{"id":"78e0710e-0e13-4d1a-90b1-a61a44347fd4","arxiv_id":"2509.05998","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors define q-cosymplectic manifolds and claim Liouville-Arnold-type and Marsden-Weinstein reduction theorems for them, with an application to fast-slow systems.","lead":"This paper introduces q-cosymplectic manifolds, a geometric structure with several time directions, and claims generalized integrability and reduction theorems, illustrated with a fast-slow oscillator. A generalist might read it to see how differential geometry could model multi-time dynamics, but the key theorems contain dimensional inconsistencies.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorems 3.4 and 3.7 apply Bogoyavlenskij's Theorem 2.2 with the wrong vector-field count: the hypotheses give r+1 commuting fields tangent to fibers of dimension 2r+1, so the torus conclusion does not follow for r>0.","rationale":"The reader's weakest-assumption analysis is correct and identifies the decisive gap. The proof of Theorem 3.4 misapplies Bogoyavlenskij's theorem by treating r+1 commuting Hamiltonian/evolution fields as if they were a full set of fiberwise commuting vector fields; the dimensional identity forces the fiber dimension to be 2r+1 under the paper's own condition, while only r+1 fields are available. This is an internal inconsistency, not merely a disagreement with another convention, so it directly invalidates the central integrability claim in the abstract and title. I see no reason to change the reader's REJECT verdict; the concern is the same one and remains unrepaired elsewhere in the manuscript. Some components (Poisson bracket, symplectization, reduction) may be salvageable, but that does not affect the verdict on the paper as written.","tokens_in":27427,"tokens_out":10310,"duration_ms":111583,"concrete_test":"Perform a dimension check on a minimal noncommutative case of Theorem 3.4: take q=1, n=2, r=1, m=3, which satisfies 2n+q-1=4=m+r. Then F=(f_2,f_3) maps the 5-manifold to R^2, so fibers have dimension 3; the proof exhibits exactly E_H and X_{f_1} (2 vector fields). Determine whether any third commuting vector field tangent to the fibers is implied by the stated hypotheses. If none is, Theorem 2.2 cannot be invoked and the T^2 conclusion is dimensionally impossible. The same count can be repeated symbolically for arbitrary r to verify that the required number of fields is 2r+1, not r+1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central integrability claim is unsupported because Theorems 3.4 and 3.7 do not satisfy the hypotheses of the quoted Theorem 2.2. In Theorem 3.4, F=(f_{r+1},...,f_m) has k=m-r components, so its fibers have dimension (2n+q)-(m-r). Using condition (1), 2n+q-1=m+r, this dimension is (m+r+1)-(m-r)=2r+1. Bogoyavlenskij's theorem requires dim M-k = 2r+1 independent pairwise commuting vector fields tangent to the fibers. The proof supplies only E_H and X_{f_1},...,X_{f_r}, i.e. r+1 vector fields. For every r>0 this is too few, and the asserted torus dimension T^{m-r} would require fibers of dimension m-r, not 2r+1; no hypothesis forces m-r=2r+1. Theorem 3.7 has the same defect after adding H to the level map: its fibers have dimension 2r while the proof supplies r+1 fields. The torus-action, constant-vector-field conjugation, and any Section 5 conclusions resting on them therefore do not follow from the assumptions. The Poisson bracket, symplectization, and reduction portions appear separate and may well stand, but the integrability theorems are a load-bearing part of the abstract and title.