{"id":"8550dbd3-84ff-47f8-bf1b-f0e0acf2fb4e","arxiv_id":"2412.11845","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For 1D NLS without external parameters, the paper constructs non-resonant infinite-dimensional Kronecker tori (almost periodic, not quasi-periodic) arbitrarily close to Kuksin-Pöschel KAM tori.","lead":"A new proof shows that a nonlinear Schrödinger equation on a circle admits infinitely many almost periodic solutions that are not quasi-periodic, with no external parameter. It does so by constructing infinite-dimensional invariant tori that accumulate near the known finite-dimensional KAM tori.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the regularizing normal form is proven with integer denominators, and the later KAM/Birkhoff and non-resonance steps hold together.","rationale":"The reader's verdict (ACCEPT, MODERATE confidence) pinpoints Theorem 2.1 as the load-bearing assumption. I agree that the entire construction depends on the regularizing normal form, but I do not find a concrete weakness there: the proof avoids small divisors because the denominators are integer squares, and the stronger growth condition on f is stated and used transparently. After checking the subsequent KAM, Birkhoff, loop, and limiting steps, I found the arguments coherent and the measure estimates sufficient. The most delicate point, the non-resonance extraction in Proposition 9.17, is correct because any two integer relations would be proportional via the integrality of c·1. The paper honestly lists its limitations (sparse tori, stronger smoothness, uncontrolled decay of radii), and these match the proof. Therefore no significant objection is justified, and the verdict remains ACCEPT.","tokens_in":89371,"tokens_out":35826,"duration_ms":296295,"concrete_test":"Independently verify the rank-one resonance module used in Proposition 9.17: for a finite truncation with Q-free ω^(nat) and ω^(∞)=ω^(nat)+a·1, compute the integer kernel K={c∈Z^M: c·ω^(∞)=0} for several random choices of a and ω^(nat); confirm that K is either {0} or Z·d for a primitive d, exploiting that c·1 is an integer. If any counterexample appears, the non-resonance step would need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the full argument, focusing on the regularizing normal form (Theorem 2.1), the KAM theorem (Theorem 5.3), the Birkhoff theorem (Theorem 6.3), the loop (Proposition 8.1), and the passage to infinite dimension and non-resonance (Section 9). The central claim is conditional on Theorem 2.1, but that theorem is proven in detail using a convergent Birkhoff procedure whose cohomological denominators are integer quantities σ1ℓ1^2+...+σqℓq^2. For any non-resonant term this integer is nonzero, hence bounded below by 1 in absolute value. Thus no small-divisor exclusion is needed in Section 2, and the strengthened entire-growth assumption on f is used exactly to make the Wick expansion converge; it is an explicit hypothesis, not a hidden gap. The KAM and Birkhoff theorems use standard measure-exclusion arguments (Lemma 4.1 and Proposition 4.2) with controlled losses, and the extremely small smallness thresholds (r^4000, ρ^N) are absorbable because the iterative scheme allows arbitrarily fast radius decay. The non-resonance construction in Proposition 9.17 is the most delicate part, but it is valid: for ω_j^(∞)=ω_j^(nat)+a, any relation c·ω^(∞)=0 implies (c·ω^(nat), c·1)=(-a(c·1), c·1); since c·1 is an integer and ω^(nat) is Q-free, the resonance module has rank at most one, so factoring out the single relation d(ξ) indeed yields Q-free frequencies ~ω. The measure estimates for the parameter sets (Lemmas 9.11, 9.16, 9.18) are consistent. I found no internal inconsistency or unproven claim that threatens Theorem 1.6(v).