{"id":"e5f9aea6-122d-41c1-a8da-b8e314c830b1","arxiv_id":"2607.07057","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Involutive systems of particular integrals define coisotropic constraint sets whose characteristic quotients carry reduced Hamiltonian flows, yielding a notion of particular Liouville integrability.","lead":"This paper shows that families of weakly-conserved quantities—'particular integrals'—can generate lower-dimensional Hamiltonian dynamics through a standard geometric reduction. It connects a broad class of restricted-integrability examples, including Eisenhart lifts, under one framework.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Core theorem is sound, but no example verifies the simplicity of the characteristic foliation needed for the quotient; the only nontrivial claimed reduction (Example 5) is asserted without checking it.","rationale":"After reading the manuscript, I find no error in the proof of Theorem 1; it is standard coisotropic reduction and the internal logic is sound. The reader's weakest assumption—regularity of the characteristic foliation in the examples—is indeed the most load-bearing gap. The theorem is conditional, and the paper's examples do not verify the condition. Section 4 lift examples do not perform the characteristic quotient at all; they only establish invariance of zero sets. Example 5 is the sole nontrivial claimed reduction and it is asserted without verification. This does not invalidate the central theorem, which is correct as stated, but it means the paper's claim of a 'bridge' to lower-dimensional Hamiltonian dynamics is demonstrated only in the trivial cyclic-coordinate case (Example 3). The appropriate response is to keep the acceptance unchanged while noting the missing verification; if the regularity condition fails in Example 5, that example's conclusion would collapse, but the theorem itself would stand. Hence UNCHANGED.","tokens_in":23589,"tokens_out":20712,"duration_ms":184956,"concrete_test":"Verify Example 5: write K_x and K_y explicitly, compute X_{K_x} and X_{K_y} and their flows, and determine the leaf space M̄ = {K_x=K_y=0}/(R² action). Check that M̄ is a smooth 4-dimensional manifold (or find singular leaves), and that H̄ and J̄_z are functionally independent on M̄ with {H̄, J̄_z}=0. If the quotient is not smooth or the functions are dependent, the example's particular-Liouville-integrability claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1 is a correct conditional statement—essentially coisotropic reduction, as Remark 2 concedes—but its conclusion (the existence of a reduced Hamiltonian system on M̄) requires the characteristic foliation ker Ω to be simple and the leaf space to be a smooth manifold. This hypothesis is never verified in any example. In the lift constructions of §4.1–4.4, the authors only check dynamic invariance of zero sets (e.g., {I=h, p_z=0}); they do not form or check the characteristic quotient. Indeed, quotienting by the full system {f−c, p_z} would give a reduced space of dimension 2(n−1), not the original 2n-dimensional system, so those examples do not instantiate Theorem 1. The only nontrivial claimed application of the reduction is Example 5 (two-body Coulomb in a magnetic field), which asserts a reduced 2-degree-of-freedom Hamiltonian system and particular Liouville integrability without computing the leaf space or verifying simplicity. If ker Ω is not simple, H̄ and the reduced Liouville tori do not exist, so the paper's central bridge is unestablished for its own examples.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces systems of particular integrals, families of functions f_1,...,f_k whose time derivatives close linearly on the family. Lemma 1 shows that the common zero set M_f is a dynamically invariant embedded submanifold under a functional-independence assumption. In the Hamiltonian setting, Theorem 1 states that if the f_i are in particular involution, then M_f inherits a presymplectic form Ω = ι*ω whose characteristic distribution is spanned by the restricted Hamiltonian vector fields X_{f_i}; if the characteristic foliation is simple and the leaf space is a smooth quotient, the restricted Hamiltonian descends to a unique Hamiltonian system on a 2(n-k)-dimensional symplectic manifold. The paper then defines particular Liouville integrability and illustrates it with the planar two-body Coulomb problem in a constant magnetic field (Example 5). The remainder constructs lifts (Eisenhart, scalar/diagonal/non-diagonal auxiliary, magnetic, polynomial) that produce systems of particular integrals and recover the original dynamics by restriction to the zero set of auxiliary momenta and projection along auxiliary variables.","tokens_in":1383,"tokens_out":2065,"duration_ms":153141,"significance":"The main theorem is proved carefully and is correct