{"id":"ee476039-d2f4-4a6d-b7a0-71bce1ebaf4b","arxiv_id":"2606.25145","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Anharmonic separable 2D oscillators support closed nonlinear Lissajous orbits only under energy-dependent resonance, with trajectory-dependent particular integrals rather than global superintegrability.","lead":"Separable anharmonic oscillators produce closed Lissajous-like orbits only when partial energies satisfy nonlinear resonance conditions; the extra conserved quantities then exist only on those trajectories. This clarifies how superintegrability reorganizes from global to particular once the potential is no longer harmonic.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The manuscript's strongest claim is elementary, self-contained, and correctly executed: for N≥2 the frequency ratio is energy-dependent, closed orbits exist only on resonant submanifolds, and the associated extra integrals are particular. The multi-valued-angle issue flagged by the reader is acknowledged in the text (circle-valued J, single-valued cos/sin representatives) and does not create an inconsistency. Explicit low-order algebraic checks (N=2) and the hyperelliptic formulation (N≥3) supply independent geometric support. No load-bearing gap requires a verdict change; the reader's ACCEPT with high confidence remains appropriate.","tokens_in":15325,"tokens_out":450,"duration_ms":4646,"concrete_test":"Independently recompute the Poisson bracket of the explicit phase-space representative I^{(0)}_{1:2} (Eq. 71) with H^{(2)}_{A=16} both on and off the equal-energy resonant shell E_x=E_y=1; confirm it vanishes identically only after restriction to the phase-locked trajectory γ^{(0)}_{1:2} and is nonzero as a global identity.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption (legitimacy of cos/sin of the resonant phase J as particular integrals once {I,H}=0 only on the resonant shell) is already handled carefully in the paper and does not undermine the central claim. Sections 3.2 and 4.3 construct J from local action-angle variables away from zero-energy degeneracies, note multi-valuedness explicitly, and replace it by single-valued trigonometric (or elliptic/hyperelliptic) representatives; the Poisson bracket is shown to vanish precisely when the nonlinear resonance condition holds. Explicit algebraic orbit equations for N=2 (via Jacobi multiplication) and hyperelliptic phase constraints for N≥3 further corroborate that the extra conserved quantities are trajectory-dependent rather than global. No hidden circularity or incorrect identity appears; the global-versus-particular distinction is correctly drawn for this separable family.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies classical trajectories of the separable family V(x,y)=½(x^{2N}+A y^{2N}) and the associated conserved quantities. For N=1 it recalls that rational frequency ratios yield global superintegrability and ordinary Lissajous figures. For N≥2 the frequencies become energy-dependent, so closed nonlinear Lissajous-type orbits exist only when a nonlinear resonance condition fixed by the partial energies holds. The authors construct the corresponding extra quantities (resonant phase combinations and, for N=2, algebraic orbit representatives) and show that their Poisson brackets with H vanish only after restriction to those resonant sets, hence are particular rather than global integrals. For N≥3 the same resonance mechanism is expressed via hyperelliptic phase constraints.","tokens_in":15494,"tokens_out":1153,"duration_ms":34723,"significance":"The work gives a clean, explicit illustration of how global superintegrability of the anisotropic harmonic oscillator reorganizes under nonlinear deformation into trajectory-dependent (particular) conservation laws selected by energy-dependent resonance. Strengths include: (i) explicit nonlinear resonance conditions (Eqs. 36, 88–90) jointly fixed by A and the partial energies; (ii) concrete algebraic orbit equations for the quartic case obtained from Jacobi multiplication at k=1/√2; (iii) direct Poisson-bracket checks for the lowest resonances showing non-vanishing off the resonant set; and (iv) a careful discussion (§4.4) distinguishing phase-locking invariants from tautological constants on a single orbit. The contribution is primarily conceptual and expository within an already-studied separable family, but the global-versus-particular distinction is drawn with useful geometric detail.","major_comments":[{"comment":"Sections 3.2–3.3: there is a mismatch in the strength of the particular-integral claim. The resonant phase J^{(p,q)}=p θ_y−q θ_x (and its single-valued cos/sin representatives) has ˙J=pΩ_y−qΩ_x, which vanishes on the entire resonant shell Ω_x/Ω_y=p/q. By contrast, the algebraic phase-space representatives I^{(0)}_{1:2} and I^{(0)}_{1:3} (Eqs. 71, 75) are only asserted to satisfy {H,I}=0 after restriction to the individual phase-locked trajectory γ^{(0)} (Eqs. 72, 