{"id":"803a468f-8aa5-4d62-b280-a20d4a61f303","arxiv_id":"1908.11072","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Hamiltonians with a finite number of zero normal frequencies, a KAM-type theorem states that for most frequencies the existence of invariant tori is decided by a single leftover constant, and this yields quasi-periodic solutions for a nonlinear Schrödinger equation with a zero mode.","lead":"The authors prove a new KAM theorem for Hamiltonian systems whose unperturbed motion has a finite number of zero frequencies. Their result gives a simple quantity that decides whether small perturbations still have regular torus-shaped motions, and they use it to show a zero-frequency Schrödinger equation has many quasi-periodic solutions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption (A) is impossible when zero normal frequencies exist: (1.8) fails identically for l=e_{j_m}, so Theorem 1.1 applies to no Hamiltonian; it must be amended to restrict l to nonzero normal modes.","rationale":"The reader's weakest assumption correctly identifies the fatal-as-stated flaw in Assumption (A): with Ω0=0, the conditions (1.7)-(1.8) cannot hold for l supported on a zero mode. This is load-bearing because the whole theorem, including the δ0 dichotomy and the NLS application, depends on a non-empty hypothesis class. My independent reading of Sections 2 and 4 confirms that the actual small-divisor analysis solves zero-mode equations via finite matrix invertibility rather than via the scalar condition ⟨l,Ω⟩≠0, so the natural repair is to restrict l to N+\\J. The paper otherwise has a plausible KAM strategy: the iteration preserves a richer normal form, the zero-mode linear and quadratic terms are kept, and the final dichotomy is argued by comparing the conserved drift with the shrinking ε_m scale. The NLS verification is lengthy and partly informal, but it does give explicit coefficient computations and parity arguments for the vanishing of the zero-mode drift; it is not obviously wrong. Because the main obstacle is a repairable statement-level gap rather than a proven contradiction inside the KAM construction, the conditional verdict remains appropriate. I found no independent reason to upgrade to rejection or acceptance.","tokens_in":68421,"tokens_out":8941,"duration_ms":93203,"concrete_test":"Directly test the printed Assumption (A) by substituting l=e_{j_m}, k=0 for one zero mode: since Ω_{j_m}=0, equation (1.8) gives 0≠0 and equation (1.7) gives the full parameter set, confirming vacuity. Then redo the formal small-divisor measure estimate of Lemma 4.1 with Z redefined as {(k,l)≠0, |l|≤2, supp(l)⊆N+\\J}; if the proof and measure bound (4.9) go through unchanged, the amended theorem is the intended one and the NLS application can be checked against the amended hypotheses; if any resonance condition involving zero-mode l is actually needed, the KAM construction is missing a small-divisor condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 sets Ω0=0 and then, in Assumption (A), equations (1.7)-(1.8), requires for all (k,l) with 1≤|l|≤2 that {ξ : ⟨k,ω(ξ)⟩+⟨l,Ω(ξ)⟩=0} have measure zero and ⟨l,Ω(ξ)⟩≠0 on Π. But Ω includes the zero modes Ω_{j_m}=0. Taking k=0 and l=e_{j_m} for any j_m∈J gives ⟨l,Ω(ξ)⟩≡0 on Π, so (1.8) is violated identically and the set in (1.7) is all of Π. Since Π is assumed to have positive measure, Assumption (A) can never be satisfied by any system with a zero normal frequency. Thus the central dichotomy, as stated, is vacuous and the NLS application cannot invoke Theorem 1.1. The rest of the paper indicates the intended hypothesis: the small-divisor conditions in (2.12) and Section 4, and the solvability analysis in (2.23)-(2.31), only use nonzero normal frequencies for the scalar Melnikov conditions, while zero-mode terms are handled through finite matrix non-resonance conditions. The fix is to require (1.7)-(1.8) only for l supported on N+\\J. This is a genuine logical gap, not merely a typo, because the theorem as printed has no instances; however, it is likely repairable, so the correct verdict is conditional rather than rejection.