{"id":"8bf8ba94-00a0-472e-be82-95a5b32141e8","arxiv_id":"2502.03345","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove that for several families of moduli, cyclically shifting the starting tuple does not change its Ducci cycle, and they tabulate many more such cases.","lead":"The paper proves several cases where the Ducci number game on lists of numbers modulo m has a rotational symmetry: cyclically shifting the starting list does not change its eventual repeating cycle. A generalist might read it because it is a clean example of how a simple repeating process can be invariant under rotation, with proofs for infinite families and computer tables suggesting broader patterns.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"H-closedness proofs rest on unpublished maximum-length theorems; if L_m(3)=l or L_m(n)=1 fails in the used ranges, Lemma 4's transfer to all cycles breaks. Theorem 2(3) also states D^{p-1}=u while its proof establishes D^{p-1}=H^2.","rationale":"I read the paper in good faith and checked the intended coefficient algebra. The circulant structure of D gives D^r(e)=(c_r,b_r,a_r) for n=3, and the identities in the proofs are consistent with that convention. Spot checks for m=4, 8, 10, and for n=4 with m=3,5,7 confirm D^2=H, D^{p-1}=H^2, and D^{m-1}=H^{-1} on the respective cycles. The false statement in Theorem 2(3) is real but does not undermine H-closedness, since H^2 generates the same cyclic shift group when n=3. The genuine soft spot is the dependence of Lemma 4 on the unpublished length theorems from [12] and [13], and the mod-6 congruences from [14]. These are load-bearing because every later identity is proved only after knowing that D^L(e) lies on the eventual cycle. My proposed test re-derives those bounds from the binomial-coefficient description, which would settle the concern without relying on the preprints. Until that is done, CONDITIONAL is the appropriate verdict, matching the reader's assessment.","tokens_in":11866,"tokens_out":42112,"duration_ms":376654,"concrete_test":"Independently derive the two imported bounds from first principles. For even n and gcd(n,m)=1, prove L_m(n)=1 by solving Dx=a(1,-1,1,-1,...,1) over Z_m and showing the only solution is a=0 for all a; this reduces to n*a=0 mod m and is decidable by linear algebra. For n=3, prove L_m(3)=l for m=2^l and m=2^l p (p≡5 mod 6) from the binomial-sum formulas for a_r, b_r, c_r, verifying that the 2-adic part forces exactly the first l steps to be preperiodic. If both derivations succeed, the dependence on [12] and [13] is discharged; if either has a counterexample in the ranges used by Theorems 2 and 3, the H-closedness proofs fail.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central proofs of Theorems 2(1), 2(3), and 3 all pass through Lemma 4, which reduces H-closedness of Z_m^n to H-closedness of the basic Ducci sequence using explicit maximum-length theorems: L_m(3)=l for m=2^l and m=2^l p, and L_m(n)=1 for even n with gcd(n,m)=1. These bounds, imported from the authors' preprints [12] and [13], determine that D^L(e) is already on the eventual cycle. The coefficient identities D^2=H, D^{p-1}=H^2, and D^{m-1}=H^{-1} are proven on the orbit of the basic tuple; Lemma 4 then transfers them to all of K. If either imported bound has an unstated restriction or fails for some edge case, those identities are not established on the eventual cycle, and H-closedness is unsupported. The paper also imports Lemma 8 of [14] for the mod-6 coefficient congruences used in Theorem 2(2) and (3). Separately, the printed Theorem 2(3) asserts D^{p-1}(u)=u, while its proof establishes D^{p-1}(u)=H^2(u); this internal inconsistency must be corrected, although the H^2 identity would still imply H-closedness.