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that the nonlinear Schrödinger equation on the circle, i∂_t u + ∂_x^2 u = f(|u|^2)u with a real entire nonlinearity f satisfying f(0)=0 and f'(0)≠0, admits non-resonant infinite-dimensional Kronecker tori, and moreover that such tori accumulate on the finite-dimensional KAM tori of Kuksin–Pöschel. The proof is organized around four main blocks: a symplectic regularizing normal form (Theorem 2.1) obtained by Wick renormalization and a convergent Birkhoff procedure whose cohomological denominators are nonzero integers; a KAM theorem (Theorem 5.3) that eliminates an enlarged adapted jet using internal parameters and measure-exclusion arguments; a Birkhoff normal form theorem (Theorem 6.3) and a one-site opening loop (Proposition 8.1) that iteratively increase the dimension of the torus while keeping the amplitude independent of the dimension; and a final passage to the limit over a sparse set S∞ of Fourier modes, followed by a non-resonance construction (Proposition 9.17) that factors out the at-most-one-dimensional resonance module. The main theorem is Theorem 1.6, whose item (v) is the genuinely new part: arbitrarily close to most Kuksin–Pöschel tori there exist non-resonant infinite-dimensional tori. This yields Corollary 1.5, the existence of almost periodic solutions that are not quasi-periodic.","tokens_in":89738,"tokens_out":18152,"duration_ms":167617,"significance":"If correct, this is a landmark result: it provides the first construction of infinite-dimensional invariant tori, and hence of genuinely almost periodic solutions, for a non-integrable Hamiltonian PDE without external parameters. The paper is honest about its limitations: the nonlinearity must be entire with the strengthened derivative bounds |F^{(p)}(0)| ≲ c^{2p}(p!)^{-1} rather than merely analytic in a neighborhood; the infinite-dimensional tori are supported on a sparse set of Fourier modes satisfying the explicit condition (7); and linear stability is lost when passing to the infinite-dimensional limit. These restrictions are stated clearly in the introduction. The proof is remarkably self-contained: the regularizing normal form is proven in detail with integer small divisors, the KAM and Birkhoff steps use explicit measure estimates, and the final non-resonance argument is based on the Q-freeness of the natural frequencies rather than on additional parameters. I found no circularity and no hidden external parameter.","major_comments":[],"minor_comments":[{"comment":"The assertion that the vectors e and ~d are collinear is not fully justified. It requires an explicit normalization: since ~d·ω^{(ri)}=0 and e·ω^{(ri)}=0, one may choose α = (Σ e_k)/(Σ ~d_k) (the denominator is nonzero, otherwise ~d would give a nontrivial relation among the shifted frequencies), and then (e−α~d)·ω^{(ri)}=0 with Σ(e_k−α~d_k)=0. This reduces to a relation among the ω^{(nat)}_{j_k}, which are Q-free, so e=α~d. Please expand this step, as it is the heart of the non-resonance claim.","section":"Section 9.5.2, Proposition 9.17"},{"comment":"The bound |H^{ℓ,σ}_n| ≲ C^{q+2n}(1∧⟨ℓ*_3⟩²/⟨ℓ*_1⟩) uses ℓ*_3 for q<3 without a convention. Please state that one sets ⟨ℓ*_j⟩=1 for j>q, or give a separate statement for q=0,1,2. The same convention issue appears when comparing this bound with the weight Θ_ℓ in Definition 3.1.","section":"Theorem 2.1 and Section 9.1.2"},{"comment":"The displayed estimate for IV contains ambiguous and partly wrong exponent notation: “5q − 4 · 5p” should be “5^q − 4·5^p”, and the lower bound “≥ 5^q − 5^{p+1}” is not a lower bound when q=p+1 (it gives zero). The intended estimate is correct because 5^{p+1}−4·5^p = 5^p > 0, but the text should be rewritten with unambiguous superscripts and a correct chain of inequalities.","section":"Lemma 9.4, proof"},{"comment":"The induction for the radii (r_p) is summarized by an unspecified function g in (144). This is acceptable because only upper bounds on r_p are needed, but it would help to state explicitly that at each step the finitely many smallness conditions on r_p are imposed after S_p, ε, and r_{p−1} have been fixed. This would make the induction formally transparent.","section":"Section 9.3.2, iteration of the loop"},{"comment":"There are a few language glitches, e.g. “admits plenty of almost periodic solutions”, “a function leaving on an infinite dimensional torus”, and “Tξ” used for two different tori in (2) and later. These do not affect the mathematics but should be corrected in a final revision.","section":"Abstract and Introduction"}],"recommendation":"minor_revision","confidential_remarks":"This