{"id":"e771fdb6-4140-4879-97e0-453de21bd696","arxiv_id":"2505.10051","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The authors give an extended proof sketch showing that 1D NLS without external parameters supports plenty of non-resonant infinite-dimensional tori, hence almost periodic non-quasi-periodic solutions.","lead":"This note sketches a proof that nonlinear Schrodinger equations on the circle, with no external parameters, admit infinite-dimensional invariant tori and therefore almost periodic solutions. It explains two new tools: a regularizing normal form and an iterative KAM argument that adds one Fourier site at a time.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Section 2 black-box smoothing (7) is not reconciled with the Section 3.7 loss s0=2δ=2; the derived sparsity condition (20) may rest on a coefficient bound weaker than (7).","rationale":"The reader's conditional verdict is appropriate: the note is an extended sketch and the decisive proofs live in [BGR24]. My read agrees that the regularizing normal form is the weak point, but I sharpen it to a specific tension: the paper states (7) as the black box, yet the KAM part needs a weaker/lossy estimate (19) and even writes a weaker coefficient bound for sextic terms. Since the twist condition is the gate for the infinite-dimensional limit, this gap is load-bearing. The proposed check targets [BGR24] directly and would either confirm the normal form and complete the derivation, or expose a fatal mismatch. I do not change the verdict: CONDITIONAL remains correct because the claim is credible but not self-contained, and the missing computation is checkable.","tokens_in":14752,"tokens_out":17622,"duration_ms":165867,"concrete_test":"In the companion paper [BGR24] (arXiv:2412.11845), locate the main normal-form theorem and extract the exact coefficient bound for the sextic monomial |u_j|²|u_ℓ|²|u_k|² with |k|≥|ℓ|≥|j|. Compare the exponent of ⟨ℓ⟩ in that bound with (7) and with ⟨ℓ⟩^{4δ} used in §3.7. If the true exponent is larger than 4δ, check that the sum in (20) still converges for the constructed S∞; if the bound is (7), re-derive (19) from it and verify whether the loss s0=2 appears. This settles whether (18) holds uniformly in p.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2 states the regularizing normal form as the coefficient bound (7): |P^{ℓ,σ}| ≲ 1 ∧ |ℓ*_3|^{2δ}/|ℓ*_1|^δ with δ=1, which is a 1-smoothing estimate. Section 3.7, however, asserts \"our case with s0=2δ=2\" and uses estimates (19) whose derivation is delegated to [BGR24]. The heuristic supporting (19) invokes the sextic term c_{k,ℓ,j}|u_j|²|u_ℓ|²|u_k|² with bound |c_{k,ℓ,j}| ≤ ⟨k⟩^{-δ}⟨ℓ⟩^{4δ}⟨j⟩^{4δ}. For k ≫ ℓ ≥ j this is far weaker than (7), which would give ≲ 1 ∧ j²/k. The twist condition (18) and the sparsity condition (20) are exactly what justify taking the limit p→∞ in §3.6; if the true normal form in [BGR24] has a loss, the note does not prove that (7) implies (19), and if it has no loss, the stated need for sparsity is inconsistent. Either way the central claim depends on an unverified link between the stated smoothing and the KAM twist estimate. The admitted simplification in §3.1 of treating the L2 norm as constant is also load-bearing, since the L2-norm dependence is what produces the loss s0=2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript states Theorem 1.2, which asserts the existence of non-resonant infinite-dimensional Kronecker tori for the nonlinear Schrödinger equation on the circle without external parameters, and Corollary 1.5, which asserts the existence of almost-periodic solutions that are not quasi-periodic. The proof is presented as an extended sketch: Section 2 describes a partially regularizing normal form whose coefficients depend on the L2 norm, and Section 3 outlines a KAM-type construction using internal parameters, iterating finite-dimensional tori in the style of Pöschel. The text repeatedly states that detailed arguments are contained in the companion paper [BGR24], and several steps are explicitly announced as omitted or simplified.","tokens_in":15033,"tokens_out":14321,"duration_ms":142055,"significance":"If the result holds, it is a significant advance: it would provide the first construction of infinite-dimensional invariant tori for a non-integrable Hamiltonian PDE without external parameters, and equivalently almost-periodic non-quasi-periodic solutions for the 1D NLS. The proposed mechanism—iterating finite-dimensional KAM tori while using internal parameters and a partially regularizing normal form—is novel and goes beyond the previously known external-parameter results. The paper also gives a clear and useful survey of the literature. However, as submitted, the manuscript is a proof sketch rather than a complete proof, and the central claim depends on the companion paper [BGR24] plus several admitted omissions. The significance of the result therefore cannot be evaluated from this manuscript alone.","major_comments":[{"comment":"Theorem 1.2 is not proved in the manuscript. The text repeatedly states that detailed proofs are contained in [BGR24] (e.g., §1, §2.2, §3.3, §3.7), and the convergence step in §3.6 is described only as \"very schematically\" using an \"infinite reservoir of smallness.