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Ducci sequences on Z_m^n under the endomorphism D=I+H, where H is the cyclic shift, and introduces the notion of H-closedness: a Ducci sequence is H-closed when every tuple in its eventual cycle is mapped into the same cycle by every power of H. The main results are Theorem 2, asserting that Z_3^m is H-closed when m is a power of 2, when m is prime congruent to 5 mod 6, and when m=2^l p with p congruent to 5 mod 6 prime, and Theorem 3, asserting that Z_n^m is H-closed for even n when m is prime and m ≡ -1 mod n. The proofs reduce to the basic Ducci sequence via Lemma 4 and then verify coefficient identities for powers of D on that sequence.","tokens_in":12208,"tokens_out":26386,"duration_ms":218983,"significance":"The reduction to the basic Ducci sequence in Lemma 4 is a useful organizing idea, and the coefficient computation for Theorem 3 is elegant and internally coherent. If the imported maximum-length and coefficient lemmas are all valid, the paper supplies the first systematic infinite families of H-closed and weakly H-closed cases, backed by extensive tables for 4 ≤ n ≤ 32. The credit is tempered, however, by the fact that the main theorems depend on several unpublished preprints by the same authors, by a false printed identity in Theorem 2(3), and by an apparently misstated coefficient lemma from [14]; as submitted, the proofs are not self-contained and one central reduction is incompletely justified.","major_comments":[{"comment":"The printed statement of Theorem 2(3) says that if u ∈ K(Z_3^m) then D^{p-1}(u)=u. The proof, however, establishes the congruences a_{l+p-1} ≡ b_l, b_{l+p-1} ≡ c_l, c_{l+p-1} ≡ a_l mod m, which give D^{p-1}(D^l(e)) = H^2(D^l(e)) on the basic cycle, i.e., D^{p-1}=H^2 on K. The identity D^{p-1}=u is inconsistent with this proof and is false in general, since H^2 is not the identity. The theorem statement must be corrected to D^{p-1}=H^2 on K; the H^2 identity is what actually implies H-closedness.","section":"Theorem 2(3), Section 3"},{"comment":"The coefficient congruences imported from Lemma 8 of [14] are load-bearing for Theorems 2(2) and 2(3), but as printed they are not consistent with the coefficients defined in Section 2. For example, D^4(0,0,1)=(6,5,5), so a_4=6, b_4=5, c_4=5; the bullet for r≡4 mod 6 states c_r=a_r+1=b_r+1, which would require 5=7=6. No modulus is specified in the bullet list, and the proof applies the relation over Z_m for arbitrary m. This is not merely a missing reference: the assertion as stated is internally inconsistent with the paper's own definitions. The proof of Theorem 2(2) needs a corrected, precisely stated lemma, with either a proof or an exact citation.","section":"Lemma 8 from [14], Section 3"},{"comment":"In the proof of Lemma 4, Eq. (2.1) postulates an unspecified β with -n < β < n and β≠0 such that D^{L+α}(e)=H^β(D^L(e)). The subsequent transfer argument proves only that D^{L+α}(u)=H^β(D^L(u)) for this single β. This is insufficient for H-closedness unless β generates the cyclic shift group. Since the hypothesis that the basic sequence is H-closed gives the case β=1, the proof should choose β=1 and then iterate; as written, the (⇐) direction of Lemma 4 is incomplete. This matters because Lemma 4 is used to pass from the basic cycle to all of K in Theorems 2(1), 2(3), and 3.","section":"Lemma 4, Section 2"},{"comment":"The maximum-length values L_m(3)=l and L_m(n)=1 are imported from Theorem 2 of [12] and Theorem 2 of [13], respectively, and the coefficient multiplication rule and the mod-6 congruences are imported as Corollary 7 and Lemma 8 of [14]. These sources are the authors' own submitted preprints and are not stated in sufficient detail in this manuscript. If any of these imported statements has an unstated hypothesis or fails in an edge case, Lemma 4's reduction and the subsequent coefficient identities are not established. The paper should either include complete proofs of these ingredients or state them as lemmas with full hypotheses and proofs.","section":"Theorems 2 and 3, dependence on [12]-[14]"}],"minor_comments":[{"comment":"The text says that the MATLAB-based determination of H-closedness will be discussed \"in Section ??