{"id":"19166f2f-f551-4de8-98ed-e52345f68526","arxiv_id":"2607.17407","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Fouling maps are generalized to non-invertible bundle maps, and traces of a flow-invariant (1,1)-tensor built from symmetric tensor fields yield polynomial first integrals for natural Hamiltonian systems.","lead":"This paper introduces \"fouling maps,\" a non-invertible kind of canonoid transformation that preserves positions, and shows how to build them from symmetric tensor fields on the configuration space. Each such map yields polynomial constants of motion, giving a systematic way to find first integrals for mechanical systems, with examples on the Euclidean plane and the 2-sphere.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.9(2) is false: for k=1, fouling requires R_T1=0 and d(♭(i_dV T1))=0; counterexample V=xy, T1=∂x⊗∂x gives non-fouling despite R_T1=0.","rationale":"The reader's weakest assumption concerned the generation of vector fields on T*Q by basic and linear Hamiltonian vector fields; that is a standard fact and is not the main vulnerability. I found a concrete, checkable error in Theorem 4.9(2): for k=1, the condition R_T1=0 is insufficient; Proposition 4.8(iii) leaves a residual term equivalent to d(♭(i_dV T_1))=0. A simple flat-space counterexample (V=xy, T_1=∂_x∂_x) confirms that Ψ_T1 fails to be fouling despite R_T1=0. This is a genuine mathematical flaw in a stated theorem, though it does not invalidate the central Theorem 4.10, which correctly includes the closure condition as (iii). The paper therefore needs a correction to Theorem 4.9(2), and the reader's CONDITIONAL verdict remains appropriate.","tokens_in":30786,"tokens_out":31134,"duration_ms":268372,"concrete_test":"Direct check: on R^2 with Euclidean metric, set V=xy, T_1=∂_x⊗∂_x, H=(p_x^2+p_y^2)/2+xy. Compute Ψ_T1(q,p)=(q,p_x,0), ω_Ψ=dx∧dp_x, X_H=p_x∂_x+p_y∂_y-y∂_{p_x}-x∂_{p_y}. Then L_{X_H}ω_Ψ=d(i_{X_H}ω_Ψ)=d(p_x dp_x+y dx)=dy∧dx≠0, so Ψ_T1 is not a fouling map, although R_{T_1}=0. This refutes Theorem 4.9(2).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The special case k=1 of Theorem 4.9 is incorrect. The theorem states that Ψ_{T_1} is a fouling map for a mechanical Hamiltonian iff R_{T_1}=0. However, Proposition 4.8(iii) for k=1 contains, after setting R_{T_1}=0, a residual term [L_Y i_dV i_♭Z T_1 - L_Z i_dV i_♭Y T_1 - i_dV i_♭[Y,Z] T_1]^{hom}. This term is not automatically zero; it equals d(♭(i_dV T_1))(Y,Z). Theorem 4.10 includes this as condition (iii), but Theorem 4.9(2) omits it. A concrete counterexample: on Q=R^2 with Euclidean metric, take V=xy and T_1=∂_x⊗∂_x. Then R_{T_1}=0 (constant coefficients), and Ψ_T1(q,p)=(q,p_x,0). For H=(p_x^2+p_y^2)/2+xy, X_H=p_x∂_x+p_y∂_y-y∂_{p_x}-x∂_{p_y} and ω_Ψ=dx∧dp_x. Then i_{X_H}ω_Ψ=p_x dp_x+y dx, so d(i_{X_H}ω_Ψ)=dy∧dx≠0. Hence Ψ_T1 is not fouling, contradicting Theorem 4.9(2). The missing condition d(♭(i_dV T_1))=0 is essential.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces 'fouling maps', non-invertible bundle maps over the identity on T*Q that preserve the Hamiltonian nature of a vector field in a generalized sense, and studies the associated (1,1)-tensor L_Ψ. It proves that if Ψ is fouling for H, then L_Ψ is invariant under X_H and the traces of its powers are first integrals (Theorem 3.6). For mechanical Hamiltonians, the paper constructs polynomial fouling maps from symmetric tensor fields on Q, culminating in Theorem 4.10, which characterizes when a finite sum Ψ_T = Σ Ψ_{T_k} is fouling in terms of the tensors R_{T_k}, the potential V, and a closure condition on ♭(i_dV T_1). Several explicit examples are given on the Euclidean plane, the 2-sphere, and a Liouville metric manifold, with associated polynomial constants of motion.","tokens_in":31222,"tokens_out":18679,"duration_ms":170599,"significance":"If correct, the framework offers a systematic, tensor-geometric method for producing polynomial first integrals for natural Hamiltonian systems, going beyond the invertible canonoid/fouling transformation literature and including the first non-trivial curved-space examples of which I am aware. The main conceptual contribution — passing from invertible fouling transformations