{"id":"8eadb2ee-810a-4f26-8692-ef2c8db0a366","arxiv_id":"2607.09937","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"T_k (kth powers) is never maximal for k≥3; V_k is maximal only for k=2,3; W_k is never maximal for k≥3, each proved by explicit supersequences whose sum-free property follows from known Diophantine results.","lead":"The paper classifies maximality for three families of integer sequences that are closed under multiplication but whose pairwise sums never land back in the sequence. Most such sequences can be properly enlarged while preserving both properties, via explicit constructions that rest on classical and recent Diophantine non-existence theorems.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the single external non-classical ingredient and correctly judges that the remainder of the proofs is elementary and classical. Because that ingredient is used only for three small explicit integers and the rest of the classification is self-contained, the load-bearing risk is low and does not warrant a change of verdict. The paper’s reductions (composite k, odd primes via Cohen, k=4 via the fourth-power equations, V_k via residue classes, W_k via Catalan) are transparent and free of free parameters or circularity. Hence the ACCEPT verdict stands.","tokens_in":7535,"tokens_out":349,"duration_ms":2920,"concrete_test":"Independently verify (or recompute via known tables / modular methods) that none of 11, 121, 1331 equals a difference of two rational fourth powers; if any of them does, re-examine only the k=4 case of Theorem 4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims (Theorems 4–5) rest on classical Diophantine non-existence (Wiles, Fermat, Selmer, Catalan/Mihăilescu, Cohen) plus elementary modular arithmetic, all of which are correctly applied. The sole non-classical citation is Ratcliffe–Xuan (arXiv:2604.15832) used only to exclude the three concrete values 11, 121, 1331 as differences of rational fourth powers; that dependence is narrow, explicit, and does not undermine the rest of the argument. No hidden assumption, gap, or internal inconsistency appears in the reductions.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies increasing sequences of positive integers that are closed under multiplication (P1) while containing no sum of two terms (P2), called GD sequences, and asks which are maximal. It treats three classical families: T_k = (n^k), V_k = numbers ≡1 (mod k), and W_k (presented as k^n−1 but proved and OEIS-linked as pure powers of k). Theorem 4 asserts that T_k is never maximal for k≥3, via enlargements by powers of 11 and reduction to a compiled non-existence theorem (Theorem 6: Wiles, Selmer, Cohen, Fermat, and a 2026 preprint of Ratcliffe–Xuan). Theorem 5 asserts that V_k is maximal precisely for k=2,3 and non-maximal for k≥4 (by the ±1 residue class), and that W_k is non-maximal for all k≥3 (by a p^2-power enlargement and Mihăilescu’s theorem). An open problem asks for explicit maximal extensions of each family.","tokens_in":7649,"tokens_out":1169,"duration_ms":50456,"significance":"If the statements are corrected and the external Diophantine inputs are accepted, the paper cleanly settles maximality for three natural multiplicative families by elementary modular arithmetic plus standard deep theorems (FLT, Catalan/Mihăilescu, Selmer, Cohen). The reductions themselves are transparent and fully written; the open problem of constructing maximal extensions is well-posed and of genuine interest in additive–multiplicative combinatorial number theory. The contribution is solid but modest: it organizes known non-existence results around a simple maximality notion rather than introducing new Diophantine technology. Credit is due for the careful pairwise-coprimality reduction before applying Cohen and for the uniform treatment of the three families.","major_comments":[{"comment":"Example 3, Theorem 5(ii), and the real/complex examples: the manuscript defines W_k = (k^n − 1) and likewise writes (x^n − 1), yet (i) OEIS A000244/A000302/A000351/A000400 are pure powers of k, (ii) the P2 argument “k^n + k^n < k^{n+1}” is the pure-power argument, and (iii) equation (8) is exactly the sum-of-two-pure-powers equation, not the equation obtained from sums of terms of the form p^{2(m−1)}k^n − 1 (which would produce a +1). As written, P1 fails for k^n − 1 (e.g. (k−1)^2 is never of