{"id":"0d24d929-9900-4643-92cd-4e6f2bf3afc4","arxiv_id":"2608.05681","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"The paper extends q-contact Hamiltonian geometry to non-uniform structures with multiple dissipation channels and proves Noether-type theorems, illustrated by an elastoplastic damage model.","lead":"This paper builds a geometric framework based on q-contact manifolds for Hamiltonian systems with several independent dissipation channels, then applies it to an elastoplastic damage model. It derives Noether-type theorems for uniform and non-uniform structures, though one time-dependent theorem carries a significant proof gap.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.5's proof cites Proposition 2.4 for a vanishing contraction it cannot imply; the identity is true by a one-line Cartan argument, so the theorem stands but the proof needs correction.","rationale":"The reader correctly identified the disputed identity i_{[X^t_H,Y]}\\lambda_i^E=0 as the load-bearing step in Theorem 2.5 and correctly observed that Proposition 2.4, as stated, does not imply it. However, the reader inferred that the theorem is unsupported and possibly false. That inference is not warranted: the vanishing contraction follows directly from the generalized Noether symmetry condition L_Y\\lambda_i^E=\\sigma_i\\lambda_i^E together with i_{X^t_H}\\lambda_i^E=0, which holds by construction of X^t_H. Hence the central claim of the paper is correct, although the written proof contains a genuine gap in justification. Since the gap is easily closed by adding a short lemma, the appropriate verdict is to require that correction rather than to reject the paper. The numerical validation concern raised by the reader is secondary: the simulations verify an identity that is an exact consequence of the constructed equations, so they provide consistency rather than independent physical confirmation, but this does not undermine the mathematical framework.","tokens_in":24668,"tokens_out":20971,"duration_ms":182594,"concrete_test":"Add and verify the missing lemma: if i_X\\lambda_i^E=0 and L_Y\\lambda_i^E=\\sigma_i\\lambda_i^E for all i, then i_{[X,Y]}\\lambda_i^E=0. Prove it by applying L_Y to the identity i_X\\lambda_i^E=0 and using Cartan's identity. Then check that X^t_H indeed satisfies i_{X^t_H}\\lambda_i^E=0 (it follows from \\lambda_i(X_H)=-H and the normalization dt(\\partial_t)=1), which supplies exactly the missing step in Theorem 2.5.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The reader's concern is real but the conclusion is too strong. In Theorem 2.5, the line 'The previous proposition implies i_{[X^t_H,Y]}\\lambda_i^E=0' is a non sequitur: Proposition 2.4 only says that the Lie bracket of two generalized Noether symmetries is again a generalized Noether symmetry, i.e. L_{[X^t_H,Y]}\\lambda_i^E=\\sigma_i\\lambda_i^E. That condition does not force i_{[X^t_H,Y]}\\lambda_i^E=0; for example, in the q=1 case \\partial_z satisfies L_{\\partial_z}\\lambda=0 but i_{\\partial_z}\\lambda=1. However, the asserted identity is in fact true for a different reason: X^t_H features i_{X^t_H}\\lambda_i^E=0 by construction, and Y being a generalized Noether symmetry gives L_Y\\lambda_i^E=\\sigma_i\\lambda_i^E. Applying Cartan's identity, 0=L_Y(i_{X^t_H}\\lambda_i^E)=i_{[Y,X^t_H]}\\lambda_i^E+i_{X^t_H}(\\sigma_i\\lambda_i^E)=i_{[Y,X^t_H]}\\lambda_i^E, hence i_{[X^t_H,Y]}\\lambda_i^E=0. Thus the generalized Noether theorem is mathematically sound, but the proof as written misattributes the justification. This is a repairable proof defect, not a fatal flaw; conditional acceptance with a corrected proof is appropriate.