under the stated regularity hypotheses; the proof is essentially standard coisotropic reduction, as the authors explicitly acknowledge in Remark 2. The paper's contribution lies in the formulation of systems of particular integrals as generators of invariant constraint ideals, and in the notion of particular Liouville integrability, which connects non-global conservation laws to presymplectic reduction. The lift constructions are concrete and may be useful for producing systems with particular integrals. However, the sole nontrivial example of particular Liouville integrability (Example 5) is asserted rather than verified in its quotient regularity, which is a load-bearing gap in the illustration of the paper's central new notion.","major_comments":[{"comment":"The claim that the original system is particularly Liouville integrable requires verifying the hypotheses of Theorem 1 on M_f = {K_x = K_y = 0}: the characteristic distribution spanned by X_{K_x}|M_f and X_{K_y}|M_f must have a simple foliation, the leaf space must be a smooth manifold with surjective submersion, and the reduced functions H_bar and J_z_bar must be functionally independent on the quotient. None of these is checked. Since this is the only example of Definition 2, the new notion is not yet substantiated. Please provide the quotient construction (or state and verify a sufficient condition, e.g. that the R^2 action generated by K_x, K_y is free and proper on M_f) and prove the independence of H_bar and J_z_bar.","section":"§3.2, Example 5"},{"comment":"The lift constructions are presented as illustrations of the framework, but they do not instantiate Theorem 1. In each case the invariant set is {f=c, p_z=0} (or with several auxiliary momenta); the characteristic distribution for the system {f-c, p_z} is two-dimensional, spanned by X_f|M_f and ∂/∂z, and the quotient by this distribution would have dimension 2(n-1), not the dimension of the original system. The paper only projects along the auxiliary coordinate z after restricting to p_z=0, which is a different procedure. The simplicity of the characteristic foliation is never checked for these families. If these sections are meant only to illustrate Lemma 1, this should be stated explicitly; if they are intended as applications of the reduction theorem, the characteristic quotient must be constructed.","section":"§4.1–4.4"}],"minor_comments":[{"comment":"The displayed formula for {H, eH} has a sign error: the right-hand side should have an overall minus sign. The conclusion is unaffected because the expression vanishes on the level p_z^2 = 2.","section":"§4.1, Example 6"},{"comment":"The local-coordinate description assumes the stronger condition {f_l, f_s}=0, whereas Theorem 1 only assumes particular involution (4), i.e. {f_i,f_j}=c^ell_{ij} f_ell. The relation between the two conditions should be clarified, since the local slice M_{f,Q} is a slice of the characteristic distribution only under the stronger involution assumption.","section":"§3.1.2"},{"comment":"The statement that 'the pair z, p_z forms a system of particular integrals' is correct, but the subsequent discussion only uses the hypersurface p_z=0, not the zero set of z. Consider rephrasing to avoid suggesting that {z=0, p_z=0} plays a role.","section":"§4.1"},{"comment":"The magnetic vector potential is denoted A_i, while the auxiliary lift functions in §4.2 are also called A(q,z); later §4.3 uses Lambda for the auxiliary function. Consider renaming to avoid notational collision.","section":"§4.2 and §4.3"},{"comment":"The involution of the pseudomomentum components K_x, K_y is stated without proof or reference. A short verification or reference would help, since the bracket is not immediately obvious for two charges in a magnetic field.","section":"Example 5"}],"recommendation":"major_revision","confidential_remarks":"The central theorem is standard coisotropic reduction, as the authors admit, so the novelty rests on the framing and the new integrability notion. The gap in Example 5 regarding the smoothness of the characteristic quotient is fixable by a computation and should be addressed before publication. The paper is otherwise well written and within the scope of the journal; I do not see a need for rejection if the example is repaired or explicitly qualified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Core take: this is a sound, modest paper. Theorem 1 is exactly what the authors say it is—standard coisotropic reduction (Remark 2)—so the real contribution is the framing: systems of particular integrals (Definition 1), their role as generators of coisotropic constraints, and the new label “particular Liouville integrability.” That framing is clean and worth having. Lemma 1 is proved carefully; Theorem 1 is correctly stated under the regularity hypotheses; Remark 2 is honest. The lift sections do a nice job showing that first integrals of a base system become components of systems of particular integrals in Eisenhart-type and auxiliary lifts.