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces q-cosymplectic manifolds, defined by a closed 2-form and q closed, pointwise independent 1-forms, and studies Hamiltonian, gradient, local-gradient, and q-evolution vector fields. It proves a Poisson-bracket structure (Theorem 3.2), a symplectization statement (Theorem 3.8), two Liouville–Arnold-type integrability theorems (Theorems 3.4 and 3.7), and a Marsden–Weinstein-type reduction theorem (Theorem 4.2). Section 5 applies the formalism to a fast–slow harmonic-oscillator system, including the averaged slow dynamics. The claimed central contributions are the integrability theorems and the reduction theorem for the new geometric structure.","tokens_in":27784,"tokens_out":8783,"duration_ms":104179,"significance":"If the main results were valid, the paper would provide a useful framework for multi-time Hamiltonian dynamics, connecting q-cosymplectic geometry with noncommutative integrability and reduction. The construction of the Poisson bracket in Theorem 3.2 and the symplectization in Theorem 3.8 are natural and appear to be correct; the Marsden–Weinstein reduction in Theorem 4.2 is also plausibly correct and is proved in detail. However, the two Liouville–Arnold-type theorems are dimensionally inconsistent and do not follow from the quoted Bogoyavlenskij theorem. Since integrability is advertised in the abstract and the title, this is a load-bearing defect. The fast–slow application is heuristic and does not repair the gap.","major_comments":[{"comment":"The hypotheses do not meet those of Bogoyavlenskij's Theorem 2.2. In Theorem 3.4 the level map is F=(f_{r+1},...,f_m), so k=m-r and the fibers have dimension dim M - k = 2n+q-(m-r). Using the stated condition 2n+q-1=m+r, this equals 2r+1. Theorem 2.2 requires n=2r+1 commuting, independent vector fields tangent to the fibers. The proof supplies only E_H and X_{f_1},...,X_{f_r}, which is r+1 fields. For every r>0, r+1<2r+1, so the hypotheses are not satisfied. Moreover the asserted torus T^{m-r} has dimension m-r=2n+q-1-2r, which is not forced to equal 2r+1. Thus the theorem's conclusion does not follow from the assumptions.","section":"Theorem 3.4, Section 3"},{"comment":"The same dimensional defect appears. Here F=(H,f_{r+1},...,f_m) has k=m-r+1 components, so its fibers have dimension dim M - k = 2n+q-(m-r+1). With 2n+q-1=m+r this is 2r. The proof supplies E'_H and X'_{f_1},...,X'_{f_r}, i.e. r+1 vector fields. For r>=2, r+1<2r, and for r=0 the nonzero field E'_H cannot be tangent to a zero-dimensional fiber. The claimed torus T^{m-r+1}=T^{2n+q-2r} also does not match the fiber dimension 2r except in special cases. Hence Theorem 3.7 is not a consequence of the stated hypotheses and cannot be rescued by the invocation of Theorem 2.2.","section":"Theorem 3.7, Section 3"},{"comment":"Because Theorems 3.4 and 3.7 are the basis for the advertised 'Liouville–Arnold-type theorems' and for the word 'Integrability' in the title, the central claim of the manuscript is unsupported. Restricting r=0 in Theorem 3.4 would recover a one-field Liouville statement, but then the noncommutative integrability content disappears; Theorem 3.7 does not reduce cleanly even to r=0. This is not a local typo but a structural mismatch in the main new theorems.","section":"Abstract and title"}],"minor_comments":[{"comment":"The form Omega = sum_{j=1}^n x_j dy_j is not closed; d(Omega)=sum_j dx_j wedge dy_j. This contradicts the q-cosymplectic requirement dOmega=0. The example should use Omega = sum_j dx_j wedge dy_j.","section":"Example 2.3"},{"comment":"The condition '{H',f_k}=0, k=1,...,q' appears to have the wrong index range: f_k is only defined for k<=m and q is unrelated to m. It should likely be k=1,...,m or k=r+1,...,m.","section":"Theorem 3.7, condition (3.15)"},{"comment":"The proof refers to 'relations (3.11) and (3.15)', but (3.15) belongs to Theorem 3.7 and is not available in Theorem 3.4. The relevant bracket identities are (3.9)-(3.10) and (3.8).","section":"Proof of Theorem 3.4"},{"comment":"The assertion that the averaging theorem applies 'because (i)-(iii)' is informal; in particular, the projection commutation in (iii) is not demonstrated. The application section is presented as an illustration, but its claims about the geometric validity of averaging would need more proof if they are meant as theorems.","section":"Section 5.1.1"}],"recommendation":"reject","confidential_remarks":"The dimensional mismatch in Theorems 3.4 and 3.7 is decisive: the stated hypotheses cannot satisfy Bogoyavlenskij's theorem, so the paper's advertised integrability results are not established. The correct-looking Poisson bracket, symplectization, and reduction parts do not compensate for this central failure. I would not recommend a standard revision cycle unless the authors substantially restrict the claims (e.g., to the r=0 case) and remove the noncommutative-integrability statements from the abstract and title."