that form). The definition, examples, and Diophantine setup must be aligned—almost certainly by deleting the spurious “−1” throughout the W-family and the real/complex illustrations—so that the objects proved about are the objects claimed.","section":null},{"comment":"Theorem 6(ii) and the k=4 case of Theorem 4: non-existence of nonzero solutions to x^4 − y^4 = 11^a z^4 for a=1,2,3 is taken solely from the absence of 11, 121, 1331 in Table 3 of the April 2026 arXiv preprint Ratcliffe–Xuan. That dependence is load-bearing for the T_4 enlargement and rests on an unpublished source. Supply an independent, self-contained argument for these three concrete equations (or an alternative prime/enlargement for T_4 that relies only on published results).","section":null}],"minor_comments":[{"comment":"In-text citation “A. Ratcliffe and N. Tho [3]” disagrees with the reference list “A. Ratcliffe and T. Xuan”; unify the author name.","section":null},{"comment":"Keywords list A016863 while Example 2 and the body cite A016861 for V_6; correct the OEIS pointer.","section":null},{"comment":"The parenthetical digression on Carmichael numbers after the proof of Theorem 6 is unrelated to the maximality claims and should be removed or moved to a remark/open question.","section":null},{"comment":"Notation: the paper uses “a ∈ S” for “a is a term of the sequence S”; a brief sentence fixing this convention early would help readers accustomed to set notation.","section":null},{"comment":"In the proof of Theorem 4 for composite k ≠ 4, the claim “k = pl with l ≥ 3” is correct but would be clearer if the case k = p^2 (p odd) were mentioned explicitly as already covered.","section":null}],"recommendation":"major_revision","confidential_remarks":"The logical skeleton is sound once W_k is read as pure powers of k; the −1 appears to be a systematic typesetting/copy-paste error rather than a conceptual mistake, because every proof equation and OEIS citation matches the pure-power interpretation. I would not reject on that ground, but it must be fixed before acceptance. The Ratcliffe–Xuan dependence is narrow; if the journal’s policy is strict about published sources for load-bearing lemmas, insist on a self-contained check of the three fourth-power differences. Scope is appropriate for a short number-theory note."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper settles maximality for three concrete families of what the authors call GD sequences (increasing positive integers closed under multiplication, sum-free). Theorems 4 and 5 are the whole contribution: T_k (kth powers) is never maximal for k≥3; V_k (numbers ≡1 mod k) is maximal only for k=2,3; W_k (k^n-1) is never maximal for k≥3. They also give explicit larger GD supersequences in each non-maximal case.\n\nWhat is new is precisely those maximality statements; nothing in the cited literature already decides them. The proofs are clean reductions. Composites reduce to smaller exponents. The 11-power enlargements of T_p and T_4 reduce to the Diophantine statements collected in Theorem 6 (Wiles, Fermat, Selmer, Cohen, plus Ratcliffe–Xuan for the three small values 11,121,1331). The V_k and W_k enlargements reduce to elementary modular arithmetic or Catalan/Mihăilescu. All of that is written carefully and correctly. The stress-test is right: no hidden gap or circularity appears.\n\nThe soft spots are real but narrow. The only non-classical dependence is the April 2026 Ratcliffe–Xuan list used solely to exclude three concrete fourth-power differences; if that list later needs correction the T_4 case would need a different argument, but everything else stands. The open problem of constructing maximal supersequences for every V_p (p prime ≥5) is left open, which is honest. Scope is modest: three families, no general theory of GD sequences, no new Diophantine results of their own.