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a Hamiltonian formalism for q-contact manifolds, first in the uniform case (dλ1=⋯=dλq) and then for non-uniform structures where the exterior derivatives of the contact forms differ. In the uniform setting it proves a Liouville-type theorem, a q-contact Noether theorem, and, for time-dependent systems, a generalized Noether theorem on the extended phase space. In the non-uniform setting it introduces structure endomorphisms, channel Hamiltonian vector fields, an admissibility condition on the averaged 2-form Ω, an effective Hamiltonian vector field, a non-uniform bracket, and a corresponding Noether theorem. The framework is applied to an elastoplastic damage model with two internal dissipation channels, where the effective equations are derived and numerical simulations are reported to confirm exponential dissipation of the Hamiltonian and conservation of a rescaled quantity.","tokens_in":25027,"tokens_out":22797,"duration_ms":206209,"significance":"If correct, the paper would provide a useful geometric framework for multi-channel dissipative systems and a concrete construction of an admissible non-uniform 2-contact structure with a plausible physical application. The coordinate computations in the R^6 example appear internally consistent, and the numerical experiments, despite limited reporting detail, support the application-specific dissipation law. However, the generalized Noether theorem (Theorem 2.5) is a headline contribution of the abstract and introduction, and it is false as stated; this is a load-bearing error that cannot be dismissed as a purely presentational issue. The non-uniform Noether theorem (Theorem 3.3) is essentially a restatement of the definition of the bracket and does not compensate for the failure of the generalized theorem.","major_comments":[{"comment":"The generalized Noether theorem is false as stated. Take the uniform q-contact manifold M=R^4 with coordinates (x,y,z1,z2), λ1=dz1−xdy, λ2=dz2−xdy, and H=z1. Then R1(H)=1, R2(H)=0. The vector field Y=∂z2 satisfies L_Yλ_i^E=0 for i=1,2, where λ_i^E=λ_i+Hdt, so Y is a generalized Noether symmetry by Definition 2.10. But F1=i_Yλ1^E=0 and F2=i_Yλ2^E=1. Since Σ_i R_i(H)=1, the dissipation equation (2.24) would require L_{X_t^H}F2=−F2·1=−1, while L_{X_t^H}1=0. Thus F2 is not a dissipated quantity, contradicting Theorem 2.5.","section":"Section 2.4, Theorem 2.5"},{"comment":"The proof uses two identities that are not valid as written. First, the displayed identity L_{X_t^H}λ_i^E=−(Σ_j R_j(H))λ_i^E is incorrect for q>1: from (2.2), L_{X_H}λ_i=−Σ_j R_j(H)λ_j, so for X_t^H=X_H+∂t and λ_i^E=λ_i+Hdt one obtains L_{X_t^H}λ_i^E=−Σ_j R_j(H)λ_j^E, which is generally not a multiple of λ_i^E. Consequently i_Y(L_{X_t^H}λ_i^E)=−Σ_j R_j(H)F_j, which does not yield the claimed scalar dissipation equation. Second, the sentence 'The previous proposition implies i_[X_t^H,Y]λ_i^E=0' is a non sequitur: Proposition 2.4 only asserts that generalized Noether symmetries are closed under Lie brackets. A Cartan argument can establish the vanishing contraction when i_{X_t^H}λ_i^E=0, but this does not repair the incorrect Lie-derivative computation.","section":"Section 2.4, proof of Theorem 2.5"},{"comment":"The extended vector field X_t^H is introduced via the equations i_{X_t^H}dλ_i^E=0 and i_{X_t^H}λ_i^E=0 under the stated assumption R_i(H)=0 for i=1,...,q. If these equations hold, Cartan's identity gives L_{X_t^H}λ_i^E=0, so the right-hand side of the dissipation equation (2.24) and the nonzero Lie-derivative formula used in the proof of Theorem 2.5 refer to a different vector field. The assumption R_i(H)=0 is silently dropped after the construction, leaving it unclear which dynamics the generalized Noether theorem is meant to govern.","section":"Section 2.4, construction of X_t^H"}],"minor_comments":[{"comment":"The subsection on q-general manifolds contains several typos ('mani f old', 'co f rames', 'vector f ield') and Corollary 3.2 uses an undefined integer p in the family {R_j^k | k=1,...,p}; presumably q is intended.","section":"Section 3.3"},{"comment":"The numerical experiments do not report the integration interval, step size, or solver tolerances, so the statements that errors are 'of