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Jo, quick read of 2509.05998. The interesting parts are Section 4 and the symplectization theorem; the advertised integrability results in Section 3 are not established.\n\nThe problem is a dimensional count. In Theorem 3.4, the level sets of F have dimension 2r+1, but the proof supplies only r+1 commuting vector fields (E_H and X_{f_1},...,X_{f_r}). Bogoyavlenskij's Theorem 2.2 requires dim(M)-k = 2r+1 independent fields tangent to the fibers. For r>0 you are short by r fields, so the conclusion that each fiber is T^{m-r} does not follow. Theorem 3.7 has the same defect. This is load-bearing: the abstract and title promise integrability theorems, and the fast-slow application leans on them.\n\nWhat does hold up? The Poisson bracket, the symplectization theorem 3.8, and the Marsden–Weinstein reduction theorem look correct to me. The reduction follows Albert's cosymplectic argument with q Reeb fields, and the proof checks out. The fast-slow example is fine as an illustration, though it doesn't need those broken integrability theorems.\n\nOn novelty: the definition of q-cosymplectic manifold, dim M = 2n+q with q closed 1-forms and a closed 2-form whose kernel is the Reeb distribution, is essentially the standard k-cosymplectic structure in the literature. The paper doesn't cite that body of work. The name 'q-cosymplectic' makes a known object sound new. That could be fixed by honest referencing, but combined with the broken theorems the claimed contribution is thin.\n\nSmaller issue: Theorem 2.5 infers that each λ_i is T^n-invariant because the product λ_1∧...∧λ_q is invariant. That inference is not valid; the product can be invariant while the individual forms mix. The theorem may be true under extra assumptions, but the proof as written doesn't establish it.\n\nWho is this for? Someone working in k-cosymplectic or multi-time mechanics might salvage the reduction theorem and produce a modest paper. As written, the central claims are unsupported. I would not publish it, but the error is subtle enough that a serious referee could give actionable feedback, and the reduction half deserves expert judgment. My recommendation: send to review if the venue wants careful verification, but with the expectation of major revision or rejection.","headline":"The paper has a real, correct core in the reduction and symplectization results, but its advertised Liouville–Arnold theorems do not go through, and the q-cosymplectic name hides a known concept.","tokens_in":28327,"tokens_out":5045,"would_cite":false,"duration_ms":58749,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D20","53D17","70H06","37J35"],"pacs":[],"model":"deepseek-v4-flash","headline":"q-cosymplectic manifolds give multitime Hamiltonian systems Poisson brackets, tori, and reduction.","keywords":["q-cosymplectic manifold","multi-time Hamiltonian dynamics","Poisson bracket","Liouville-Arnold integrability","Marsden-Weinstein reduction","Reeb vector fields","fast-slow systems"],"falsifier":"Directly check the count in Theorem 3.4: take n=2, q=2, r=1; the condition forces m=4, so the level sets of (f_2,f_3,f_4) are 3-dimensional and the theorem asserts T^3, but only two commuting fields E_H and X_{f_1} are tangent to the fibers. Constructing such a system on the explicit R^{2n+q} q-cosymplectic manifold of Example 2.3 and asking whether its compact connected level set is a 3-torus would decide the claim.","tokens_in":27274,"feed_emoji":"⏱️","tokens_out":13320,"duration_ms":132503,"temperature":0.7,"pith_summary":"The paper introduces q-cosymplectic manifolds—spaces with one closed two-form and q closed one-forms—and argues they are the right arena for Hamiltonian mechanics with q independent time directions. From this data it constructs