\" The reader is not given a proof of the KAM steps, the measure estimates, or the limit p→∞. As a standalone paper, this is a load-bearing omission. The authors should either include the complete proof or precise theorem-by-theorem reductions to [BGR24], or clearly re-frame the paper as an extended abstract of [BGR24] rather than as a proof of Theorem 1.2.","section":"§3, especially §3.6"},{"comment":"The regularizing normal form bound (7) is not reconciled with the derivative estimate (19) used for the twist condition. Section 2 states the coefficient bound |P^{ℓ,σ}| ≲ 1 ∧ |ℓ*_3|^{2δ}/|ℓ*_1|^δ with δ=1 for the normal form after a symplectic change and a gauge transform, while §3.7 asserts \"our case with s0=2δ=2\" and uses estimate (19). The derivation of (19) from (7) is not given and is delegated to [BGR24]. This is load-bearing because (18)–(20) are exactly what allow the limit p→∞ in §3.6. Moreover, the heuristic discussion in §3.7 for the sextic term c_{k,ℓ,j}|u_j|^2|u_ℓ|^2|u_k|^2 suggests a bound |c_{k,ℓ,j}| ≤ ⟨k⟩^{-δ}⟨ℓ⟩^{4δ}⟨j⟩^{4δ}, which for k≫ℓ≥j appears weaker than the 1-smoothing bound in (7), so the stated loss s0=2δ=2 is not evidently consistent with (7).","section":"§3.7 and §2"},{"comment":"The L2 norm is treated as a constant in the KAM part, but Section 2 builds the normal form with coefficients depending on ‖u‖^2_{L2}, and §3.7 attributes the loss s0=2 to this dependence. The text says \"to simplify our presentation, in this sketch of proof we'll consider ‖u‖^2_{L2} as a constant.\" This simplification affects the frequency modulation in (17) and the twist condition in (18), since the internal parameters ξ enter both through the amplitudes and through the L2-norm argument of the coefficients. The admitted simplification is therefore load-bearing and must be either removed or fully justified.","section":"§3.1"},{"comment":"The twist condition is only stated for differentiable λ, but the text explicitly notes: \"in fact ξ ↦ λ will be only Lipschitz and we have to adapt the twist condition but we omit this problem here.\" Since the twist condition (18) and the sparsity condition (20) are essential for taking p→∞, the Lipschitz adaptation must be supplied or precisely referenced. As written, the proof of the twist condition is incomplete.","section":"§3.7"}],"minor_comments":[{"comment":"The symbol s0 is used with two different meanings: in §2 it denotes a Sobolev regularity exponent in (1/2, s), while in §3.7 it denotes the weight exponent in the ℓ^1_{s0} spaces, where s0=2. Please introduce distinct notation to avoid confusion.","section":"§2 and §3.7"},{"comment":"The sextic integrable term is written as Z6 = ∑_{k≠ℓ} |u_k|^2|u_ℓ|^4/(k−ℓ)^2 in §2.3 and as Z6 = ∑_{k≠ℓ} |u_k|^4|u_ℓ|^2/(k−ℓ)^2 in §3.7. Although the sums are symmetric under exchange of k and ℓ, the notation should be made consistent to avoid confusion.","section":"§2.3 and §3.7"},{"comment":"The sentence \"we will ignore it\" refers to removing a small part of parameters during the Birkhoff step. If this removal is not described, the measure estimate for O^{(2)}_ε stated in §3.5 has no visible justification in the note; a precise reference or a brief explanation should be added.","section":"§3.4"},{"comment":"There are several typographical issues, including \"extern al\" in the abstract, \"withδ = 1\" after (7), and inconsistent spacing in displayed formulas. A careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"This manuscript is essentially an extended abstract of the companion paper [BGR24]. The referee may wish to consider whether the journal publishes research announcements; if so, the authors should be asked to state precisely which statements are proved here and which are proved in [BGR24], with theorem numbers. If the journal expects self-contained research papers, then the current manuscript is not acceptable because the main theorem is not proved. I found no indication of inappropriate citation practice; the self-citation to [BGR24] is explicit and appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The central theorem of this note is not new—it is proved in the authors' companion paper BGR24 (arXiv:2412.11845)—and the note is an extended sketch, not a self-contained proof. Taken for what it is, it is a well-written, honest roadmap of a significant result: infinite-dimensional non-resonant tori for NLS on the circle without external parameters, giving almost periodic solutions that are not quasi-periodic.\n\nWhat is genuinely new here is the combination of ideas. The regularizing normal form that makes the nonlinearity smoothing up to a gauge transform, by letting coefficients depend on the L2 norm, is the key ingredient. The iterative 'add one site' KAM scheme, using third and fourth Melnikov conditions to kill the adapted jet and an infinite reservoir of smallness to take the limit, is clever and credible. The note is explicit about its status: it repeatedly says 'we will ignore it' and 'we omit this problem here', and Theorem 1.2 is stated without proof, pointing to BGR24. That honesty is a real strength.