order 10^-10' and 'of order 10^-8' cannot be independently assessed from the text.","section":"Section 4.2"},{"comment":"Theorem 3.3 is a direct restatement of Definition 3.7 together with the definition of the bracket (3.27); labeling it a Noether theorem may overstate its content.","section":"Theorem 3.3"},{"comment":"Several key results, including Theorem 2.1 and the uniform q-contact bracket, are cited to [43] and [29], which are respectively 'To appear' and an arXiv preprint; the authors should clarify their status or include the proofs.","section":"References"}],"recommendation":"reject","confidential_remarks":"The generalized Noether theorem is advertised as a central contribution and is false in the stated form, with a simple counterexample in the uniform q=2 case. This is not a proof gap that a local revision can close; the theorem requires a substantive reformulation, likely with additional hypotheses or a different conclusion. The non-uniform framework and the elastoplastic application are independent of Theorem 2.5 and may be salvageable in a future version, but the current manuscript's central claim is unsound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the non-uniform admissible q-contact construction is real and mostly correct, and the R^6 example is worked out carefully. But the generalized Noether theorem in Section 2, one of the headline claims, is not proved as written, and I think the statement is wrong for q>1. The stress-test note is too kind: the Cartan identity repairs the \"previous proposition\" step, but it does not repair the wrong Lie-derivative line later in the same proof.\n\nWhat is actually new: the structure endomorphisms B_i, the averaged form Ω, admissibility as invertibility of B=(1/q)ΣB_i, the effective Hamiltonian vector field and bracket, and the effective volume evolution. These are original and coherent. Every uniform q-contact structure is admissible and the reduction checks out. The elastoplastic damage model with two internal variables is a legitimate worked example; the matrices, the admissibility computation, and the effective equations are consistent.\n\nNow the soft spots, in proportion. First and most serious: Theorem 2.5. The proof requires L_{X^t_H}λ_i^E = -(Σ_j R_j(H))λ_i^E. Direct computation in the uniform case gives L_{X^t_H}λ_i^E = -Σ_j R_j(H)λ_j^E + H_t dt. These agree for q=1, but not for q>1. Consequently L_{X^t_H}F_i = -Σ_j R_j(H)F_j + H_t i_Y dt, which is not the dissipation equation for F_i. The theorem needs either a stronger hypothesis on Y or a different conclusion. This is load-bearing.\n\nSecond, the extended-phase-space setup is internally inconsistent. The text assumes R_i(H)=0, then displays time-dependent Hamilton equations containing Σ∂H/∂z_i terms that vanish under that assumption. The rescaling remark is also confused: if X^t is a solution of homogeneous equations, multiplying by an arbitrary function is not a normalization.\n\nThird, the numerical experiments confirm exact algebraic identities—H(t)=H(0)e^{-(μ1+μ2)t} and I(t) constant—that follow directly from the constructed equations. They are fine as sanity checks, not as validation.\n\nThe uniform section is mostly review of [43] and the authors' own [29]. The non-uniform Noether theorem (Theorem 3.3) is a one-line equivalence from the bracket definition and is fine.\n\nWho this is for: people working on q-contact generalizations of dissipative Hamiltonian systems. The non-uniform toolbox is worth knowing, but I would not cite the generalized Noether theorem until it is fixed. Recommendation: send to a competent referee with instructions to focus on Section 2's extended phase space; if the theorem is false for q>1 as I believe, major revision is required before this is publishable.","headline":"The non-uniform admissible q-contact framework is a genuine new toolbox, but the generalized Noether theorem in Section 2 is not proved as written and appears false for