76). For the central claim of particular superintegrability it should be stated clearly which objects are integrals on the full resonant submanifold and which are merely constants along a single closed orbit; otherwise the algebraic I’s risk looking tautological in the sense the authors themselves warn against in §4.4.","section":null},{"comment":"Section 4.3 and the abstract: the phrase “particular superintegrability in the Liouville sense” is slightly imprecise. Liouville integrability is already guaranteed globally by the two partial energies. What is particular is the third (phase) integral on the resonant shell. A short, explicit count—dimension of the resonant set, number of independent particular integrals thereon, and how this exceeds the restricted Liouville bound—would make the terminology load-bearing rather than decorative.","section":null}],"minor_comments":[{"comment":"Figures 1, 2 and 5 are helpful but purely qualitative. A brief numerical check that the constructed I’s remain constant along a resonant orbit while drifting off-resonance would strengthen the Poisson-bracket claims for readers who do not recompute them.","section":null},{"comment":"Notation: the same symbol H^{(N)} is used for the full Hamiltonian and, with subscripts, for partial energies; a consistent H_x^{(N)}, H_y^{(N)} (already used in places) throughout would avoid momentary ambiguity.","section":null},{"comment":"Section 2: the global cubic and quartic integrals for A=4 and A=9 are standard; a pointer to a classical reference (or a one-line derivation sketch) would help non-specialists.","section":null},{"comment":"Eq. (13) and the subsequent polynomial orbit equations for N=1 are written for the zero-relative-phase branch only; the text already notes this, but a single sentence that other Δ yield different algebraic curves of the same degree would prevent misreading.","section":null},{"comment":"References: the self-citation cluster on particular integrability is appropriate as background, but a short comparison with other treatments of resonant tori / action-angle locking in nearly integrable systems (beyond the KAM citations) would situate the geometric claims more broadly.","section":null},{"comment":"Typos / style: “polyno-mial” line break in the abstract; occasional missing spaces before citations; “Bˇ rehov´ a” in the affiliation should be checked for encoding.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is correct and well written; the main risk is incremental novelty relative to the authors’ own prior papers on particular integrability. I view the explicit nonlinear-Lissajous geometry and the global-versus-particular contrast for this concrete family as sufficient for a math-ph journal, provided the clarification of shell-versus-orbit conservation is made. No integrity or scope concerns."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper does one useful thing well: it takes the familiar separable family V = ½(x^{2N} + A y^{2N}) and shows, with explicit formulas, how closed nonlinear Lissajous orbits appear only on energy-dependent resonant shells once N ≥ 2, and how the extra conserved quantities live only on those shells.\n\nWhat is actually new are the concrete realizations. For the quartic case they give the algebraic orbit equations obtained from Jacobi cn-multiplication formulas (the 1:2 and 1:3 cases are written out cleanly). For N ≥ 3 they replace the polynomial orbit equation by the natural hyperelliptic phase constraint. They also write the corresponding particular integrals (phase combinations or their cos/sin representatives) and check that the Poisson brackets vanish only after restriction to the resonant set. The global-versus-particular distinction is drawn correctly and without circularity: the two partial energies are always global; the third object is not.\n\nThe math is elementary and correctly executed. Everything follows from Hamilton’s equations, energy conservation, and standard elliptic/hyperelliptic identities. The multi-valuedness of the angle variables is noted and handled by taking single-valued trigonometric (or elliptic) representatives; that is the only soft spot, and it is minor. The paper does not claim a new abstract theorem, nor does it need to. Self-citations supply the background definitions of particular integrability and are not load-bearing for the new orbit equations.