\"; no such section appears in the manuscript. The computational method behind Figures 3-5 should be described, or the dangling reference should be removed.","section":"Section 2"},{"comment":"The proof begins \"We want to show D^{m-1}(u)=H(u)\", but the theorem statement and the calculation require D^{m-1}(u)=H^2(u); this is a typo that should be corrected.","section":"Section 3, proof of Theorem 2(2)"},{"comment":"There are multiple typos in the first lines: \"p ≡ 5 mod m\" should be \"p ≡ 5 mod 6\", \"D^{p-1}(u)=H(u)\" should be \"D^{p-1}(u)=H^2(u)\", and \"L_m = l\" should be \"L_m(3)=l\".","section":"Section 3, proof of Theorem 2(3)"},{"comment":"The sentence \"For s ≠ 0, n\" should read \"For s ≠ 1, n\", since s ranges from 1 to n and the exceptional cases are s=1 and s=n.","section":"Section 4, proof of Theorem 3"},{"comment":"The condition involving H^{L+β} after the tables appears to be a typo: it should compare D^{L+α}(e) with H^β(D^L(e)), not H^{L+β}(e), in order to match the notation used elsewhere in the paper.","section":"Section 4, computational description"}],"recommendation":"major_revision","confidential_remarks":"The editor may wish to verify the availability and status of references [12]-[14], since the main theorems depend on maximum-length and coefficient results from those preprints. The manuscript fits the journal's scope, but as submitted it is not self-contained and contains a false statement in Theorem 2(3) that will need correction before the claims can be evaluated cleanly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take: this is a legitimate, incremental paper in the Ducci-sequence subfield. The new notion is H-closed sequences (the eventual cycle is invariant under cyclic shift), and the paper proves three infinite families for n=3 and one even-n family. The workhorse Lemma 4 — H-closedness of Z_m^n is equivalent to H-closedness of the basic Ducci sequence — is genuinely useful and correct: it uses the standard decomposition of any tuple into shifts of (0,...,0,1) and linearity/commutation of D,H. That is the cleanest part of the paper.\n\nThe coefficient calculations in Sections 3 and 4 check out. Theorem 2(1) (m=2^l gives D^2=H on K) follows from the binomial identities; Theorem 2(2) (m≡5 mod 6 prime, D^{m-1}=H^2) uses Fermat's little theorem and the mod-6 coefficient congruences; Theorem 3 (n even, m≡-1 mod n prime, D^{m-1}=H^{-1}) is a neat application of Lemma 5 and the fact that only i=0 and i=δn-1 terms survive. I agree with the reader: the printed Theorem 2(3) is wrong as stated (D^{p-1}=u), while the proof establishes D^{p-1}=H^2 on K. That is a typo, not a conceptual flaw, but it has to be fixed.\n\nReal soft spots:\n- The transfer lemma depends on maximum-length theorems imported from the authors' two submitted preprints ([12], [13]). If L_m(3)=l or L_m(n)=1 fails in any used range, Lemma 4 does not land. The companion papers may well be right, but the present paper is not self-contained on its load-bearing assumption. A referee needs those preprints or a proof sketch.\n- The computational survey (Figures 3-5) is unreproducible as written: Section 4 references 'Section ??', and no MATLAB code or parameters are given. The tables are exploratory, not load-bearing for the theorems — but they are presented as data, and the paper should say how they were produced.\n- Minor: the modular coefficient congruences (Lemma 8 of [14]) are also imported. Again, likely true, but on a chain.\n\nWho it's for: people working on Ducci sequences or modular maps on cyclic groups. It will not pull in outsiders. But within that niche it is a serviceable extension with a reusable lemma.\n\nRecommendation: send it to review. I'd require the Theorem 2(3) fix, an appendix or explicit references for the imported length theorems, and a brief computational methods paragraph. With those, I'd accept it for a specialist journal.","headline":"Correct but under-polished extension of the Ducci-cycle literature; the H-closed reduction is the useful idea, and the main theorems hold up modulo a typo and heavy reliance on the authors' own preprints.","tokens_in":12685,"tokens_out":3140,"would_cite":true,"duration_ms":25492,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20D60","11B83","11B50"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that Ducci cycles on $\\mathbb{Z}_m^n$ are closed under cyclic rotation for $n=3$ with $m$ a power of $2$, $m\\equiv 5\\pmod 6$ prime, or $m=2^\\ell p$ with $p\\equiv 5\\pmod 6$ prime, and for even $n$ with $m\\equiv -1\\pmod n$…","keywords":["Ducci sequences","H-closed","modular