is a strong and important paper. The main theorem is believable and the proof is remarkably detailed and coherent; the stress points I checked (integer denominators in Section 2, the measure estimates in Section 4, and the resonance-module argument in Section 9.5) all hold up. The paper is extremely long, but given the depth and the novelty of the result, the length is justified. The only issues I found are local presentation gaps, mainly the missing normalization step in Proposition 9.17, which should be fixed before publication. I do not have concerns about novelty, attribution, or fit with the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper delivers what it claims: the first infinite-dimensional invariant tori for NLS without external parameters. Theorem 1.6(v) places non-resonant infinite-dimensional Kronecker tori arbitrarily close to the Kuksin–Pöschel KAM tori, and Corollary 1.5 gives almost periodic solutions that are not quasi-periodic. That answers a well-known open question, and it is the real news here.\n\nWhat is genuinely new is the combination of a symplectic regularizing normal form (Theorem 2.1) with a Pöschel-style iterative loop that opens one site at a time and preserves the twist condition using internal parameters. The regularizing normal form is load-bearing, and it is the right place to focus attention. The stress-test note confirms the key point: the cohomological denominators in Section 2 are nonzero integers, hence bounded below by 1 in absolute value, so no small-divisor exclusions are needed there. The strengthened whole-function growth assumption on f is used exactly to make the Wick expansion converge; it is an explicit hypothesis, not a hidden gap. The non-resonance construction in Section 9 is delicate but valid — for ω^(nat) Q-free and a(ξ) common to all frequencies, the resonance module has rank at most one. I read the KAM/Birkhoff loop as coherent, and the r^4000 smallness conditions are absorbable because the sequence of radii can be chosen to decay arbitrarily fast.\n\nThe soft spots are real but proportionate. The proof is 93 pages of estimates with many unquantified smallness conditions; no one can verify it in one sitting, and the authors openly say the radii sequence is not controlled. The stronger smoothness assumption on f is a genuine restriction compared to Kuksin–Pöschel, and the tori are sparse in Fourier space, so this does not settle typicality of almost periodic behavior. These limitations are stated in the paper, not concealed.\n\nWho should read this: anyone working on KAM for Hamiltonian PDEs, normal forms, or almost periodic solutions. It deserves a serious referee with time to check the loop and the normal form theorem carefully. My own verdict is moderately confident accept; I would send it to peer review rather than desk-reject.","headline":"First construction of infinite-dimensional invariant tori for NLS without external parameters; the 93-page proof is long and conditional on a regularizing normal form, but the main steps hold up under scrutiny.","tokens_in":90283,"tokens_out":1694,"would_cite":true,"duration_ms":20151,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B15","35Q55","37K55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves existence of infinite-dimensional invariant tori, and hence almost-periodic non-quasi-periodic solutions, for parameter-free NLS on the circle.","keywords":["nonlinear Schrödinger equation","infinite-dimensional invariant tori","Kronecker tori","almost periodic solutions","KAM theory","regularizing normal form","Hamiltonian PDE","small divisors"],"falsifier":"Compute the degree-six remainder of the regularizing normal form for a polynomial nonlinearity $f(z)=z$ (the cubic NLS): the paper itself notes all such terms can be written explicitly and give the improved estimate $\\sum_{k\\neq \\ell}|u_k|^4|u_\\ell|^2/(k-\\ell)^2$. If a single allowed nonlinearity in the paper's class produced a degree-six coefficient that violates the bound $|H_n^{\\ell,\\sigma}| \\leq C^{q+2n}(1\\wedge\\langle\\ell^*_3\\rangle^2/\\langle\\ell^*_1\\rangle)$, then Theorem 2.1, and with it the whole iterative