\n\nThis is for people who like additive-multiplicative combinatorics on concrete sequences and OEIS-style questions. It is correctly proved, clearly written, and free of free parameters or self-referential definitions. A serious editor should send it to referees; I would accept it after ordinary checks on the external citations. I would not cite it myself in the next year, but I would bring it to a reading group if we were looking at sum-free multiplicative sets.","headline":"Clean, correctly proved maximality classification for three natural families of multiplicative-closed sum-free sequences; modest scope, solid execution.","tokens_in":8243,"tokens_out":558,"would_cite":false,"duration_ms":4612,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B83","11D41","11D72"],"pacs":[],"model":"grok-4.5","headline":"Three classical families of multiplicative integer sequences are shown to be non-maximal for most parameters, via Diophantine non-existence results.","keywords":["integer sequences","GD sequences","maximal sequences","generalized Fermat equation","Catalan's conjecture","multiplicative closure","sum-free sets"],"falsifier":"An explicit nonzero integer solution to x^4 - y^4 = 11^a z^4 for some a=1,2 or 3, or a verified rational representation of 11, 121 or 1331 as a difference of two fourth powers, would collapse the maximality argument for fourth powers.","tokens_in":8420,"feed_emoji":"🔢","tokens_out":617,"duration_ms":4774,"temperature":0.7,"pith_summary":"The paper studies increasing sequences of positive integers that are closed under multiplication yet contain none of their pairwise sums (called GD sequences). It asks which of three classical families—kth powers, numbers congruent to 1 mod k, and numbers of the form k^n-1—are maximal, meaning they cannot be properly enlarged while preserving both properties. The authors prove that every sequence of kth powers with k≥3 can be strictly enlarged, that the congruence family is maximal only for k=2 and k=3, and that the exponential family is never maximal. The enlargements are constructed by adjoining carefully chosen multiples of powers of 11 or of large primes, and the proofs that the larger sequences remain free of sums rest on classical and recent theorems about superelliptic equations. A natural open problem is left: produce an explicit maximal GD sequence containing each of the original families.","feed_headline":"Most classical multiplicative sequences can be enlarged","feed_subtitle":"kth powers, residue classes, and exponential sequences fail maximality for almost all k","key_machinery":"GD sequences (increasing positive integers closed under multiplication but free of pairwise sums) together with explicit enlargements whose sum-free property is guaranteed by the non-existence of nonzero solutions to the generalized Fermat equations x^p + y^p = 11^a z^p and x^4 ± y^4 = 11^a z^4, and by Mihăilescu’s theorem.","core_discovery":"For every integer k≥3 the sequence of kth powers is not maximal as a GD sequence; the sequences of integers congruent to 1 modulo k are maximal precisely when k=2 or 3 and are non-maximal for all k≥4; and the sequences of the form k^n-1 are non-maximal for every k≥3.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["kth powers fail maximality as GD sequences for all k≥3","1 mod k residue classes maximal only for k=2 and 3","Sequences k^n-1 non-maximal for every k≥3","Classical multiplicative sequences often enlargeable","Powers residues and exponentials rarely maximal GD"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The claim that the three numbers 11, 121 and 1331 cannot be written as a difference of two rational fourth powers, which is taken from a recent computational table rather than a self-contained proof.","fun_headline_variants_meta":{"raw":{"variants":["kth powers fail maximality as GD sequences for all k≥3","1 mod k residue classes maximal only for k=2 and 3","Sequences k^n-1 non-maximal for every k≥3","Classical multiplicative sequences often enlargeable","Powers residues and exponentials rarely maximal GD"]},"model":"grok-4.5","effort":"low","cost_usd":0.005372,"raw_usage":{"total_tokens":1326,"prompt_tokens":598,"num_sources_used":0,"completion_tokens":84,"cost_in_usd_ticks":53720000,"prompt_tokens_details":{"text_tokens":598,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":644,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":598,"tokens_out":84,"duration_ms":4531,"temperature":1.0,"reasoning_tokens":644,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T14:27:39.101171+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"An explicit nonzero integer solution to x^4 - y^4 = 11^a z^4 for some a=1,2 or 3, or a verified rational representation of 11, 121 or 1331 as a difference of two fourth powers, would collapse the maximality argument for fourth powers.","supporting_citations":[],"review_version":1}