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a KAM theorem for infinite-dimensional Hamiltonian systems with finitely many zero normal frequencies. The main result (Theorem 1.1) states that, under nondegeneracy, spectral asymptotics, and regularity assumptions, the Hamiltonian can be conjugated to a normal form containing zero-mode terms, and the outcome is governed by the constant quantity δ0 = sqrt(|N^{z0}(ξ)|^2 + |N^{bar z0}(ξ)|^2): if δ0 = 0, a rotational torus persists for most parameters; if δ0 > 0, no torus exists in a certain shrinking domain. As an application, the authors show that the periodic NLS equation iu_t - u_xx + |u|^2u = 0 with an even symmetry condition possesses many quasi-periodic solutions despite having a zero normal frequency.","tokens_in":68722,"tokens_out":6360,"duration_ms":62139,"significance":"If the hypothesis issue identified below is repaired, the paper would address a recognized open problem in KAM theory: Kuksin's degenerate case of zero normal frequencies ([21] mentions that no preservation theorem is known in this case). The paper's new technical content includes the treatment of zero-mode terms via finite matrix non-resonance conditions, the derivation of the δ0 criterion, and a detailed verification of the structural assumptions for the NLS application. Strengths are the explicit decomposition of the homological equations into four types with solution estimates (Section 2), the iterative measure estimates (Section 4), and the careful structural verification in Section 7, where the zero-mode coefficients are shown to vanish at every step. There is no circularity: δ0 is read off from the limit normal form and is not used to construct the tori. However, as printed, Theorem 1.1 applies to no Hamiltonian with zero normal frequencies, so the advertised applications do not currently follow.","major_comments":[{"comment":"Assumption (A) is not satisfiable when zero normal frequencies are present. Since Ω_{j_m}=0 for j_m∈J, taking k=0 and l=e_{j_m} (with |l|=1) gives ⟨l,Ω(ξ)⟩ ≡ 0 on Π, so (1.8) fails identically and the set in (1.7) is all of Π. Because Π is assumed to have positive measure, the assumption can never hold for any Hamiltonian with a zero normal frequency. Consequently Theorem 1.1 as stated is vacuous, and the NLS application in Section 7, which has Ω^0_0=0, cannot invoke it. The intended hypothesis is evidently to restrict (1.7)-(1.8) to l supported on the nonzero normal frequencies N+\\J, while zero-mode resonances are controlled by the finite matrix conditions (2) and (4) in Section 2.2, as the solvability analysis in (2.23)-(2.31) and the small-divisor sets in (2.12) and Section 4 already do. This is a load-bearing logical gap rather than a typo, though the repair is local and the rest of the proof appears designed for it.","section":"Theorem 1.1, Assumption (A), eqs. (1.7)-(1.8)"},{"comment":"The proof of the first iterative step is entirely delegated to [23] with the sentence \"These results can be seen clearly in [23].\" This is the base step of the KAM scheme and is central to the theorem. Since [23] does not treat zero normal frequencies, and the present paper's novelty is precisely the handling of zero modes, the lemma should at least state which results from [23] are used and how they are adapted, especially the measure estimate (2.17) and the solution estimates (2.14)-(2.16) in the presence of the zero-mode terms. As written, the proof of the key lemma is not self-contained in a way that supports the intended extension.","section":"Lemma 2.1, proof"},{"comment":"The proof that δ0 > 0 implies nonexistence of tori shows that any solution of H_m starting in Ξ_m leaves Ξ_m by time 1, using the estimates (6.4)-(6.5). This does establish that no invariant torus is contained in Φ_{m-1}(Ξ_m × {ξ}). However, the domain Ξ_m depends on m and shrinks as ε_m → 0, so the conclusion is only local in the shrinking neighborhoods. The statement in Theorem 1.1 is formally correct as written (\"there is no torus in the domain Φ_{m-1}(Ξ_m × {ξ})\"), but the authors should clarify in Section 6 whether a fixed-size neighborhood is intended, and if so, provide the additional argument needed to extend the nonexistence to that fixed neighborhood.","section":"Section 6, nonexistence branch"}],"minor_comments":[{"comment":"\"Furan University\" should read \"Fudan University.\"","section":"Affiliations"},{"comment":"\"If a curse I → l^{a,p}\" should read \"If a curve I → l^{a,p}.\"","section":"Section 7, Lemma 7.1"},{"comment":"The symbol ⋖ is used throughout without being defined; it should be introduced as \"a ≤ c b for a constant c depending on n and τ\" or replaced by explicit inequalities.","section":"Notation"},{"comment":"In the final part of Theorem 1.1, \"a district Ξ_m\" should be \"a domain Ξ_m.