\n\nThe soft spots are exactly where the note is an outline rather than a proof. The most specific concern is the link between the coefficient bound (7) in Section 2 and the loss parameter s0=2 in Section 3.7. Section 2 claims a 1-smoothing estimate: for a term with modes k≫ℓ≥j, the coefficient is bounded by about 1∧j²/k. Section 3.7, however, uses a much weaker heuristic bound, roughly ⟨k⟩^{-1}⟨ℓ⟩^4⟨j⟩^4, to justify the sparsity condition (20). It is not shown how (7) implies (19), nor why the L2-norm dependence—which Section 3.1 simply treats as constant—produces the loss s0=2. A referee reading only this note cannot verify the central mechanism. This is not a fatal flaw in the note itself, because the proofs are deferred, but it is exactly the place where the companion paper needs to be checked.\n\nThe self-citation is not a problem: the note is an extended announcement, not an attempt to claim a new theorem. It should be judged as an expository contribution. Its value is real for anyone who wants the proof strategy before reading BGR24.\n\nBottom line: I would send this to referees, but only with the understanding that the actual mathematical verification happens in BGR24. The referee should be asked to confirm that (7) really is compatible with the loss and sparsity used in Section 3.7. For my own reading group, I would pair it with the companion paper rather than read it alone.","headline":"A clear, honest extended sketch of a major result, but it is not the proof—the theorem lives in the companion BGR24 paper, and the sketch leaves an unverified gap between its stated smoothing estimate and the sparsity condition.","tokens_in":15625,"tokens_out":6246,"would_cite":true,"duration_ms":61016,"reading_group":"maybe","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 claims that one-dimensional nonlinear Schrödinger equations on the circle admit infinite-dimensional non-resonant invariant tori without external parameters, and hence almost periodic solutions that are not quasi-periodic.","keywords":["infinite dimensional tori","KAM theory","regularizing normal form","almost periodic solutions","nonlinear Schrödinger equation","non-resonant Kronecker tori","internal parameters","sparsity condition"],"falsifier":"Compute, for cubic NLS, the terms generated by the normal form at order eight and check the claimed bound $|P^{\\ell,\\sigma}| \\lesssim 1 \\wedge (|\\ell^*_3|^2/|\\ell^*_1|)$ on every monomial whose highest mode is paired as an action; a single monomial whose coefficient stays of order one while $|\\ell^*_1|$ grows would falsify the smoothing estimate that the KAM part uses as a black box.","tokens_in":14500,"feed_emoji":"🌊","tokens_out":16054,"duration_ms":146520,"temperature":0.7,"pith_summary":"This note claims that the nonlinear Schrödinger equation on the circle, with a real entire nonlinearity whose cubic term is present and no external forcing or potential, has infinite-dimensional non-resonant invariant tori, and therefore almost periodic solutions that are not quasi-periodic. The proof strategy is iterative: instead of constructing the infinite-dimensional torus in one step, it builds a convergent sequence of finite-dimensional KAM tori, each obtained from the previous one by opening a new Fourier mode and using the mode's squared amplitude as an internal parameter. The key input is a regularizing normal form that makes the nonlinearity gain one derivative up to an $L^2$-norm phase factor, which is what allows the frequencies to be twisted in the limit of infinitely many modes. This is the first such construction for a non-integrable Hamiltonian PDE with no external parameter.","feed_headline":"NLS has almost periodic solutions that are not quasi-periodic","feed_subtitle":"Infinite-dimensional tori are built without external parameters, accumulating on known finite-dimensional ones.","key_machinery":"The load-bearing object is the regularizing normal form of Section 2, used as a black box in Section 3. After a symplectic change of variables, the Hamiltonian is $\\mathrm{Z}_2$ plus a remainder whose coefficients obey $|P^{\\ell,\\sigma}| \\lesssim 1 \\wedge (|\\ell^*_3|^2/|\\ell^*_1|)$ (equation (7) with $\\delta=1$), which makes the vector field one-derivative smoothing except for an $L^2$-norm phase rotation, a gauge transform. This smoothing is what upgrades the KAM twist condition to the diagonal-dominance form (18) and yields the sparsity condition (20) on the infinite mode set $S_\\infty$. Around this, the proof builds a loop: open one new site, run a normal-form procedure to shrink the adapted jet to $r_p^{4000}$, then run a KAM step with the current radius as the small parameter; because $r_{p+1}$ can be chosen much smaller than $r_p$, the loop has an infinite reservoir of smallness.","core_discovery":"On the paper's own terms, the discovery is Theorem 1.2: there exist non-resonant infinite-dimensional Kronecker tori for (NLS), meaning