to non-invertible fouling maps and deriving invariant tensor fields — is natural and potentially useful. The derivations are mostly coordinate tensor computations, and Theorem 4.10 has the form of a complete characterization. However, the paper currently contains a false special-case theorem (Theorem 4.9(2)) and internally inconsistent sphere examples. These are local but load-bearing for the claimed completeness, so the manuscript needs revision before the results can be relied on.","major_comments":[{"comment":"Theorem 4.9(2) is false as stated. For k=1, after imposing R_{T_1}=0, Proposition 4.8(iii) leaves the residual term [L_Y i_{dV} i_{♭Z} T_1 - L_Z i_{dV} i_{♭Y} T_1 - i_{dV} i_{♭[Y,Z]} T_1]^{hom}, which equals d(♭(i_{dV}T_1))(Y,Z). This is not automatically zero. The proof's assertion that '(b) is always satisfied' for k=1 is incorrect. Concrete counterexample: on Q=R^2 with Euclidean metric, V=xy and T_1=∂_x⊗∂_x. Then R_{T_1}=0, but Ψ_{T_1}(x,y,p_x,p_y)=(x,y,p_x,0). For H=(p_x^2+p_y^2)/2+xy, one computes X_H=p_x∂_x+p_y∂_y-y∂_{p_x}-x∂_{p_y} and ω_Ψ=dp_x∧dx, so i_{X_H}ω_Ψ=-y dx-p_x dp_x and d(i_{X_H}ω_Ψ)=dx∧dy≠0; hence Ψ_{T_1} is not fouling. The correct condition is R_{T_1}=0 together with d(♭(i_{dV}T_1))=0, which is exactly the condition that appears in Theorem 4.10(iii). This error should be corrected and Theorem 4.9(2) should be reconciled with Theorem 4.10.","section":"Theorem 4.9(2)"},{"comment":"The sphere examples contain an internal inconsistency. In Section 6.1 the closedness condition for T_0 is correctly derived as (6.7): ∂_ϕ A_0 - sin^2 θ ∂_θ B_0 - B_0 sin(2θ)=0, and the bundle map is written with fiber coordinate B_0 sin^2 θ. However, in the final maps (6.17) and (6.20) the stated condition is '∂B_0/∂θ - ∂A_0/∂ϕ = 0', which is the Euclidean condition, not (6.7). Moreover, in (6.17) the vertical component P_ϕ is written as B_0 + ... rather than B_0 sin^2 θ + ... . If B_0 sin^2 θ is intended, the condition must be (6.7); if the Euclidean-looking condition is intended, the semi-basic form is different. As written, the examples are not verified to be fouling maps, and the final 'with ∂B_0/∂θ - ∂A_0/∂ϕ = 0' is not equivalent to the closedness condition used earlier. This must be fixed for the curved-space examples to be valid.","section":"Section 6.1"}],"minor_comments":[{"comment":"The local expression of a semi-basic 1-form is written as θ_i(q,p) dp_i; it should be θ_i(q,p) dq^i. This typo is confusing because semi-basic forms have no dp terms.","section":"Remark 3.2"},{"comment":"The statement that Hamiltonian vector fields of basic functions and linear functions 'generate the space of vector fields' is terse. What is needed is that they span T_m(T^*Q) at each point, or generate X(T^*Q) as a C^∞-module; this is true (e.g., X_{q^i∘π}=-∂_{p_i} and X_{p_i}=∂_{q_i}). The current wording may mislead readers into thinking a Lie-algebra generation is being claimed.","section":"Section 3"},{"comment":"There is a missing parenthesis/typo: d(♭(i_dV T_1) should be d(♭(i_dV T_1)) = a_1 d(dV)=0.","section":"Eq. (6.9)"},{"comment":"In equations (6.21) and (6.22), the variable ψ is used in expressions such as sin^2 ψ and cos^2 ψ, although the potential and coordinates are in (θ,ϕ). This appears to be a typesetting artifact and should be corrected to ϕ.","section":"Examples 6.1,6.2"},{"comment":"The statement refers to 'The author' in the singular, although the paper has three authors; the intended author should be named or rephrased.","section":"AI Use Statement"}],"recommendation":"major_revision","confidential_remarks":"The most serious issue is the false statement of Theorem 4.9(2); it is a clear mathematical error with a simple counterexample, but it is local and the paper's main theorem (Theorem 4.10) already contains the missing closure condition. The sphere examples also need a consistency fix. Neither issue seems to invalidate the overall method or the main characterization theorem if corrected. I recommend major revision rather than rejection, with the expectation that the authors can repair the stated theorem and examples without changing the core framework."