{"id":"4c1fd827-bddb-44e0-b74c-81989c9e6518","arxiv_id":"2607.10259","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Full-dimensional KAM tori with super-polynomial but sub-exponential radius decay exist for infinite-dimensional long-range Hamiltonian systems under a strengthened Bourgain-type Diophantine condition.","lead":"The paper proves that infinite-dimensional mechanical systems with long-range (all-to-all) interactions still possess full-dimensional KAM tori whose action radii decay only like e to a negative power of log of the site index. This sits between earlier exponential-decay results and the still-open polynomial-decay regime raised by Kuksin.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the openly stated frequency-set restriction.","rationale":"The reader correctly isolates the only genuine limitation: the Diophantine set is uncountable but not shown to have positive product measure, and d=1 is out of reach. That limitation is stated by the authors in §1.5 and proved carefully in §5 via box-dimension covering and the non-resonance bound (5.7). The analytic core (weighted-norm Poisson bracket, Lie-series flow, truncation that exploits σ>2, and the standard Newton scheme) is written out in full and appears consistent. No hidden circularity or estimate loss that would invalidate the claim under the stated hypotheses was found. Therefore the CONDITIONAL verdict, already reflecting the restricted frequency set, needs no adjustment.","tokens_in":16007,"tokens_out":614,"duration_ms":7584,"concrete_test":"Independently recompute the product bound after (3.16): verify that ∑_{j∈k} ln⌊ j⌋ ≤ C(σ,w) Bs (ln Bs)^{1−σ} still yields η^{-1} ∏ (1+k_j^{2} ⌊ j⌋^{4})^{C0(d)} ≪ ε_s^{-0.01} for the retained modes when σ>2; if the exponent fails to stay o(ln ε_s^{-1}), the truncation step loses control and the iteration does not close.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1.2 / Corollary 1.3 holds under the hypotheses the authors actually use. The strengthened Diophantine condition (1.8) that drops the maximal index and requires C0(d) ≥ (4d^{2}+8d+1)/(4d(1−d)) for box dimension d∈(0,1) is the price paid for controlling long-range interactions; the paper proves only that uncountably many such frequencies exist (Theorem 5.1) and explicitly excludes d=1. That restriction is already the reader’s weakest_assumption and is not hidden. The iterative estimates (Poisson bracket Lemma 2.2, flow Lemma 2.3, truncation (3.16)–(3.17), remainder and frequency-shift bounds in the Iterative Lemma) close under the weighted norms with σ>2 and the saving factors δs, so no additional load-bearing gap appears in the written argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves existence of full-dimensional KAM tori for infinite-dimensional mechanical systems with long-range (all-to-all) interactions of the form (1.1)–(1.5). Under a strengthened Bourgain-type Diophantine condition (Assumption 1.1, (1.8)) that omits the maximal multi-index and requires C0(d) ≥ (4d^{2}+8d+1)/(4d(1−d)) for box dimension d∈(0,1), and under smallness of the weighted norm 9R9_ι,r,h ≤ ε (Definition 1.1), Theorem 1.2 constructs a real-analytic symplectic map Φ close to the identity conjugating H to a normal form N* plus a remainder R* containing only terms of order |α|≥2 in the actions. Corollary 1.3 then yields invariant tori with radii I_j ∼ exp(−ln^σ |j|), σ>2. The argument consists of Poisson-bracket and flow estimates (Section 2, Appendix A), a rapidly convergent KAM iteration with truncation (3.16)–(3.17) and Iterative Lemma 3.1 (Sections 3–4), and a measure estimate via box dimension showing that uncountably many frequencies satisfy (1.8) (Theorem 5.1, Section 5).","tokens_in":16284,"tokens_out":1443,"duration_ms":13404,"significance":"The result sits between the exponential-decay tori of Dolgopyat–Fayad–Paradela and the polynomial-decay regime posed by Kuksin for PDEs, and it extends Cong’s recent PDE work (I_j ∼ exp(−ln^σ |j|), σ>2) to long-range mechanical systems. The technical contribution is the adaptation of the weighted norm and the strengthened Diophantine condition that removes the maximal index, together with a careful reduction of the infinite union over maximal indices via box dimension (Lemma 5.2). The paper supplies a complete iterative scheme, explicit estimates, and an honest statement that only uncountably many frequencies are obtained and that d=1 lies outside the argument. These are genuine advances for infinite-dimensional KAM with long-range interactions, even though the frequency set is not of positive product measure.","major_comments":[{"comment":"The Diophantine condition (1.8) and the restriction d<1 are load-bearing for the whole argument (see the truncation