\"","section":"Theorem 1.1, conclusion"},{"comment":"The normal form in (7.9) omits the zero-mode term Ω^0_0 z0 \\bar z0 (with Ω^0_0=0) that appears in (7.12); this is likely intentional but should be stated explicitly to avoid confusion.","section":"Section 7, eq. (7.9)"},{"comment":"Reference [1] is incomplete: it lists only a title and an arXiv number, with no authors; the full citation should be supplied.","section":"References"},{"comment":"The exponent of K_1 in (2.50) appears as (10b^2+2)τ+10b^2 in the display but as (10b^2+2)τ+10b^2-1 in one place in the text; the exponent should be consistent.","section":"Section 2.2, eq. (2.50)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is an early preprint (2019) with many typos and some imprecise notation. The core mathematical issue is the vacuous Assumption (A); this is a genuine but local gap, and the rest of the paper suggests the intended repair. There is no indication of circularity or of parameters fitted to data; the constant δ0 is defined from the limit normal form. The paper is potentially suitable for a dynamical systems or mathematical physics journal if the authors fix the assumption and clarify the scope of the first-step proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First: the paper is aimed at a real open problem Kuksin flagged—preserving lower-dimensional tori when the normal frequency has zeros—and the main theorem as printed is vacuous. The stress-test note is right: Assumption (A) requires <l,Omega(xi)> != 0 for all 1 <= |l| <= 2, but Omega contains the zero modes, so l = e_{j_m} gives <l,Omega> identically zero. The theorem has no instances. The intended restriction to l supported on N+\\J is visible in Section 2, where the scalar Melnikov conditions are used only for nonzero normal frequencies and the zero-mode terms are handled by the finite matrix conditions (2)-(4). But that restriction has to be stated in Theorem 1.1; without it, the main result cannot be cited.\n\nWhat is genuinely new: the paper does not just re-label Pöschel-Kuksin. It keeps the resonant zero-mode terms in the normal form and solves the non-diagonal homological equations using Kronecker products and determinant small divisors. The delta_0 dichotomy—existence of a torus if the drift vanishes, nonexistence in a shrinking domain otherwise—is a reasonable way to express what the singularity does. The NLS application is also on-point: it exhibits a concrete Hamiltonian with one zero normal frequency and verifies the vanishing coefficients that land in the delta_0 = 0 case. The verification is long and not easy to check, but it is an honest attempt, not a handwave.\n\nSoft spots beyond the main assumption: the paper leans heavily on [23] and [28] for the iterative estimates and convergence, which is normal for this area, but it means a referee has to trust those transfers. The NLS section has typos and some notational slips (\"Furan University\", \"a curse\" for \"a curve\", repeated mislabeled indices), and the coefficient-vanishing proof is compressed enough that I would want it reworked. None of this is fatal if the assumption is fixed.\n\nBottom line: as it stands, the theorem is not usable, but the skeleton is right and the intended repair is clear. I would send it to a serious referee—ask specifically whether the repaired non-resonance conditions suffice—rather than desk reject. This is a contribution worth engaging if the authors amend it.","headline":"A serious attempt at a genuinely open KAM problem, but Theorem 1.1 as printed is vacuous because Assumption (A) fails on the zero modes; the intended fix is clear and the paper deserves a careful referee.","tokens_in":69277,"tokens_out":3302,"would_cite":false,"duration_ms":36754,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37K55","35B15","35Q55","37J40","70H08"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single constant quantity decides whether a Hamiltonian with finitely many zero normal frequencies has KAM tori.","keywords":["KAM theory","zero normal frequencies","quasi-periodic solutions","nonlinear Schrödinger equation","Hamiltonian PDEs","small divisors","invariant tori","normal forms"],"falsifier":"Work out the first Newton step for the minimal case of one zero normal frequency ($b=1$) and a perturbation whose only low-order term is a nonzero constant coefficient $\\hat R^{z_0}(0,\\xi)$. If the iteration produces $\\delta_0>0$ and the flow estimate for $|z_0(1)|$ stays below $\\varepsilon_m^{7/6}$ for initial data with $\\|z^*(0)\\|_{a,p}+\\|\\bar z^*(0)\\|_{a,p}\\le \\varepsilon_m^{7/6}$, the no-torus conclusion fails; if the computed drift exceeds $\\varepsilon_m^{7/6}$, the dichotomy is confirmed in the cleanest possible case.","tokens_in":68169,"feed_emoji":"🌀","tokens_out":10407,"duration_ms":98294,"temperature":0.7,"pith_summary":"KAM persistence is usually blocked when the unperturbed system has normal frequencies equal to zero, because the first and second