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nRead this one for Theorem 4.10, not Theorem 4.9. The paper's substantive contribution is a tensorial characterization of polynomial fouling maps for natural Hamiltonian systems. That result, including the non-invertible definition and the sphere examples, appears to be new, and the proof chain is coherent. The trace-of-powers invariant trick is inherited from Azuaje–Escobar-Ruiz, but the application to symmetric tensor fields and the curved-space examples are genuine extensions.\n\nThe main thing you need to know: Theorem 4.9(2) is false as stated. For k=1 it claims Ψ_{T1} is fouling iff R_{T1}=0, but Proposition 4.8 leaves a residual term that is exactly d(♭(i_{dV} T1)). The counterexample works: Q=R^2 with Euclidean metric, V=xy, T1=∂x⊗∂x. Then R_{T1}=0, but Ψ_T1(q,p)=(q,p_x,0), and i_{X_H}ωΨ = p_x dp_x + y dx has differential dy∧dx≠0. The missing closedness condition is included in Theorem 4.10(iii), so the main characterization for polynomial sums is consistent. The bug is isolated to the single-tensor statement, but it is in a numbered theorem and has to be fixed.\n\nAlso check the sphere examples. They state the T0 condition as ∂B0/∂θ − ∂A0/∂φ = 0, but the correct condition (6.7) is ∂φA0 − sin^2θ ∂θB0 − B0 sin(2θ)=0. That looks like a transcription typo rather than a wrong computation, but as printed the examples do not match their own equations.\n\nRemark 6.3 overreaches: claiming no non-trivial fouling or canonoid transformation on a curved space exists in the literature is a strong negative claim with no citation support. It should be softened to 'to the best of our knowledge' with a genuine literature check.\n\nMinor: the reduction to the three bracket conditions in (3.2) uses the fact that basic and linear Hamiltonian vector fields generate all vector fields on T*Q. Standard, but unproved. Fine for a paper at this level.\n\nNet: the central theorem holds, the examples are mostly right, and the flaws are local. This deserves a serious referee. Send it out, but the referee should be asked to verify Theorem 4.9 against Theorem 4.10 and re-check the S^2 example conditions.","headline":"Theorem 4.10 is the real result and looks right; Theorem 4.9(2) is false as stated, and the S^2 examples have a condition typo—fixable, not fatal.","tokens_in":31650,"tokens_out":4413,"would_cite":true,"duration_ms":40834,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["70G45","70H15","53Z05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that polynomial fouling maps for mechanical Hamiltonians are completely characterized by tensor equations and that each such map induces an invariant tensor whose trace powers are polynomial first integrals.","keywords":["fouling maps","canonoid transformations","polynomial constants of motion","mechanical Hamiltonian","symmetric tensor fields","cotangent bundle","invariant tensor fields","Liouville metric"],"falsifier":"Take a fouling map built from symmetric tensors that satisfies Theorem 4.10 on a specific manifold (e.g., the 2-sphere example with V = sin^2θ) and check that the computed first integrals f1 and f2 are indeed constant along numerical solutions of Hamilton's equations; a nonzero Lie derivative along X_H would disprove the claim.","tokens_in":30708,"feed_emoji":"⚙️","tokens_out":3553,"duration_ms":31271,"temperature":0.7,"pith_summary":"The paper introduces fouling maps: bundle maps over the identity on a cotangent bundle that, without being invertible, satisfy the same invariance condition that defines canonoid transformations. The central claim is that for mechanical Hamiltonians (kinetic energy plus potential on a semi-Riemannian manifold), every polynomial fouling map built from symmetric tensor fields is completely characterized by a short list of tensor equations involving a curvature-like tensor R_T and the potential's differential. From any such map the authors construct a (1,1)-tensor that is invariant under the Hamiltonian flow, so the traces of its powers are polynomial first integrals. This gives a systematic tensorial method to produce explicit polynomial constants of motion, with worked examples on the Euclidean plane and the 2-sphere.","feed_headline":"Symmetric tensors yield new polynomial first