{"id":"45d793ac-ce97-41d4-af7c-449c127d74c5","arxiv_id":"2502.03348","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For n=m=p prime, the only Ducci periods are 1, the order of 2 modulo p (for constant tuples), and p times that order; for n=3 and m odd prime, all non-exceptional tuples realize the maximum period.","lead":"The paper studies a modular arithmetic game on a circle of n numbers, where each turn replaces every entry by the sum of itself and its next neighbor modulo m. It proves exactly which cycle lengths can occur for three entries modulo an odd prime, and for cycles whose length equals the modulus when that number is an odd prime.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader focused on the black-box L_m(3) = 0 in Theorem 3. I agree that is the weakest point for that theorem, but for the stated strongest claim (Theorem 4) the argument is largely self-contained and the few black-box uses are removable. The terse derivation of P_p(p) = pδ omits the non-multiple-of-p case, but the omitted case is easy to supply and does not change the conclusion. I find no load-bearing flaw; the verdict can remain conditional because of the unreproduced MATLAB census in Section 5, but not because of an error in Theorem 4.","tokens_in":17331,"tokens_out":32669,"duration_ms":285059,"concrete_test":"For p = 7, construct the reduced matrix B from the equations D^6 u = u, verify det(B) ≡ ±1 mod 7, and then enumerate all 7^7 tuples mod 7 to confirm the period multiset is exactly {1, 3, 21}. This checks both the matrix-nonsingularity step and the maximal-period computation P_7(7) = 21.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The central classification in Theorem 4 is internally sound: over F_p^p one has D^p = 2I, so the basic tuple returns first at pδ (any α = qp + r with 0 < r < p forces the last coordinate 2^q = 1 and hence α ≥ pδ); the divisor analysis reduces to d | δ or d = p·r with r < δ; and the latter is impossible for nonzero u since D^{pr}u = 2^r u. The remaining fixed-point system D^{p-1}u = u has only uniform solutions, as the reduced (p−1)×(p−1) matrix is the nonsingular Lemma 11 matrix. The black-box facts from [13] used for Theorem 3 and for the zero-period clause of Theorem 4 are true in the needed cases and can be replaced by the short invertibility argument D = I + H, so they do not threaten the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the period structure of the Ducci map D(x_1,...,x_n)=(x_1+x_2,...,x_n+x_1) on Z_m^n. The main results are Theorem 3, which states that for n=3 and m an odd prime, every tuple that satisfies neither the sum condition nor the uniformity condition has maximal period P_m(3), and Theorem 4, which states that for n=m=p an odd prime, the only possible periods are 1 (for the zero tuple), δ (the multiplicative order of 2 modulo p, for uniform nonzero tuples), and pδ (for all other tuples). The paper also contains auxiliary lemmas about uniform tuples, sum-condition tuples, and period lifting under divisors of the modulus, as well as computational tables for small prime n,m and a discussion of symmetry groups of exceptional periods.","tokens_in":17458,"tokens_out":15647,"duration_ms":139582,"significance":"If the results are correct, the paper gives a satisfyingly complete picture of possible periods in the n=3 odd-prime case and in the n=m=p case, with the latter being a sharp three-value classification. The methods are elementary but the coefficient machinery a_{r,s} is useful, and the computational tables provide concrete data that could guide future work. The paper is modest in scope but represents a solid contribution to the Ducci-sequence literature. I explicitly note that Theorem 4's classification is falsifiable and the proof is checkable; the main obstacles are the false statement of Lemma 5 for composite moduli and the reliance on same-author preprints for load-bearing facts, both of which are fixable.","major_comments":[{"comment":"Lemma 5 as stated is false for composite odd m_1. For example, take m=9 (so l=0, m_1=9) and x=3. The order of 2 modulo 9 is δ=6, but 2^2·3 ≡ 3 mod 9, so D^2(3,3,...,3)=(3,3,...,3) and the period is 2, not 6. The proof implicitly treats the condition 2^r x ≡ x mod m_1 as equivalent to 2^r ≡ 1 mod m_1, which is only valid when gcd(x,m_1)=1. The lemma should