bound (3.17) and the covering argument of Lemma 5.2). The paper correctly proves only uncountability (Theorem 5.1) and explicitly excludes d=1 (Section 1.5). This is not a hidden gap, but the abstract and introduction should state more prominently that the result is for a zero-product-measure set of frequencies; otherwise readers may over-interpret the existence claim relative to classical KAM measure statements.","section":null},{"comment":"In the Iterative Lemma 3.1 the frequency-shift estimate (3.14) and the inverse-function step that produces ω*_s+1 rely on the a-priori bound (3.8) remaining strictly less than 1. The constants 0.58, 0.55, 0.5 appearing in (3.14)–(3.15) and (3.20) are chosen so that the series of ε_s^{0.5} converge, but the dependence of ε* on η, σ, w, ι is left completely implicit. A short remark quantifying how small ε must be relative to η (or at least that the iteration closes for sufficiently small ε) would make the existence statement fully checkable.","section":null}],"minor_comments":[{"comment":"Definition 1.1 of the weighted norm 9R9_ι,r,h uses the product over j∈n of e^{ι w ln^σ ⌊ j⌋}; the same product appears with different exponents in the Poisson-bracket proof (2.10)–(2.11). A one-line reminder that the factor e^{-ι w ln^σ ⌊ℓ⌋} is absorbed by the (1-ι) weight would improve readability.","section":null},{"comment":"In (3.16) the truncation threshold B_s = (ln ε_s^{-1})/(w δ_s) is introduced without an explicit comparison to the Diophantine product; the subsequent estimate that the product is ≪ ε_s^{-1} for σ>2 is correct but terse. Adding the intermediate bound on ∑_{j∈k} ln ⌊ j⌋ would help.","section":null},{"comment":"Several references appear with future arXiv numbers (DFP26, LWYZ26). If the paper is accepted, the final versions or DOIs should be inserted when available.","section":null},{"comment":"Typographical: “bω” is used both for the fixed Diophantine vector and for the limiting frequency; a consistent notation (e.g., ω_* versus ω̂) would avoid confusion in Sections 3–4.","section":null},{"comment":"The constant C(σ,w,h) in the estimate after (3.16) is never named; writing “there exists C=C(σ,w,h)>0 such that …” would match the style of Section 5.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a solid, carefully written extension of DFP26 and Cong’s PDE work. The frequency-set restriction is openly acknowledged and does not invalidate the existence claim. I see no reason to reject; minor revision is appropriate mainly for presentational clarity on the measure-zero nature of the frequency set and for a brief quantitative remark on ε*. The paper fits a serious math-ph or dynamical-systems journal."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper fills a concrete intermediate slot between Dolgopyat–Fayad–Paradela’s exponential-decay long-range tori and Kuksin’s still-open polynomial-decay question. The new statement is existence of full-dimensional invariant tori with radii I_j ∼ exp(−ln^σ |j|), σ>2, for infinite-dimensional mechanical systems with all-to-all interactions. That decay is slower than pure exponential and matches Cong’s recent PDE rate, but the setting is long-range mechanical rather than short-range PDE.\n\nWhat they do well is write a complete, standard Newton scheme. The weighted norm (Definition 1.1) is adapted from DFP, the Poisson-bracket and flow estimates (Lemmas 2.2–2.3, Appendix) close cleanly under σ>2, and the truncation (3.16)–(3.17) plus the Iterative Lemma control the small divisors once the maximal index is dropped from the Diophantine condition. The measure argument in Section 5 follows DFP’s box-dimension strategy and correctly shows only uncountably many frequencies, not positive product measure; they flag that d=1 is out of reach. Citations are appropriate and the self-references supply background techniques, not the target claim.\n\nThe soft spots are exactly the ones the authors advertise: the Diophantine lower bound (1.8) is stronger than classical Bourgain (no maximal index, C0(d) depending on box dimension d<1), and the frequency set is uncountable rather than large-measure. That is a genuine restriction, not a hidden flaw; the iterative estimates themselves look solid under those hypotheses. No circularity, no invented constants, no load-bearing gap visible in the written argument.