\n\nThis is for people who work on classical superintegrability, particular integrals, or nonlinear oscillators and want clean, explicit examples they can later quantize or deform. It is not for someone looking for a structural advance in the theory of integrable systems. I would send it to a serious referee; the calculations check out and the geometric point is worth having on the record. Engage if the topic is already on your desk; otherwise it is a solid reference rather than required reading.","headline":"Clean, explicit classical examples that make the global-vs-particular superintegrability distinction concrete for a standard family of separable oscillators; solid and useful, not a conceptual breakthrough.","tokens_in":16075,"tokens_out":482,"would_cite":true,"duration_ms":6367,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37J35","70H06","70H12"],"pacs":[],"model":"grok-4.5","headline":"Closed nonlinear Lissajous orbits in anharmonic oscillators are carried by trajectory-dependent particular integrals, not by global superintegrability.","keywords":["Lissajous figures","nonlinear oscillators","integrability","superintegrability","particular integrals","resonance conditions","hyperelliptic phases"],"falsifier":"Compute the Poisson bracket of one of the explicit quartic representatives (for example the phase-locked 1:2 or 1:3 quantity) with the Hamiltonian off the resonant energy shell; if the bracket vanishes identically rather than only after restriction, the particular-versus-global distinction collapses.","tokens_in":16244,"feed_emoji":"〰️","tokens_out":629,"duration_ms":6060,"temperature":0.7,"pith_summary":"This paper studies two-dimensional separable oscillators with potentials that are pure powers of the coordinates, from the familiar harmonic case up through higher even powers. In the harmonic case, rational frequency ratios produce closed Lissajous figures because an extra integral of motion exists everywhere in phase space. Once the potential becomes anharmonic, the frequencies themselves depend on how the energy is shared between the two directions, so closed orbits appear only when the initial data satisfy a nonlinear resonance condition. The authors construct the extra conserved quantities that live on those resonant orbits and show that their Poisson brackets with the Hamiltonian vanish only after restriction to those orbits. They call these quantities particular integrals. The geometric description also changes: quartic resonances give algebraic curves via elliptic multiplication formulas, while higher powers require hyperelliptic phase constraints. The result matters because it explains how remnants of superintegrability can survive after a nonlinear deformation destroys the global symmetry.","feed_headline":"Closed anharmonic Lissajous orbits need particular integrals","feed_subtitle":"Global superintegrability dies with the harmonic potential; only resonant trajectories keep extra conserved quantities.","key_machinery":"Particular integrals: phase combinations J = p θ_y − q θ_x (or their single-valued cosine/sine representatives, or the corresponding algebraic/hyperelliptic orbit constraints) whose Poisson bracket with the Hamiltonian vanishes only after restriction to the nonlinearly resonant trajectories.","core_discovery":"For the family V = ½(x^{2N} + A y^{2N}), N ≥ 2, closed configuration-space orbits exist only on resonant submanifolds fixed by the energy-dependent frequency ratio; the associated extra integrals satisfy {I, H} = 0 solely on those submanifolds and are therefore particular, not global.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Anharmonic Lissajous orbits require particular integrals only","Nonlinear Lissajous figures signal particular superintegrability","Closed orbits in anharmonic potentials need trajectory-dependent integrals","Particular integrals enable resonant Lissajous paths for N≥2","Global superintegrability fails; resonant orbits keep particular integrals"],"cache_read_input_tokens":128,"weakest_assumption_plain":"That single-valued trigonometric functions of the resonant phase combination count as genuine particular integrals once their Poisson bracket vanishes only on the resonant shell, relying on local action-angle variables away from zero-energy points.","fun_headline_variants_meta":{"raw":{"variants":["Anharmonic Lissajous orbits require particular integrals only","Nonlinear Lissajous figures signal particular superintegrability","Closed orbits in anharmonic potentials need trajectory-dependent integrals","Particular integrals enable resonant Lissajous paths for N≥2","Global superintegrability fails; resonant orbits keep particular integrals"]},"model":"grok-4.5","effort":"low","cost_usd":0.005156,"raw_usage":{"total_tokens":1432,"prompt_tokens":763,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":51560000,"prompt_tokens_details":{"text_tokens":763,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":601,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":763,"tokens_out":68,"duration_ms":4889,"temperature":1.0,"reasoning_tokens":601,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T12:21:44.840921+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute the Poisson bracket of one of the explicit quartic representatives (for example the phase-locked 1:2 or 1:3 quantity) with the Hamiltonian off the resonant energy shell; if the bracket vanishes identically rather than only after restriction, the particular-versus-global distinction collapses.","supporting_citations":[],"review_version":2}