arithmetic","cyclic shift","n-number game","binomial coefficients","eventual cycles","Fermat's little theorem"],"falsifier":"Compute the Ducci cycle of the basic tuple $(0,0,1)$ in $\\mathbb{Z}_8^3$: if any cycle point $v$ has $D^2(v)\\neq H(v)$, the power-of-two case fails. Similarly, compute the cycle of $(0,0,0,1)$ in $\\mathbb{Z}_7^4$ and check whether $D^6(v)=H^{-1}(v)$ for every cycle point; a counterexample would falsify the even-$n$ prime case.","tokens_in":11702,"feed_emoji":"🔄","tokens_out":16428,"duration_ms":123363,"temperature":0.7,"pith_summary":"This paper asks when the Ducci map $D$ on $\\mathbb{Z}_m^n$, which sends each coordinate to its sum with the next coordinate modulo $m$, produces cycles that are unchanged by the cyclic shift $H$. The authors call such a modulus pair $H$-closed and prove it for several families: $n=3$ with $m$ a power of $2$, with $m$ a prime congruent to $5 \\bmod 6$, or with $m=2^\\ell p$ where $p\\equiv 5 \\bmod 6$ is prime, and even $n$ with $m\\equiv -1 \\bmod n$ prime. The key reduction is that $\\mathbb{Z}_m^n$ is $H$-closed exactly when the basic Ducci sequence starting from $(0,\\ldots,0,1)$ is $H$-closed, and the cycle is described by explicit identities such as $D^2=H$, $D^{m-1}=H^2$, and $D^{m-1}=H^{-1}$. These identities give a precise sense in which rotation is a hidden symmetry of the iterated process.","feed_headline":"Ducci cycles stay rotation-invariant for several new moduli","feed_subtitle":"New identities tie the Ducci map to cyclic rotation, covering n=3 and even n with m≡-1 (mod n).","key_machinery":"The load-bearing reduction is Lemma 4: $\\mathbb{Z}_m^n$ is $H$-closed if and only if its basic Ducci sequence, the orbit of $e=(0,\\ldots,0,1)$ under $D$, is $H$-closed. This works because $u=\\sum_{s=1}^n x_s H^{-s}(e)$ for every $u=(x_1,\\ldots,x_n)$, and $D$ commutes with $H$, so an identity $D^{\\alpha}(e)=H^{\\beta}(e)$ propagates to all tuples. To prove such identities, the paper tracks the coefficients $a_{r,s}$ that give the $s$-th entry of $D^r(e)$, using the Pascal-style identity $a_{r,s}=\\sum_i \\binom{t}{i}a_{r-t,s-i}$. For $n=3$, these reduce to three interlocking sequences $a_r,b_r,c_r$ with $a_r+b_r+c_r=2^r$ and a mod-$6$ congruence pattern, and Fermat's Little Theorem closes the prime moduli cases.","core_discovery":"The paper's central claim is that for the listed $(n,m)$, the eventual Ducci cycle is closed under the cyclic shift $H$, witnessed by fixed power identities on the cycle subgroup $K(\\mathbb{Z}_m^n)$. For $n=3$, the intended identities are $D^2=H$ when $m$ is a power of $2$; $D^{m-1}=H^2$ on all of $\\mathbb{Z}_m^3$ when $m\\equiv 5\\pmod 6$ is prime; and $D^{p-1}=H^2$ on the cycle when $m=2^\\ell p$ with $p\\equiv 5\\pmod 6$ prime. For even $n$ with $m\\equiv -1\\pmod n$ prime, the identity is $D^{m-1}=H^{-1}$ on the cycle. The proof obtains these by reducing to the basic Ducci sequence—every tuple is a linear combination of cyclic shifts of $(0,\\ldots,0,1)$, and $D$ commutes with $H$—and then verifying the identities with the binomial coefficients that appear in the iterates of $(0,\\ldots,0,1)$.","pith_inferences":["Editorial note: the printed statement of Theorem 2(3) says $D^{p-1}(u)=u$, but the proof in the same section establishes $D^{p-1}(u)=H^2(u)$; the intended identity appears to be the latter.","The same coefficient method should be able to prove the paper's stated conjecture that for even $n$ and $m=p^l$ with $p\\equiv -1\\pmod n$ prime, $D^{\\varphi(m)}=H^{(-1)^l}$ on the cycle, extending $H$-closedness to prime powers.","The tables suggest a parity principle worth testing: for even $n$ the cycle action is an inverse rotation, while for $n=3$ it is a forward rotation $H^2$; determining whether the sign of the rotation depends only on $n$'s parity is a natural next step.","A concrete extension is to check composite moduli with the same residues, such as $m=2^\\ell p^j$ with $p\\equiv 5\\pmod 6$, since the paper's theorems cover only the first power of each prime factor."],"forward_implications":["For $n=3$ and $m$ a power of $2$, every Ducci cycle is rotation-invariant: from any cycle point $u$, $D^2(u)=H(u)$, so one rotation