\n\nWhere the paper is soft: none of the examples actually verifies that the characteristic foliation is simple. The stress-test note is right about this. Example 3 works because the reduction is visible in the chosen coordinates, but the authors don't say so. Example 5, the only nontrivial claimed reduction, asserts the 2-DOF reduced system and its Liouville integrability without checking that the leaf space is a smooth manifold. That is a real gap in the illustration, not in the theorem. Also, the lift examples in Section 4.2 don't instantiate Theorem 1 at all—quotienting by {f−c, p_z} would give a different space than the original dynamics; they are only dynamic invariance plus dropping the auxiliary coordinate. The paper doesn't explicitly claim otherwise, but a reader could be misled. The local-coordinate section assumes strict involution rather than the weaker particular involution, which is fine but narrower than the theorem.\n\nCitation pattern is normal; the prior-work references are the natural source for the scalar notion, and the generalization is acknowledged. No fitted parameters, no circularity.\n\nWho is this for? Geometric mechanicians who work with constrained dynamics, presymplectic reduction, or Eisenhart lifts. It will be a useful reference for the particular-integral framing, not for new reduction theorems. Send it to a competent referee; with minor revisions—mostly verifying or qualifying the examples—it is publishable.","headline":"Sound but modest paper: the reduction theorem is honestly labeled standard coisotropic reduction; the new value is the particular-integral framing and the Liouville-type definition, though the examples skip the foliation check.","tokens_in":24308,"tokens_out":7913,"would_cite":true,"duration_ms":69180,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37J35","70H06","53D20","70H45"],"pacs":[],"model":"deepseek-v4-flash","headline":"Systems of particular integrals — functions whose time derivatives close linearly on the family — define invariant submanifolds, and their characteristic quotients carry reduced Hamiltonian flows. The paper proves this and introduces partic","keywords":["particular integrals","Hamiltonian reduction","presymplectic reduction","Liouville integrability","invariant submanifolds","auxiliary-coordinate lifts","magnetic Hamiltonian systems","polynomial Hamiltonians"],"falsifier":"Take the two-body Coulomb system in a constant magnetic field, restricted to K_x=K_y=0 as in the paper, and compute the foliation generated by the Hamiltonian vector fields of K_x and K_y on that invariant set. If the leaf space is not a smooth manifold, the theorem's regularity hypothesis fails and the claimed reduced two-degree system is not defined for this example, marking the boundary of the claim.","tokens_in":1376,"feed_emoji":"⚛️","tokens_out":3340,"duration_ms":117961,"temperature":0.7,"pith_summary":"The paper aims to show that a family of phase-space functions whose time derivatives close linearly on the family itself — a system of particular integrals — defines a dynamically invariant submanifold, even when no single member is conserved. In Hamiltonian mechanics, if the family is additionally in particular involution and functionally independent on its common zero set, the restricted dynamics becomes presymplectic, and quotienting by the characteristic distribution produces a genuine Hamiltonian system with fewer degrees of freedom. The authors name this outcome particular Liouville integrability and illustrate it with mechanical examples and with auxiliary-coordinate lifts of natural Hamiltonian systems. The broader point is that non-global, partial conservation laws can organize lower-dimensional Hamiltonian dynamics through a mechanism that directly generalizes symplectic reduction.","feed_headline":"Particular integrals shrink Hamiltonian systems by k dimensions","feed_subtitle":"The joint zero set of such families carries a reduced Hamiltonian flow with fewer degrees of freedom.","key_machinery":"The workhorse is a system of particular integrals: a k-tuple of functions f_1,...,f_k whose time derivatives along the Hamiltonian flow close linearly on the tuple, f_dot_i = a_i^j f_j. The common zero set of such a tuple is dynamically invariant by uniqueness of solutions of linear ODEs. When the tuple is in particular involution, the zero set is coisotropic, and the pullback of the ambient symplectic form is a presymplectic form whose characteristic distribution is exactly the span of the restricted Hamiltonian vector fields of the f_i. Quotienting by that distribution, when it is regular, produces the reduced symplectic manifold; this is the