Melnikov conditions fail at $k=0$. This paper removes that blockage by refusing to eliminate the zero modes: it proves a dichotomy governed by the constant $\\delta_0 = \\sqrt{|\\breve N^{z_0}(\\xi)|_2^2 + |\\breve N^{\\bar z_0}(\\xi)|_2^2}$ assembled from the linear zero-mode coefficients of the limiting normal form. If $\\delta_0=0$, a rotational torus with frequency $\\omega_*(\\xi)$ persists for parameters in a large-measure Cantor set; if $\\delta_0>0$, the theorem produces a definite shrinking domain that contains no invariant torus. The method matters because it turns the zero-frequency obstruction into a computable final-state quantity, and the paper uses it to show that the periodic nonlinear Schr\\\"odinger equation $iu_t-u_{xx}+|u|^2u=0$, which has a zero mode, still possesses many quasi-periodic solutions.","feed_headline":"One constant decides whether zero-frequency KAM tori exist","feed_subtitle":"When δ0 = 0 tori survive; when δ0 > 0 none remains. That yields many quasi-periodic NLS solutions.","key_machinery":"The central object is the constant quantity $\\delta_0=\\sqrt{|\\breve N^{z_0}(\\xi)|_2^2+|\\breve N^{\\bar z_0}(\\xi)|_2^2}$, computed from the coefficients of the $z_0$ and $\\bar z_0$ linear terms in the final normal form; it is invariant under the iteration by construction and decides between torus existence and torus absence. The argument that carries the proof is a non-standard KAM step in which the five zero-mode coefficient classes $\\hat R^{z_0}(0,\\xi)$, $\\hat R^{\\bar z_0}(0,\\xi)$, $\\hat R^{z_0z_0}(0,\\xi)$, $\\hat R^{z_0\\bar z_0}(0,\\xi)$, $\\hat R^{\\bar z_0\\bar z_0}(0,\\xi)$ are promoted into the next normal form instead of being eliminated. The resulting homological equation splits into four types, and the zero-mode blocks are inverted by Kronecker products and column straightening, with new small-divisor conditions requiring non-vanishing determinants of finite matrices such as $i\\langle k,\\omega_m\\rangle I_{3b^2}-B_{1m}(\\xi)$. The normal form preservation is what gives $\\delta_0$ a meaning at the limit of the iteration.","core_discovery":"On its own terms the paper's discovery is Theorem 1.1: for a parameter-dependent Hamiltonian $N+R$ whose normal form is $\\langle\\omega(\\xi),y\\rangle+\\langle\\Omega_0(\\xi)z_0,\\bar z_0\\rangle+\\langle\\Omega(\\xi)z,\\bar z\\rangle$ with $\\Omega_0\\equiv 0$ on a finite block, and under assumptions of nondegeneracy, spectral asymptotics, regularity and smallness, a Newton-type KAM iteration conjugates $H$ to a normal form that keeps the zero-mode terms. Whether $\\delta_0$ is zero or positive then decides the geometry: when $\\delta_0=0$, the zero block drops out of the linear flow and the embedded torus survives with shifted frequency; when $\\delta_0>0$, the zero-mode linear term acts as a persistent drift that forces every trajectory starting near the torus out of the shrinking domain $\\Phi_{m-1}(\\Xi_m\\times\\{\\xi\\})$, so no invariant torus exists there. The paper also establishes that in the NLS application the quantities $\\breve N^{z_0}$ and $\\breve N^{\\bar z_0}$ vanish at every iteration, so the application lands on the existence side of the dichotomy.","pith_inferences":["Editorial inference: $\\delta_0$ should be computable at first order as the projection of the perturbation onto the zero-mode linear terms; if that is true, one can decide torus persistence for concrete PDEs by a one-step calculation before running the full Newton scheme.","Editorial inference: the same normal-form strategy may transfer to finite blocks with small nonzero normal frequencies or with eigenvalue limit points, since only finite-dimensionality of the exceptional block is used; the finite-limit-point shallow-water results could be interpreted as a special case of this mechanism.","Editorial inference: in generic perturbations with a zero mode but no symmetry, $\\delta_0$ should be nonzero, so torus persistence is the exceptional, symmetry-forced outcome; testing a family of perturbations that breaks the parity symmetry would locate the transition at $\\delta_0=0$.","Editorial inference: one could try to read off $\\delta_0$ from the original perturbation as $\\hat R^{z_0}(0,\\xi)^2+\\hat R^{\\bar z_0}(0,\\xi)^2$ plus higher-order corrections; the paper's verifications suggest the leading term is often the exact vanishing condition."],"forward_implications":["Zero normal frequencies need no longer be excluded from KAM theorems: any system satisfying the stated assumptions with a finite zero block has its torus problem settled by the constant $\\delta_0$.","In the $\\delta_0=0$ case the