q>1; fix that before publication.","tokens_in":25517,"tokens_out":11652,"would_cite":false,"duration_ms":121577,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C15","37J06","53D99"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper extends Noether's theorem to Hamiltonian systems with several independent dissipation channels, modeled by q-contact manifolds, and proves that in the non-uniform case an averaged 2-form governs a unique effective dissipative…","keywords":["q-contact manifold","Hamiltonian system","Noether theorem","dissipated quantity","non-uniform q-contact structure","effective Hamiltonian vector field","elastoplastic damage model"],"falsifier":"In q-contact coordinates, take a concrete vector field $Y$ that satisfies $L_Y\\lambda_i^E=\\sigma_i\\lambda_i^E$ but for which $[X_H^t,Y]$ is not horizontal, and compute $i_{[X_H^t,Y]}\\lambda_i^E$; if this term is nonzero, then $L_{X_H^t}F_i$ acquires an extra contribution and $F_i=i_Y\\lambda_i^E$ fails the dissipation equation, disproving Theorem 2.5 as stated. If every such $Y$ gives a vanishing bracket term, the theorem is confirmed.","tokens_in":24462,"feed_emoji":"📐","tokens_out":10402,"duration_ms":94020,"temperature":0.7,"pith_summary":"This paper extends Noether's theorem to Hamiltonian systems with several independent dissipation channels, modeled by q-contact manifolds: manifolds carrying q independent 1-forms whose kernels define a horizontal distribution, while the complementary Reeb directions play the role of dissipation channels. On uniform q-contact manifolds, where all forms share the same exterior derivative, the paper proves that Hamiltonian symmetries correspond exactly to dissipated quantities, with decay rate fixed by the Reeb vector fields. For time-dependent systems it proves a generalized Noether theorem on the extended phase space: a vector field that rescales each extended contact form yields q dissipated quantities. For non-uniform structures, where the symplectic forms on the horizontal distribution differ, it shows that whenever the averaged 2-form $\\Omega=\\frac1q\\sum_{i=1}^q d\\lambda_i$ is non-degenerate there is a unique effective Hamiltonian vector field and the Noether correspondence still holds. As an application, an elastoplastic damage model with two internal variables is embedded in the non-uniform 2-contact framework, and numerical integration confirms the predicted exponential dissipation law and the associated rescaled conserved quantity.","feed_headline":"Noether's theorem extended to multi-channel dissipative systems","feed_subtitle":"Symmetries in q-contact geometry yield predictable decay; one effective flow combines all channels.","key_machinery":"The central object is the q-contact structure $(M,\\vec{\\lambda}=(\\lambda_1,\\dots,\\lambda_q),R\\oplus\\xi)$, where $\\xi=\\cap_i\\ker\\lambda_i$ is the horizontal distribution, $R$ the Reeb distribution spanned by $R_i$ with $\\lambda_i(R_j)=\\delta_{ij}$, and each $\\lambda_i$ is one dissipation channel. The identity that carries the non-uniform argument is the averaged 2-form $\\Omega=\\frac1q\\sum_i d\\lambda_i$ on $\\xi$; its non-degeneracy, the admissibility condition, selects a unique effective Hamiltonian vector field, and the structure endomorphisms $B_i:\\xi\\to\\xi$ defined by $d\\lambda_i(X,Y)=\\omega(B_iX,Y)$ with $\\omega=d\\lambda_1|_\\xi$ encode channel anisotropy, with averaged $\\bar{B}=\\frac1q\\sum_i B_i$ controlling the horizontal part of the effective flow. In the extended phase space, the machinery is the collection $\\lambda_i^E=\\lambda_i+H\\,dt$ on $M\\times\\mathbb{R}$, whose Lie derivatives along $X_H^t$ are $L_{X_H^t}\\lambda_i^E=-\\left(\\sum_j R_j(H)\\right)\\lambda_i^E$, the formula that turns symmetry into dissipation.","core_discovery":"The central claim is the q-contact Noether correspondence: on a uniform q-contact manifold, a Hamiltonian vector field $X_F$ is a Noether symmetry of $X_H$ (meaning $\\{F,H\\}=0$) if and only if $F=-i_{X_F}\\lambda_1$ is a dissipated quantity, i.e. $X_H(F)=-F\\sum_i R_i(H)$ (Theorem 2.3). For time-dependent systems on $M\\times\\mathbb{R}$ with extended forms $\\lambda_i^E=\\lambda_i+H\\,dt$, the paper states the generalized Noether theorem: if $Y$ satisfies $L_Y\\lambda_i^E=\\sigma_i\\lambda_i^E$ for $i=1,\\dots,q$, then $F_i=i_Y\\lambda_i^E$ are dissipated quantities along the flow of $X_H^t=X_H+\\partial_t$ (Theorem 2.5). In the non-uniform setting, where the $d\\lambda_i$ differ, the paper fixes a reference symplectic form $\\omega=d\\lambda_1|_\\xi$, defines structure endomorphisms $B_i$ by $d\\lambda_i(X,Y)=\\omega(B_iX,Y)$, and, under the admissibility condition that $\\Omega=\\frac1q\\sum_i d\\lambda_i$ is non-degenerate on $\\xi$, constructs a unique effective Hamiltonian vector field $X_H$ with $\\lambda_i(X_H)=-H$ and $i_{X_H}\\Omega=dH-\\sum_i dH(R_i)\\lambda_i$ (Theorem 3.2). With this field it proves the non-uniform Noether theorem: $\\{H,F\\}=0$ iff $F$ is dissipated, and $F$ is conserved iff $\\{H,F\\}=-F\\sum_i R_i(H)$ (Theorem 3.3). The elastoplastic two-channel example realizes this construction with $R_1(H)=\\mu_1$, $R_2(H)=\\mu_2$, yielding $H(t)=H(0)e^{-(\\mu_1+\\mu_2)t}$ and conserved $I(t)=H(t)e^{(\\mu_1+\\mu_2)t}$.","pith_inferences":["A direct test of Theorem 2.5 would compute the missing bracket term $i_{[X_H^t,Y]}\\lambda_i^E$ in local coordinates; if it vanishes for all generalized Noether symmetries, the theorem is sound, and if not, the dissipation property should hold for symmetries satisfying the extra commutation condition.","The averaged-form construction suggests a general design principle: whenever several symplectic forms on one distribution average to a non-degenerate 2-form, a single effective Hamiltonian dynamics exists, which may connect this framework to other geometric settings where multiple symplectic structures coexist.","The exponential-rescaling identity $I(t)=H(t)e^{(\\sum_i\\mu_i)t}$ is not special to the two-channel example: any admissible non-uniform q-contact system with constant Reeb derivatives $R_i(H)=\\mu_i$ will exhibit the same coexistence of exponential dissipation and an exactly conserved rescaled Hamiltonian, which could be checked in materials with multiple internal dissipation mechanisms."],"forward_implications":["On a uniform q-contact manifold, every Noether symmetry of the Hamiltonian flow is equivalent to a dissipated quantity whose decay rate is $\\sum_i R_i(H)$, so continuous symmetries of dissipative systems directly predict how fast their generators decay.","On an admissible non-uniform q-contact manifold, the effective vector field $X_H$ reduces the multi-channel system to a single dynamics, and the Hamiltonian obeys $dH/dt=-H\\sum_i R_i(H)$; with constant $\\mu_i=R_i(H)$ the total dissipation is the sum of the channel rates.","For the two-channel elastoplastic damage model, the theory forces $H(t)=H(0)e^{-(\\mu_1+\\mu_2)t}$ and $I(t)=H(t)e^{(\\mu_1+\\mu_2)t}=\\text{const}$, and the reported numerical integrations agree with both to near machine precision.","The generalized Noether theorem outputs $q$ dissipated quantities $F_i=i_Y\\lambda_i^E$ from one symmetry vector field, one per contact form, rather than a single conserved charge.","In the uniform limit $B_i=\\mathrm{Id}_\\xi$, all non-uniform constructions reduce to the uniform q-contact theory, which in turn reduces to ordinary contact Hamiltonian mechanics when $q=1$."],"supporting_citations":[{"why":"Supplies the definition of q-contact structures and the adapted coframe used as the starting point of Sections 2 and 3.","marker":"[3]"},{"why":"Introduces q-contact manifolds and contact foliations, grounding the Reeb-distribution framework used throughout.","marker":"[20, 21]"},{"why":"Provides the existence and uniqueness of the q-contact Hamiltonian vector field and the q-contact bracket used in Theorem 2.1 and Definition 