presymplectic/coisotropic reduction mechanism t","core_discovery":"The central claim is Theorem 1: for a Hamiltonian system on a 2n-dimensional symplectic manifold, a system of k particular integrals — functions satisfying {f_i,H}=a_i^j f_j — that is in particular involution ({f_i,f_j}=c_ij^l f_l) and functionally independent on its common zero set M_f makes M_f a coisotropic, dynamically invariant presymplectic submanifold of dimension 2n−k. The kernel of the restricted form is spanned by the Hamiltonian vector fields of the f_i on M_f, the restricted Hamiltonian is constant along the characteristic leaves, and when the characteristic foliation is simple and the quotient is smooth, the quotient carries a unique symplectic form and Hamiltonian whose flow is","pith_inferences":["The paper does not iterate the reduction; a natural extension is to ask whether the reduced Hamiltonian system can itself admit particular integrals, producing a tower of reductions that ends at a Liouville-integrable core.","The linear-closure condition is an algebraic closure condition on functions, so one could search systematically for systems of particular integrals on polynomial Hamiltonians; the paper demonstrates the phenomenon but does not attempt a classification.","The regularity of the quotient is never verified in the examples; applying singular-reduction techniques, or finding an example where the characteristic foliation is not simple, would sharpen the exact boundary of the theorem.","If the framework is correct, the number of particular integrals needed to reach a Liouville-integrable core could serve as a quantitative measure of partial integrability for a Hamiltonian system."],"forward_implications":["If a Hamiltonian system admits k particular integrals in particular involution, its dynamics on their common zero set is presymplectic; under the additional regularity, it becomes a Hamiltonian system on a 2(n−k)-dimensional phase space.","Systems of particular integrals produce dynamically invariant submanifolds even when none of the functions is conserved; ordinary first integrals appear as the special case with all coefficients zero.","A first integral of a base natural Hamiltonian system becomes, in the scalar, diagonal, and non-diagonal auxiliary lifts, part of a system of particular integrals of the lifted system; setting the auxiliary momenta to zero and projecting away the auxiliary coordinates recovers the original dynamics.","Particular Liouville integrability means that the reduced Hamiltonian system is completely Liouville integrable, so on the reduced phase space the standard consequences of Liouville integrability — invariant tori and quasi-periodic motion under the usual compactness and regularity assumptions — apply.","The reduction mechanism extends beyond mechanical systems to Hamiltonian systems with magnetic vector potentials and to general polynomial Hamiltonians, as long as the auxiliary terms vanish on the constraint set."],"fun_headline_variants":["k particular integrals collapse Hamiltonian phase space","Particular integrability yields reduced Hamiltonian flow","Presymplectic reduction via particular integrals","Particular integrals: a route to lower-dimensional Hamiltonians","k particular integrals trim Hamiltonian dimensions"],"cache_read_input_tokens":25728,"weakest_assumption_plain":"The reduction depends on the characteristic foliation of the restricted two-form being simple and its leaf space being a smooth manifold; if either fails, the reduced Hamiltonian system is not defined, and the paper's examples do not verify this condition.","fun_headline_variants_meta":{"raw":{"variants":["k particular integrals collapse Hamiltonian phase space","Particular integrability yields reduced Hamiltonian flow","Presymplectic reduction via particular integrals","Particular integrals: a route to lower-dimensional Hamiltonians","k particular integrals trim Hamiltonian dimensions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001084,"raw_usage":{"total_tokens":4307,"prompt_tokens":621,"completion_tokens":3686,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":365,"completion_tokens_details":{"reasoning_tokens":3630}},"tokens_in":365,"tokens_out":3686,"duration_ms":22747,"temperature":1.0,"reasoning_tokens":3630,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T08:07:13.807170+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the two-body Coulomb system in a constant magnetic field, restricted to K_x=K_y=0 as in the paper, and compute the foliation generated by the Hamiltonian vector fields of K_x and K_y on that invariant set. If the leaf space is not a smooth manifold, the theorem's regularity hypothesis fails and the claimed reduced two-degree system is not defined for this example, marking the boundary of the claim.","supporting_citations":[],"review_version":2}