a Poisson bracket, Hamiltonian, gradient and evolution vector fields, with the q-evolution field E_f=Σ_i R_i+X_f combining the q clock directions and the Hamiltonian flow. It then proves two families of results: Liouville–Arnold-type integrability theorems in which invariant level sets are tori and the evolution field becomes constant, and a Marsden–Weinstein reduction theorem in which the quotient of a symmetric q-cosymplectic system is again q-cosymplectic. A fast–slow oscillator with two clocks (fast t, slow τ) illustrates the theory: action–angle averaging gives an adiabatic invariant and a slow drift, and symmetry reduction collapses the fast oscillator.","feed_headline":"One Poisson bracket governs Hamiltonian systems with q time directions","feed_subtitle":"q-cosymplectic manifolds make multi-time dynamics integrable and reducible, with fast-slow clocks as a test case.","key_machinery":"The object doing the work is the q-cosymplectic structure itself: q closed, everywhere-independent 1-forms λ_1,...,λ_q select q Reeb vector fields R_i, while the closed 2-form Ω is symplectic on the horizontal distribution ξ=∩ker λ_i and has kernel exactly span{R_i}. The bundle isomorphism b(v)=i_vΩ+Σλ_i(v)λ_i converts functions into Hamiltonian and gradient vector fields, and the q-evolution field E_f=ΣR_i+X_f is the object whose first integrals and commuting flows drive the integrability theorems. Reduction is carried by the same data: the Reeb fields are tangent to momentum level sets and push forward to q commuting Reeb fields on the quotient.","core_discovery":"The central claim is that the structure (M,Ω,λ_1,...,λ_q), with closed Ω and closed independent λ_i such that ker Ω is spanned by Reeb vector fields R_i and Ω is symplectic on ∩ker λ_i, supports a Hamiltonian formalism parallel to symplectic geometry. The map b(v)=i_vΩ+Σλ_i(v)λ_i is an isomorphism; Hamiltonian vector fields X_f=b^{-1}(df−ΣR_i(f)λ_i) are horizontal, the gradient field is ∇f=X_f+ΣR_i(f)R_i, and the q-evolution field E_f=ΣR_i+X_f encodes all q time directions. The bracket {f,g}=Ω(X_f,X_g) is a Poisson bracket, and a q-cosymplectic manifold symplectizes into (R^q×M, pr*ω+Σ ds_i∧pr*λ_i) with a Poisson morphism. Under involutivity and the dimension condition 2n+q−1=m+r, Theorems 3","pith_inferences":["The symplectization theorem suggests a route to multitime Hamilton–Jacobi theory: solve a Hamilton–Jacobi equation on R^q×M and pull back solutions to M, which the paper does not explicitly do.","Reduction at the fast action J=c in the example could be viewed as a geometric counterpart of averaging; making that identification explicit might turn the ε→0 limit into an exact quotient.","The framework is plausibly compatible with singular reduction or non-Abelian cocycle twists, but neither is explored in this paper."],"forward_implications":["Any q-cosymplectic manifold carries a Poisson bracket, so multitime Hamiltonian systems form a corank-q Poisson system rather than a mere collection of one-time equations.","Under the paper's dimension condition, a q-evolution field with enough commuting first integrals has compact level sets diffeomorphic to tori and is conjugate to a constant field, extending action-angle coordinates to multitime dynamics.","In Liouville coordinates the forms λ_i are torus-invariant, so the clock directions remain aligned with the integrable fibration.","A Hamiltonian group action with a q-cosymplectic momentum map admits a reduced q-cosymplectic space at every weakly regular value, with reduced Reeb fields and forms pushed forward from the original ones.","The fast-slow oscillator example shows that action-angle averaging and symmetry reduction in the q-cosymplectic setting reproduce the standard slow drift equations."],"supporting_citations":[{"why":"Supplies the extended integrability criterion (Theorem 2.2) used to conclude that fibers are tori and the flow is conjugate to a constant field.","marker":"[7]"},{"why":"Supplies Liouville's theorem and the torus-coordinate setup underlying the paper's action-angle statements.","marker":"[19]"},{"why":"Supplies the fundamental