construction, would fail.","tokens_in":89171,"feed_emoji":"🌊","tokens_out":9484,"duration_ms":92722,"temperature":0.7,"pith_summary":"The paper proves that the nonlinear Schrödinger equation on the circle, with no external parameters, has plenty of almost periodic solutions. More precisely, it shows that arbitrarily close to most of the small finite-dimensional invariant tori that KAM theory had already produced for this equation, there are infinite-dimensional Kronecker tori: homeomorphic images of a countable product of circles, on which the flow is a translation with rationally independent frequencies. The proof's engine is a symplectic normal form that makes the nonlinearity smoothing, followed by an iterative procedure that adds one Fourier mode at a time and takes an infinite-dimensional limit. If the construction is correct, the equation has global solutions that are almost periodic but not quasi-periodic, and the known finite-dimensional tori are not isolated in phase space.","feed_headline":"Parameter-free NLS gains almost periodic, non-quasi-periodic solutions","feed_subtitle":"The proof places infinite-dimensional rotation tori arbitrarily close to the known finite-dimensional ones, settling a long-open question.","key_machinery":"Two objects carry the proof. First is the regularizing normal form (Theorem 2.1): a symplectic change of variables, obtained by Wick renormalization plus a convergent Birkhoff procedure, writes the NLS Hamiltonian as $H_1^{(0)}$ plus a remainder whose coefficients obey $|H_n^{\\ell,\\sigma}| \\leq C^{q+2n}(1\\wedge\\langle\\ell^*_3\\rangle^2/\\langle\\ell^*_1\\rangle)$, where $\\ell^*_1,\\ell^*_3$ are the largest and third-largest Fourier indices of the monomial. The factor $(1\\wedge\\langle\\ell^*_3\\rangle^2/\\langle\\ell^*_1\\rangle)$ is what makes the nonlinearity one derivative smoother. Second is the loop (Proposition 8.1): starting from a Hamiltonian in normal form on a finite set of modes, the paper opens one new Fourier site, applies a Birkhoff normal form step to shrink the terms that would enter the next KAM step, applies a KAM theorem that eliminates a modified adapted jet (roughly between the 3-jet and the 4-jet of the perturbation), and repeats. Eliminating this larger jet prevents the creation of large non-integrable quartic terms that would destroy the twist condition, the invertibility property of the frequency map that makes small-divisor estimates possible with internal rather than external parameters.","core_discovery":"The paper's central claim is Theorem 1.6(v): for every nonempty finite set $S_1$ of Fourier modes and for almost every parameter in the Cantor set of the classical finite-dimensional KAM construction, each finite-dimensional KAM torus $T^\\varepsilon_\\xi$ is accumulated by non-resonant infinite-dimensional Kronecker tori $T^{\\varepsilon,\\rho}_\\xi$ within Hausdorff distance $\\rho$. Each such torus is the image of $\\mathbb{T}^{\\mathbb{N}}$ under a homeomorphism, and the motion on it is a translation by a sequence of rationally independent frequencies; consequently each orbit is dense in the torus and the corresponding solution is almost periodic without being quasi-periodic. The tori are not maximal: they are built on a sparse set of Fourier modes containing $S_1$, and they are not linearly stable in the limit.","pith_inferences":["The same scheme should transfer to other one-dimensional periodic dispersive Hamiltonian PDEs once a regularizing normal form is available; the authors explicitly expect the approach to be robust rather than specific to NLS.","The strengthened entire-growth condition on the nonlinearity is a concrete price of the method; a polynomial nonlinearity might still be handled by truncating the Wick expansion, which would remove the $(p!)