invariant sets homeomorphic to $\\mathbb{T}^{\\mathbb{N}}$ on which the flow is conjugate to translation by a rationally independent frequency vector, with every orbit dense. By Corollary 1.5 these tori support almost periodic solutions that are not quasi-periodic. The construction actually produces a family of such tori accumulating on the finite-dimensional KAM tori of [KP96]: one fixes a finite set of modes, builds a torus around it, then repeatedly adds one mode, applies a normal-form step that drives the unwanted low-order terms down to size $r_p^{4000}$, and performs a KAM step whose small parameter is the current radius $r_p$ rather than the global amplitude. The limit $p\\to\\infty$ is controlled by a sparsity condition on the final set of modes, and the whole scheme works only because the regularized nonlinearity is strong enough to keep the twist map invertible as the dimension grows.","pith_inferences":["A natural next test is whether the sparsity condition (20) is genuinely necessary; the discussion in the note suggests that a regularizing normal form with better smoothing (larger $\\delta$, or no loss $s_0=0$) would relax it, and the sixth-order term $Z_6$ is the obvious place to look.","The same 'add one mode at a time' iteration should transfer to other one-dimensional Hamiltonian PDEs once an analogue of the regularizing normal form is available; this is a direct test of the method's generality.","A quantitative version that controls the decay of the radii $r_p$ rather than just choosing them successively smaller would make the tori concrete and could connect the construction to long-time stability theorems."],"forward_implications":["The NLS on the circle has global solutions that are almost periodic but not quasi-periodic, for every nonlinearity satisfying the paper's assumptions: real entire, $f(0)=0$, $f'(0)\\neq0$, and at most exponential growth.","The constructed tori form a family that accumulates on the finite-dimensional KAM tori of [KP96], so the almost periodic solutions occur arbitrarily close to known finite-dimensional invariant tori.","Each non-resonant Kronecker torus carries dense orbits, so the corresponding solutions are ergodic on the torus with respect to the product measure.","The proof uses only internal parameters, namely squared moduli of Fourier coefficients, so no external potential or forcing term is needed."],"supporting_citations":[{"why":"Supplies the full proof of the theorem and of the regularizing normal form, which this note only sketches.","marker":"[BGR24]"},{"why":"Provides the finite-dimensional KAM tori on which the new tori accumulate and the observation that the NLS frequency shift acts like a potential $V_j=|u_j|^2$.","marker":"[KP96]"},{"why":"Introduces the iterative scheme of building finite-dimensional tori and passing to the limit, which Section 3 adapts.","marker":"[Pos02]"},{"why":"Establishes the rational normal form and explicit sixth-order terms that motivate the coefficient bound and the gauge-dependent smoothing.","marker":"[BFG20]"},{"why":"Gives the Birkhoff normal form in energy space and the small-divisor estimates used for the Melnikov conditions.","marker":"[BG22]"}],"fun_headline_variants":["NLS gets almost periodic solutions without extra parameters","Infinite-dimensional tori give NLS almost periodic solutions","NLS almost periodic solutions not quasi-periodic, no external parameters","Non-resonant tori for NLS yield almost periodic orbits","NLS almost periodic solutions accumulate on known tori"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that after a carefully chosen change of variables the nonlinearity really does gain one derivative, except for an irrelevant phase rotation, with bounds that do not degrade as the mode numbers grow; if that fails, the sparsity condition cannot save the iteration.","fun_headline_variants_meta":{"raw":{"variants":["NLS gets almost periodic solutions without extra parameters","Infinite-dimensional tori give NLS almost periodic solutions","NLS almost periodic solutions not quasi-periodic, no external parameters","Non-resonant tori for NLS yield almost periodic orbits","NLS almost periodic solutions accumulate on known tori"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000676,"raw_usage":{"total_tokens":3005,"prompt_tokens":807,"completion_tokens":2198,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":423,"completion_tokens_details":{"reasoning_tokens":2116}},"tokens_in":423,"tokens_out":2198,"duration_ms":14772,"temperature":1.0,"reasoning_tokens":2116,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:17:05.157837+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for cubic NLS, the terms generated by the normal form at order eight and check the claimed bound $|P^{\\ell,\\sigma}| \\lesssim 1 \\wedge (|\\ell^*_3|^2/|\\ell^*_1|)$ on every monomial whose highest mode is paired as an action; a single monomial whose coefficient stays of order one while $|\\ell^*_1|$ grows would falsify the smoothing estimate that the KAM part uses as a black box.","supporting_citations":[],"review_version":1}