either assume gcd(x,m_1)=1 or restrict m_1 to be prime; the statement and proof must be corrected. This does not invalidate Theorem 4 because there m_1=p is prime and every nonzero x is invertible modulo p, but the lemma as written is a serious mathematical error.","section":"Section 2, Lemma 5"},{"comment":"The derivation of P_p(p)=pδ contains a garbled citation: the text says 'because of Corollary 6 in [12] a_{p,1}=2 and a_{p,s}=binom(p,s-1)', but binom(p,0)=1 for s=1, which contradicts a_{p,1}=2. Moreover, the binomial formula a_{r,s}=binom(r,s-1) in Theorem 5 of [12] is stated only for 0≤r<n, whereas here r=n=p, so the formula does not apply as written. The intended conclusion D^p(u)=2u is correct and follows directly from D=I+H and the binomial theorem over F_p: D^p=(I+H)^p≡I^p+H^p=2I mod p. Please replace the incorrect citation with this direct proof.","section":"Section 6, proof of Theorem 4"},{"comment":"The proofs of Theorem 3 and Theorem 4 depend on load-bearing facts quoted from the same authors' unpublished preprints: Theorem 7 of [13] (L_m(3)=0 for odd prime m) is used in the proof of Theorem 3 to justify equations (4.1)-(4.3), and Theorem 2 of [13] (L_p(p)=0) is used in Theorem 4, along with Corollary 6 of [12]. Since [12] and [13] are listed as 'Submitted for Publication' and are not yet peer-reviewed, the paper should either prove these facts (for example, L_m(3)=0 for odd prime m follows because D=I+H and det(I+H)=2 is invertible over F_m) or cite a published source. This is not a stylistic request: without L_m(3)=0, the equations D^d(u)=u used in the proof of Theorem 3 would not follow for tuples with a nonzero preperiod.","section":"Theorem 3 and Theorem 4"}],"minor_comments":[{"comment":"In the proof of part (5), the sentence 'if x_1,x_2,x_3 ∈ {0,m/2}, then Per(u)=3' should explicitly exclude the all-equal tuple (m/2,m/2,m/2), which has D(m/2,m/2,m/2)=(0,0,0) and period 1, not 3.","section":"Lemma 2(5)"},{"comment":"The case m=3 is dismissed with 'if you plot out the Ducci sequences for all 27 tuples'; for a formal proof, please include a short table or a direct argument showing that only uniform tuples have period less than 6.","section":"Theorem 3 proof"},{"comment":"The computational claims summarized in Figures 3-5 are not accompanied by the MATLAB code or a precise description of the algorithm beyond the n=5,m=7 example; for reproducibility, please include the program as an ancillary file or provide pseudocode.","section":"Section 5"},{"comment":"The induction proof for the residues 2, 3, 4, 5 modulo 6 is only sketched with 'We can repeat this pattern'; please expand this step so the reader can verify the claimed congruences without reconstructing the induction.","section":"Lemma 8"},{"comment":"The line 'We can now prove Theorem ??' before the proof of Theorem 4 contains a placeholder and should read 'Theorem 4'.","section":"Section 6"},{"comment":"In the proof of Lemma 5, the phrase 'Since δ is the smallest value where this can happen' is only true when x is invertible modulo m_1; the corrected statement and proof should make the coprimality assumption explicit.","section":"Section 2, Lemma 5"}],"recommendation":"major_revision","confidential_remarks":"The central theorems (Theorem 3 and Theorem 4) appear to be correct, and the errors I found are fixable. However, the false Lemma 5 is a substantive mathematical error in the manuscript as written, and the proof of Theorem 4 contains an incorrect citation for a key computation. Additionally, the paper leans heavily on two same-author preprints ([12], [13]) for load-bearing facts; the editor may wish to verify the status of those preprints or ask the authors to include the short proofs. The computational section is somewhat informal but acceptable if code or more detailed algorithms are provided. Overall, this is a promising paper that needs a revision before it can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe thing to know: Theorem 4 is the real contribution—a complete period trichotomy for n=m=p prime—and it is correct. Theorem 3 gives the n=3 classification and is also sound, with a real but replaceable dependence on prior work. This is solid subfield-level progress, not a breakthrough.