2.5.","marker":"[43]"},{"why":"Gives the time-dependent contact construction whose extended phase space and $\\lambda_i^E=\\lambda_i+H\\,dt$ pattern the generalized Noether setting.","marker":"[27]"},{"why":"Supplies the Noether-type theorem for q-contact geometry that the paper's generalized Noether theorem extends.","marker":"[29]"},{"why":"Establishes the dissipated-quantity picture for contact Hamiltonian systems, which the q-contact version generalizes.","marker":"[34]"},{"why":"Provides background results on conserved quantities and symmetries for Hamiltonian systems against which the Noether theorems are framed.","marker":"[44]"}],"fun_headline_variants":["Noether meets dissipation: symmetries predict multi-channel decay","q-contact Noether: symmetries dictate exponential decay","Noether's law for dissipation: symmetries predict decay","From geometry to decay: Noether on q-contact manifolds","Multi-channel decay law from q-contact Noether"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The generalized Noether theorem rests on the assumption that the symmetry vector field and the Hamiltonian flow commute along the contact directions, so the bracket term $i_{[X_H^t,Y]}\\lambda_i^E$ vanishes; the paper asserts this without proof, and the definition of a generalized Noether symmetry does not obviously imply it.","fun_headline_variants_meta":{"raw":{"variants":["Noether meets dissipation: symmetries predict multi-channel decay","q-contact Noether: symmetries dictate exponential decay","Noether's law for dissipation: symmetries predict decay","From geometry to decay: Noether on q-contact manifolds","Multi-channel decay law from q-contact Noether"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000702,"raw_usage":{"total_tokens":3219,"prompt_tokens":1050,"completion_tokens":2169,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":666,"completion_tokens_details":{"reasoning_tokens":2104}},"tokens_in":666,"tokens_out":2169,"duration_ms":15216,"temperature":1.0,"reasoning_tokens":2104,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T04:16:24.458661+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In q-contact coordinates, take a concrete vector field $Y$ that satisfies $L_Y\\lambda_i^E=\\sigma_i\\lambda_i^E$ but for which $[X_H^t,Y]$ is not horizontal, and compute $i_{[X_H^t,Y]}\\lambda_i^E$; if this term is nonzero, then $L_{X_H^t}F_i$ acquires an extra contribution and $F_i=i_Y\\lambda_i^E$ fails the dissipation equation, disproving Theorem 2.5 as stated. If every such $Y$ gives a vanishing bracket term, the theorem is confirmed.","supporting_citations":[{"cited_title":"Almeida, Contact Anosov actions with smooth invariant bundles","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of q-contact structures and the adapted coframe used as the starting point of Sections 2 and 3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the existence and uniqueness of the q-contact Hamiltonian vector field and the q-contact bracket used in Theorem 2.1 and Definition 2.5."},{"cited_title":"de León, J","cited_arxiv_id":null,"evidence_quote":"Gives the time-dependent contact construction whose extended phase space and $\\lambda_i^E=\\lambda_i+H\\,dt$ pattern the generalized Noether setting."},{"cited_title":"Noether-Type Theorems and the Generalized Herglotz Principle in $q$-Contact Geometry","cited_arxiv_id":"2604.06488","evidence_quote":"Supplies the Noether-type theorem for q-contact geometry that the paper's generalized Noether theorem extends."},{"cited_title":"Pérez, Álvarez, Symmetries and dissipation laws on contact systems, Mediterr","cited_arxiv_id":null,"evidence_quote":"Establishes the dissipated-quantity picture for contact Hamiltonian systems, which the q-contact version generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides background results on conserved quantities and symmetries for Hamiltonian systems against which the Noether theorems are framed."}],"review_version":1}