conservation property used to show that the forms λ_i are invariant in Liouville coordinates.","marker":"[36]"},{"why":"The classical symplectic reduction theorem that the paper generalizes to q-cosymplectic manifolds.","marker":"[24]"},{"why":"Provides the cosymplectic reduction and weak-regular-value/cocycle machinery adapted in Section 4.","marker":"[2]"},{"why":"Provides the noncommutative integrability framework whose analogue Theorems 3.4 and 3.7 state.","marker":"[25]"},{"why":"Supplies the horizontal Moser correction used to put the fast-slow system into action-angle form.","marker":"[14]"},{"why":"Supplies the averaging theorem used to justify the averaged q-evolution field in Section 5.","marker":"[5]"}],"fun_headline_variants":["Multi-time Hamiltonians tamed by q-cosymplectic geometry","Q-time Poisson brackets yield integrability and reduction","Fast-slow systems get a unifying geometric framework","Reduction and integrability for multi-dimensional time","One Poisson bracket to rule q time directions"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The integrability theorems hinge on a counting assumption: the number of commuting vector fields must equal the dimension of the level sets. Under the paper's condition, the level-set dimension becomes 2r+1 while the available commuting fields number r+1, so the matching holds only when r=0; if that count misfires, the torus conclusion does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Multi-time Hamiltonians tamed by q-cosymplectic geometry","Q-time Poisson brackets yield integrability and reduction","Fast-slow systems get a unifying geometric framework","Reduction and integrability for multi-dimensional time","One Poisson bracket to rule q time directions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000149,"raw_usage":{"total_tokens":1007,"prompt_tokens":702,"completion_tokens":305,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":446,"completion_tokens_details":{"reasoning_tokens":230}},"tokens_in":446,"tokens_out":305,"duration_ms":3901,"temperature":1.0,"reasoning_tokens":230,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T04:41:24.638314+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly check the count in Theorem 3.4: take n=2, q=2, r=1; the condition forces m=4, so the level sets of (f_2,f_3,f_4) are 3-dimensional and the theorem asserts T^3, but only two commuting fields E_H and X_{f_1} are tangent to the fibers. Constructing such a system on the explicit R^{2n+q} q-cosymplectic manifold of Example 2.3 and asking whether its compact connected level set is a 3-torus would decide the claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the extended integrability criterion (Theorem 2.2) used to conclude that fibers are tori and the flow is conjugate to a constant field."},{"cited_title":"Liouville, Note sur l’integration des equations differentielles de la dynamique","cited_arxiv_id":null,"evidence_quote":"Supplies Liouville's theorem and the torus-coordinate setup underlying the paper's action-angle statements."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the fundamental conservation property used to show that the forms λ_i are invariant in Liouville coordinates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The classical symplectic reduction theorem that the paper generalizes to q-cosymplectic manifolds."},{"cited_title":"Albert, Le théorème de réduction de Marsden-Weinstein en géométrie cosymplec- tique et de contact, J","cited_arxiv_id":null,"evidence_quote":"Provides the cosymplectic reduction and weak-regular-value/cocycle machinery adapted in Section 4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the noncommutative integrability framework whose analogue Theorems 3.4 and 3.7 state."},{"cited_title":"de León, J","cited_arxiv_id":null,"evidence_quote":"Supplies the horizontal Moser correction used to put the fast-slow system into action-angle form."},{"cited_title":"Arnold, V.V","cited_arxiv_id":null,"evidence_quote":"Supplies the averaging theorem used to justify the averaged q-evolution field in Section 5."}],"review_version":1}