T^{\\mathbf A}$, commutativity with $d$, $L_X$, $\\iota_X$, and the trace convention used throughout."},{"cited_title":"Morimoto, Prolongations of connections to bundles of infinitely near p oints, J","cited_arxiv_id":null,"evidence_quote":"provides the prolongation of connections to bundles of infinitely near points, used for Levi-Civita, geodesic, and parallelism claims."},{"cited_title":"Okayama, Contact structures on Weil bundles , Tohoku Math","cited_arxiv_id":null,"evidence_quote":"establishes the canonical contact lift on Weil bundles, used for Theorem D."},{"cited_title":"Bouesso et al, Symplectic structure and applications on Weil bundles, Afr","cited_arxiv_id":null,"evidence_quote":"shows symplectic structures induce symplectic structures on Weil bundles, the starting case extended by the paper."},{"cited_title":"Tchuiaga, Towards the cosymplectic topology , Complex Manifolds 9(1) (2022), 230–246","cited_arxiv_id":null,"evidence_quote":"supplies the submersion/section lemmas used to relate forms and vector fields along the Weil projection."}],"review_version":1}