persistence is quantitative: the torus embedding is $\\varepsilon$-close to the identity, the frequency shift is $O(\\varepsilon)$, and the good parameter set has measure $\\mathrm{Meas}\\,\\Pi\\,(1-O(\\gamma))$.","In the $\\delta_0>0$ case the theorem gives a certified torus-free region $\\Phi_{m-1}(\\Xi_m\\times\\{\\xi\\})$ rather than merely failing to construct a torus, because the linear zero-mode term produces a flow that escapes the shrinking domain.","The periodic nonlinear Schr\\\"odinger equation $iu_t-u_{xx}+|u|^2u=0$ with periodic boundary conditions and even symmetry has many quasi-periodic solutions, despite the $q_0$ zero mode, because the relevant Fourier coefficients vanish at every iteration.","The normal form reached by the iteration contains only quadratic or higher terms besides the zero-mode block, so the dynamics near the survived torus is governed by the constants $\\breve N^{z_0}$, $\\breve N^{\\bar z_0}$ and the quadratic zero-block matrix."],"supporting_citations":[{"why":"Supplies the nondegenerate KAM iteration, Melnikov conditions and measure estimates that the proof extends to the preserved zero-mode blocks.","marker":"[28]"},{"why":"Provides the homological-equation framework, weight norms, and the NLS Birkhoff normal-form computation used in Sections 2 and 7.","marker":"[23]"},{"why":"Introduces the Fourier cut-off $K_m$ used at each Newton step to keep the resonance sets finite.","marker":"[2]"},{"why":"States the open problem of zero or multiple normal frequencies that Theorem 1.1 is designed to answer.","marker":"[21]"},{"why":"Establishes the infinite-dimensional KAM setting for lower-dimensional tori that this paper generalizes.","marker":"[20]"}],"fun_headline_variants":["One constant decides zero-frequency KAM tori existence","Zero-frequency KAM tori: a single constant is the arbiter","δ0=0 yields KAM tori; δ0>0 destroys them","New KAM theorem: zero-mode frequency decides tori survival","NLS with zero frequency: many quasi-periodic solutions via KAM"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem's load-bearing premise is the non-resonance assumption (A): if the index $l$ is allowed to be supported entirely on the zero-mode coordinates with $k=0$, then $\\langle l,\\Omega(\\xi)\\rangle\\equiv 0$, which contradicts the required $\\langle l,\\Omega(\\xi)\\rangle\\neq 0$; hence the theorem is only non-vacuous when (A) is read as restricted to the nonzero normal frequencies.","fun_headline_variants_meta":{"raw":{"variants":["One constant decides zero-frequency KAM tori existence","Zero-frequency KAM tori: a single constant is the arbiter","δ0=0 yields KAM tori; δ0>0 destroys them","New KAM theorem: zero-mode frequency decides tori survival","NLS with zero frequency: many quasi-periodic solutions via KAM"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0003,"raw_usage":{"total_tokens":1708,"prompt_tokens":894,"completion_tokens":814,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":510,"completion_tokens_details":{"reasoning_tokens":722}},"tokens_in":510,"tokens_out":814,"duration_ms":7702,"temperature":1.0,"reasoning_tokens":722,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:25:40.302417+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Work out the first Newton step for the minimal case of one zero normal frequency ($b=1$) and a perturbation whose only low-order term is a nonzero constant coefficient $\\hat R^{z_0}(0,\\xi)$. If the iteration produces $\\delta_0>0$ and the flow estimate for $|z_0(1)|$ stays below $\\varepsilon_m^{7/6}$ for initial data with $\\|z^*(0)\\|_{a,p}+\\|\\bar z^*(0)\\|_{a,p}\\le \\varepsilon_m^{7/6}$, the no-torus conclusion fails; if the computed drift exceeds $\\varepsilon_m^{7/6}$, the dichotomy is confirmed in the cleanest possible case.","supporting_citations":[{"cited_title":"P¨ oschel","cited_arxiv_id":null,"evidence_quote":"Supplies the nondegenerate KAM iteration, Melnikov conditions and measure estimates that the proof extends to the preserved zero-mode blocks."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the homological-equation framework, weight norms, and the NLS Birkhoff normal-form computation used in Sections 2 and 7."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Fourier cut-off $K_m$ used at each Newton step to keep the resonance sets finite."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the open problem of zero or multiple normal frequencies that Theorem 1.1 is designed to answer."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the infinite-dimensional KAM setting for lower-dimensional tori that this paper generalizes."}],"review_version":1}