\n\nThis is for people already working in infinite-dimensional KAM or long-range localization. A serious referee should see it; the result is real within its stated limits and the proof is fully written. I would send it out.","headline":"Solid incremental KAM result: full-dimensional tori with super-polynomial (but sub-exponential) decay for genuine long-range mechanical systems, under a strengthened Bourgain-type condition that only guarantees uncountably many frequencies.","tokens_in":16933,"tokens_out":526,"would_cite":true,"duration_ms":5109,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37K55","70H08"],"pacs":[],"model":"grok-4.5","headline":"Full-dimensional KAM tori exist for infinite mechanical systems with long-range interactions, with torus radii decaying only like exp(−ln^σ |j|) for σ>2.","keywords":["full dimensional KAM tori","infinite dimensional Hamiltonian systems","long range interactions","Bourgain Diophantine condition","weighted norms","slow decay radii"],"falsifier":"Exhibit a concrete long-range Hamiltonian whose frequencies satisfy the stated Diophantine bound yet whose KAM iteration diverges, or prove that the set of admissible frequencies is empty for some interval [a,b]^N.","tokens_in":16861,"feed_emoji":"🌀","tokens_out":679,"duration_ms":7225,"temperature":0.7,"pith_summary":"Classical KAM theory and its early infinite-dimensional extensions demanded that interactions decay rapidly in space, so that distant degrees of freedom barely affect one another. Recent work relaxed that requirement to long-range (all-to-all) interactions, but only for tori whose action radii shrink exponentially fast. This paper shows that the same long-range systems still possess full-dimensional invariant tori whose radii decay far more slowly—only like exp(−ln^σ |j|) with σ>2. The result is obtained under a strengthened Bourgain-type Diophantine condition on the frequencies and a weighted analytic norm that keeps the long-range couplings under control. A reader who cares about infinite-dimensional Hamiltonian dynamics therefore learns that full-dimensional quasiperiodic motion survives even when every particle interacts with every other and the torus itself is only moderately thin.","feed_headline":"Slow-decay KAM tori survive long-range interactions","feed_subtitle":"Full-dimensional quasiperiodic motion exists even when every particle couples to every other","key_machinery":"A weighted Fourier–Taylor norm that multiplies each multi-index by an exponential factor e^{ι w ln^σ ⌊j⌋} together with a KAM iteration that solves the homological equation under the strengthened small-divisor bound (1.8); the weight absorbs the long-range interactions while the divisor condition guarantees that the generating function remains small.","core_discovery":"Under a Bourgain-type Diophantine condition that omits the maximal multi-index and a smallness assumption on a weighted Hamiltonian norm, any real-analytic long-range perturbation of an infinite-dimensional mechanical system admits a real-analytic symplectic conjugacy that reduces the Hamiltonian to a pure frequency normal form plus a remainder of order at least two in the actions; the zero-action torus is therefore invariant and its pre-image is a full-dimensional KAM torus whose radii satisfy I_j ∼ exp(−ln^σ |j|) for σ>2.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Full-dim KAM tori exist in infinite systems with long-range forces","Bourgain condition yields full KAM tori for long-range Hamiltonians","Slow-decay full tori persist under long-range particle couplings","Infinite mechanical systems admit full-dimensional KAM tori","Long-range interactions preserve full-dim invariant KAM tori"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The frequencies must obey a Diophantine lower bound strong enough that the maximal multi-index never appears, and only uncountably many such frequencies are known to exist; the argument fails for box dimension one.","fun_headline_variants_meta":{"raw":{"variants":["Full-dim KAM tori exist in infinite systems with long-range forces","Bourgain condition yields full KAM tori for long-range Hamiltonians","Slow-decay full tori persist under long-range particle couplings","Infinite mechanical systems admit full-dimensional KAM tori","Long-range interactions preserve full-dim invariant KAM tori"]},"model":"grok-4.5","effort":"low","cost_usd":0.006008,"raw_usage":{"total_tokens":1480,"prompt_tokens":622,"num_sources_used":0,"completion_tokens":78,"cost_in_usd_ticks":60080000,"prompt_tokens_details":{"text_tokens":622,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":780,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":622,"tokens_out":78,"duration_ms":6310,"temperature":1.0,"reasoning_tokens":780,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T13:05:23.930335+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a concrete long-range Hamiltonian whose frequencies satisfy the stated Diophantine bound yet whose KAM iteration diverges, or prove that the set of admissible frequencies is empty for some interval [a,b]^N.","supporting_citations":[],"review_version":1}