^{-1}$ growth requirement.","A numerical check of the first loop step for the cubic NLS could test how much of the $r^{4000}$ smallness in the adapted-jet assumption is actually needed; the exponents in the paper are far from sharp.","The construction implies a hierarchy of invariant sets: almost periodic solutions accumulate on quasi-periodic ones, rather than appearing in isolation."],"forward_implications":["Corollary 1.5 follows: the equation has global almost periodic solutions that are not quasi-periodic.","The previously known finite-dimensional KAM tori are not isolated; arbitrarily close to almost each of them lie infinite-dimensional invariant tori.","The infinite-dimensional tori carry only a sparse set of Fourier modes; they are not full-dimensional, so the result does not yet say that almost periodic motion is typical among small solutions.","The limiting tori are Kronecker tori, not linearly stable KAM tori; linear stability is lost when the dimension goes to infinity.","Because the construction allows the added-mode amplitudes to decrease extremely fast, quantitative control of the Fourier decay of the resulting solutions is abandoned."],"supporting_citations":[{"why":"Supplies the finite-dimensional KAM tori on which the new infinite-dimensional tori accumulate, and the observation that NLS frequencies can be modulated by action variables.","marker":"[KP96]"},{"why":"Proposes the scheme of a convergent sequence of finite-dimensional tori of increasing dimension that the paper makes work for NLS via its regularizing normal form.","marker":"[Pos02]"},{"why":"Provides the dispersive smoothing techniques that the regularizing normal form adapts in a symplectic setting.","marker":"[ErTz13a]"},{"why":"Supplies the nonlinear smoothing estimates for periodic NLS used in the normal-form construction.","marker":"[ErTz13c]"},{"why":"Supplies further dispersive smoothing tools for the normal-form step.","marker":"[McC22]"},{"why":"Provides the Wick-ordered polynomial expansion and coefficient estimates behind Proposition 2.9.","marker":"[DNY24]"},{"why":"Gives the small-divisor arguments used to impose the fourth Melnikov conditions in the KAM and Birkhoff theorems.","marker":"[BG22]"},{"why":"Documents the regularization-modulo-gauge phenomenon already present in the integrable cubic case, guiding Theorem 2.1.","marker":"[KST17]"},{"why":"Supplies the iterative small-divisor improvement used to turn local non-resonance bounds into uniform ones.","marker":"[BG21]"}],"fun_headline_variants":["Parameter-free NLS admits infinite-dimensional tori","Infinite tori arise near finite KAM tori in NLS","Dense-orbit tori: new solutions for parameter-free NLS","Almost periodic solutions without quasi-periodicity in NLS","Long-asked NLS question settled by infinite tori"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction rests on the regularizing normal form of Theorem 2.1, which requires the nonlinearity to satisfy the strengthened entire-growth bound $|F^{(p)}(0)| \\lesssim c^{2p}(p!)^{-1}$; if that normal form fails, the one-derivative smoothing that makes the twist condition work is lost and the KAM loop cannot be repeated indefinitely.","fun_headline_variants_meta":{"raw":{"variants":["Parameter-free NLS admits infinite-dimensional tori","Infinite tori arise near finite KAM tori in NLS","Dense-orbit tori: new solutions for parameter-free NLS","Almost periodic solutions without quasi-periodicity in NLS","Long-asked NLS question settled by infinite tori"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000825,"raw_usage":{"total_tokens":3540,"prompt_tokens":808,"completion_tokens":2732,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":424,"completion_tokens_details":{"reasoning_tokens":2648}},"tokens_in":424,"tokens_out":2732,"duration_ms":20611,"temperature":1.0,"reasoning_tokens":2648,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:31:10.626841+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the degree-six remainder of the regularizing normal form for a polynomial nonlinearity $f(z)=z$ (the cubic NLS): the paper itself notes all such terms can be written explicitly and give the improved estimate $\\sum_{k\\neq \\ell}|u_k|^4|u_\\ell|^2/(k-\\ell)^2$. If a single allowed nonlinearity in the paper's class produced a degree-six coefficient that violates the bound $|H_n^{\\ell,\\sigma}| \\leq C^{q+2n}(1\\wedge\\langle\\ell^*_3\\rangle^2/\\langle\\ell^*_1\\rangle)$, then Theorem 2.1, and with it the whole iterative construction, would fail.","supporting_citations":[],"review_version":1}