integrals","feed_subtitle":"A non-invertible version of canonical transformations produces explicit conserved quantities for mechanical systems, including on the sphere","key_machinery":"The central object is the (1,1)-tensor L_Ψ = ♭_{ω_Ψ} ∘ #_{ω_Q} on T*Q, where ω_Ψ = Ψ*ω_Q and #,♭ are the musical isomorphisms induced by the metric. Because L_Ψ commutes with the Hamiltonian flow, its trace powers are first integrals. The paper's tensorial method uses the one-to-one correspondence between homogeneous bundle maps over the identity and symmetric tensor fields on the base, reducing the fouling condition to algebraic equations on the tensors.","core_discovery":"The main result (Theorem 4.10) says that a polynomial bundle map Ψ = Σ Ψ_{T_k}, with each T_k a symmetric (k+1,0)-tensor, is a fouling map for H = (g*)^hom + V if and only if R_{T_k} = i_{dV} T_{k+2} for k=0,...,s-2, R_{T_s}=R_{T_{s-1}}=0, and d(♭(i_{dV}T_1))=0. Here R_{T_k} is a (k+2,0)-tensor built from covariant derivatives of T_k. When these equations hold, the associated bundle map preserves the Hamiltonian nature of the dynamics, and the induced operator L_Ψ is invariant under X_H; consequently, for each positive integer l, the trace of L_Ψ^l is a polynomial constant of motion.","pith_inferences":["The trace powers of L_Ψ may not exhaust the first integrals; the eigenvalues of L_Ψ are also invariant, as the sphere examples show, suggesting a fuller algebra of invariants.","The characterization could be extended to time-dependent or Poisson settings, but the paper does not do so.","The conditions R_T = i_dV T have a flattening interpretation: the tensor T must be 'Killing up to a contraction with dV', which might connect to deformed Killing tensors in superintegrability."],"forward_implications":["Any fouling map for a mechanical Hamiltonian yields a family of polynomial constants of motion, one for each power of L_Ψ.","The characterization turns the search for polynomial integrals into a linear PDE problem on symmetric tensors, solvable in examples.","For zero potential, fouling maps reduce to tensors with R_T=0, giving a new class of integrals for geodesic flows.","Examples on the 2-sphere provide the first non-trivial fouling maps for natural Hamiltonians on curved spaces, with functionally independent integrals.","The method extends to Liouville metrics, producing explicit independent invariants."],"fun_headline_variants":["Polynomial integrals from symmetric tensor fields","Fouling maps unlock polynomial first integrals","Symmetric tensors produce new conserved polynomials","Mechanical systems get new integrals via tensor maps","Non-invertible transformations yield conserved quantities"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof that the three local conditions in (3.2) capture the full fouling property relies on the standard fact that the Hamiltonian vector fields of coordinate and momentum functions generate every vector field on the cotangent bundle; if only a smaller subalgebra were generated, the conditions would be weaker than invariance.","fun_headline_variants_meta":{"raw":{"variants":["Polynomial integrals from symmetric tensor fields","Fouling maps unlock polynomial first integrals","Symmetric tensors produce new conserved polynomials","Mechanical systems get new integrals via tensor maps","Non-invertible transformations yield conserved quantities"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000473,"raw_usage":{"total_tokens":2179,"prompt_tokens":729,"completion_tokens":1450,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":473,"completion_tokens_details":{"reasoning_tokens":1394}},"tokens_in":473,"tokens_out":1450,"duration_ms":9622,"temperature":1.0,"reasoning_tokens":1394,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T18:02:18.760627+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a fouling map built from symmetric tensors that satisfies Theorem 4.10 on a specific manifold (e.g., the 2-sphere example with V = sin^2θ) and check that the computed first integrals f1 and f2 are indeed constant along numerical solutions of Hamilton's equations; a nonzero Lie derivative along X_H would disprove the claim.","supporting_citations":[],"review_version":1}