equals two Ducci steps.","For prime $m\\equiv 5\\pmod 6$, $D^{m-1}=H^2$ holds on every tuple in $\\mathbb{Z}_m^3$, so rotation symmetry is global rather than confined to the eventual cycle.","For even $n$ with prime $m\\equiv -1\\pmod n$, $D^{m-1}=H^{-1}$ on the cycle, making $m-1$ Ducci iterations equivalent to one backward rotation.","Because $H$-closedness reduces to the basic sequence, checking a single orbit decides the property, and the paper's tables classify all tested $4\\le n\\le 12$, $3\\le m\\le 12$ pairs as $H$-closed, weakly $H$-closed, or neither.","When $n$ and $m$ are both powers of $2$, every Ducci sequence is trivially $H$-closed since the only cycle is the all-zero tuple."],"supporting_citations":[{"why":"Defines the basic Ducci sequence and supplies Lemma 1 bounding length and period, which drives the reduction to the basic sequence.","marker":"[1]"},{"why":"Provides the binomial expansion for iterates and an earlier proof that D^2=H on the cycle for n=3, m=2^l used as a starting point.","marker":"[7]"},{"why":"Establishes that D and H commute and gives the decomposition of arbitrary tuples into cyclic shifts of the basic tuple.","marker":"[11]"},{"why":"Supplies the maximum-length theorem L_m(3)=l for m=2^l, used to place the basic sequence on its cycle in Theorem 2 cases (1) and (3).","marker":"[12]"},{"why":"Supplies the theorem L_m(n)=1 for even n with gcd(n,m)=1, which is the starting point of the Theorem 3 proof.","marker":"[13]"},{"why":"Provides the mod-6 congruences for the n=3 coefficients and the product formulas used to prove the three cases of Theorem 2.","marker":"[14]"},{"why":"Original study of Ducci processes on Z_m^n, giving the coefficient sum formula and the degenerate both-powers-of-2 H-closed case.","marker":"[18]"},{"why":"Supplies Fermat's Little Theorem, used to evaluate 2^{m-1} mod m in the prime-modulus arguments.","marker":"[5]"}],"fun_headline_variants":["Rotation-proof Ducci cycles for more moduli","Ducci cycles resist cyclic rotation in new cases","H-invariant Ducci cycles proved for n=3 and even n","New moduli make Ducci cycles rotation-stable"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof imports from earlier papers the assertions that the basic tuple has already entered its cycle after exactly $l$ steps when $m=2^l$, and after one step for even $n$ with $\\gcd(n,m)=1$, together with a mod-6 coefficient congruence; if any of these imports is wrong, the identities defining $H$-closedness are not established.","fun_headline_variants_meta":{"raw":{"variants":["Rotation-proof Ducci cycles for more moduli","Ducci cycles resist cyclic rotation in new cases","H-invariant Ducci cycles proved for n=3 and even n","New moduli make Ducci cycles rotation-stable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000659,"raw_usage":{"total_tokens":3050,"prompt_tokens":1018,"completion_tokens":2032,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":634,"completion_tokens_details":{"reasoning_tokens":1967}},"tokens_in":634,"tokens_out":2032,"duration_ms":15326,"temperature":1.0,"reasoning_tokens":1967,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T05:02:41.326723+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Ducci cycle of the basic tuple $(0,0,1)$ in $\\mathbb{Z}_8^3$: if any cycle point $v$ has $D^2(v)\\neq H(v)$, the power-of-two case fails. Similarly, compute the cycle of $(0,0,0,1)$ in $\\mathbb{Z}_7^4$ and check whether $D^6(v)=H^{-1}(v)$ for every cycle point; a counterexample would falsify the even-$n$ prime case.","supporting_citations":[{"cited_title":"Leveraging Print Debugging to Improve Code Generation in Large Language Models","cited_arxiv_id":"2401.05319","evidence_quote":"Supplies the maximum-length theorem L_m(3)=l for m=2^l, used to place the basic sequence on its cycle in Theorem 2 cases (1) and (3)."},{"cited_title":"& Tefft, S.M","cited_arxiv_id":null,"evidence_quote":"Provides the mod-6 congruences for the n=3 coefficients and the product formulas used to prove the three cases of Theorem 2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Fermat's Little Theorem, used to evaluate 2^{m-1} mod m in the prime-modulus arguments."}],"review_version":1}