\n\nWhat is new: the complete set of periods for n=m=p and the n=3, m odd prime classification appear in no cited prior paper. The proof strategy is clean: over F_p^p, D^p = 2I, so periods divide pδ; a divisor analysis plus invertibility of the (p−1)×(p−1) matrix in Lemma 11 leaves only 1, δ, pδ. The determinant computation is correct, and the fixed-point argument rules out intermediate periods.\n\nThe n=3 theorem also holds up. It does invoke Theorem 7 of [13] to get L_m(3)=0 for m odd prime before writing equations (4.1)–(4.3). That is a genuine black-box dependency, but the stress-test note is right that the fact is true and can be derived in a couple lines from D = I + H, so it is not a fatal gap. Still, the paper should either cite precisely or include the short proof. There is a similar, less serious reliance on Corollary 6 of [12] in the proof of Theorem 4; also easy to prove from binomial coefficients.\n\nThe weakest section is 5. The computational census is presented as evidence, but the MATLAB code and data are not shipped, and the search stops at periods over 15 million. That is a real limitation, but it is not load-bearing for Theorems 3 and 4. For a paper that uses computation to support patterns, shipping code would be a meaningful improvement.\n\nThe self-citation pattern is fair; the cited facts are needed and appear correct. I did not find inflated claims or circularity.\n\nWho it is for: people working on Ducci sequences or discrete maps over finite abelian groups. The paper should be refereed, not desk-rejected. I would ask for the black-box lemmas to be proved or quoted exactly, and for the computational code to be made available. A minor revision should be enough.","headline":"Theorem 4 is a correct and clean period trichotomy for n=m=p; the paper is solid and deserves review, with the self-cited lemmas and code made explicit.","tokens_in":18001,"tokens_out":3399,"would_cite":true,"duration_ms":34576,"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":"For the Ducci map on $\\mathbb{Z}_p^p$ with $p$ an odd prime, the paper classifies every possible cycle length: $1$ for the zero tuple, the order of $2$ modulo $p$ for constant tuples, and the maximum $p$ times that order for every other…","keywords":["Ducci sequence","period","modular arithmetic","basic Ducci sequence","n-Number Game","binomial coefficients","coefficient matrix"],"falsifier":"Enumerate all $5^5=3125$ tuples of $\\mathbb{Z}_5^5$ and compute their periods under $D$. Theorem 4 predicts exactly one tuple with period $1$ (the zero tuple), four with period $4$ (the nonzero constant tuples), and all $3120$ remaining tuples with period $20$. Finding any nonconstant nonzero tuple that returns to itself in fewer than $20$ steps would refute the classification.","tokens_in":17102,"feed_emoji":"🔁","tokens_out":8387,"duration_ms":75856,"temperature":0.7,"pith_summary":"The Ducci map sends a cyclic tuple of residues modulo $m$ to the tuple of sums of neighboring entries modulo $m$. Every tuple eventually enters a cycle, and this paper asks which cycle lengths can actually occur. For the square prime case $n=m=p$, the answer is complete: the only periods are $1$ for the zero tuple, $\\delta$ (the multiplicative order of $2$ modulo $p$) for nonzero constant tuples, and the full period $p\\delta$ for every other tuple. For three coordinates with odd prime modulus, it proves that any tuple that is neither zero, constant, nor sum-zero has the maximum period $P_m(3)$. These results give full period spectra for natural infinite families of Ducci processes.","feed_headline":"Odd-prime Ducci cycles have exactly three lengths","feed_subtitle":"Zero, constant tuples, and everything else: that is the whole period spectrum of the neighbor-sum map.","key_machinery":"The workhorse is the family of coefficients $a_{r,s}$: the coefficient of $x_s$ in the first coordinate of $D^r(x_1,\\ldots,x_n)$. For $r<n$ these are binomial coefficients $\\binom{r}{s-1}$, and they satisfy a convolution identity $a_{r+t,s}=\\sum_i a_{t,i}a_{r,s-i+1}$. When $n=p$ is prime, the binomial coefficients collapse modulo $p$: $a_{p,s}\\equiv 0$ for $s\\neq 1$ and $a_{p,1}=2$, so $D^p(\\mathbf{u})=2\\mathbf{u}$ for every tuple. This identifies the full period as $p\\delta$ and reduces the search for shorter periods to tuples fixed by $D^{p-1}$; the proof then shows $D^{p-1}(\\mathbf{u})=\\mathbf{u}$ forces $\\mathbf{u}$ to be zero or constant by inverting the alternating-sign coefficient matrix of Lemma 11, whose determinant is $\\pm 1$.","core_discovery":"The central claim is Theorem 4: if $n=m=p$ is an odd prime and $\\delta$ is the order of $2$ modulo $p$, then the only possible values of $\\mathrm{Per}(\\mathbf{u})$ on $\\mathbb{Z}_p^p$ are $1$, $\\delta$, and $p\\delta$. The zero tuple alone has period $1$; a tuple $(x,\\ldots,x)$ with $x\\neq 0$ has period $\\delta$; every other tuple has period $P_p(p)=p\\delta$. The paper also establishes Theorem 3 for $n=3$: for odd prime $m$, every tuple $\\mathbf{u}=(x_1,x_2,x_3)$ whose entries are not all equal and whose sum is not $0$ modulo $m$ has period $P_m(3)$, while nonzero sum-zero tuples have period at most $6$ by Lemma 2.","pith_inferences":["The methods point toward a conjecture the authors do not make: for $n=m=p^k$ with $p$ an odd prime, the period spectrum should be governed by the order of $2$ modulo $p^k$ and by lifts of the prime-power factors, and Lemma 6 provides the divisibility half of such a classification.","The observed stabilizers for exceptional periods (dihedral groups of order $2n$ and a Frobenius group of order $21$) suggest that counting tuples with a given exceptional period can be reduced to counting orbits of the symmetric group on solution sets of $D^d(\\mathbf{u})=\\mathbf{u}$, a count the paper does not attempt.","For a fixed dimension $n$, the tables show the sum condition is not by itself a period shortcut; a general theorem would need to read the coefficients $a_{d,s}$ modulo $m$, since the sums of coordinates alone cannot predict which short period will appear."],"forward_implications":["For $n=m=p$, the maximum period is exactly $p\\delta$, and it is achieved by every tuple except the zero tuple and the nonzero constant tuples; in particular there is no intermediate period.","For $n=3$ with odd prime modulus, the sum condition $x_1+x_2+x_3\\equiv 0$ is the only source of short periods among nonconstant tuples, and those tuples have period $6$ when nonzero.","Uniformly constant tuples always have period equal to the order of $2$ modulo the odd part of $m$, so they provide a recurring short period in every dimension.","If $m_1\\mid m$, every period realized in $\\mathbb{Z}_{m_1}^n$ is also realized in $\\mathbb{Z}_m^n$, so the set of possible periods is monotone in the modulus.","For the prime pairs tabulated in Figures 3 through 5, the exceptional short periods are rare and, in several cases, the tuples of a given exceptional period form orbits under a dihedral or Frobenius symmetry group."],"supporting_citations":[{"why":"Supplies Lemma 1, which says $\\mathrm{Per}(\\mathbf{u})\\mid P_m(n)$ and bounds lengths, making $P_m(n)$ the maximum period and restricting possible periods to its divisors.","marker":"[1]"},{"why":"Supplies the coefficient arithmetic for $a_{r,s}$, including the binomial formula for small $r$ and the consequence that $D^p(\\mathbf{u})=2\\mathbf{u}$ when $n=p$ is prime.","marker":"[12]"},{"why":"Supplies the companion result that the length (preperiod) is zero in the relevant odd-prime cases, so the equations $D^d(\\mathbf{u})=\\mathbf{u}$ are valid for the tuples considered.","marker":"[13]"},{"why":"Supplies the binomial congruence $\\binom{p-1}{s-1}\\equiv(-1)^{s-1}$ mod $p$, used to set up the $D^{p-1}(\\mathbf{u})=\\mathbf{u}$ system that the determinant lemma solves.","marker":"[14]"}],"fun_headline_variants":["Odd-prime Ducci map: only three period values exist","Ducci cycles on prime grids have exactly three lengths","Prime p Ducci periods: 1, order of 2, and p times that","Neighbor-sum map on Z_p^p: periods are one of three","Odd-prime Ducci cycles have exactly three period values"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of Theorem 3 and the no-preperiod steps in Theorem 4 assume the companion result that for odd prime modulus every tuple already lies on its Ducci cycle rather than having a preperiod; without that, the equations $D^d(\\mathbf{u})=\\mathbf{u}$ used throughout the argument are not guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["Odd-prime Ducci map: only three period values exist","Ducci cycles on prime grids have exactly three lengths","Prime p Ducci periods: 1, order of 2, and p times that","Neighbor-sum map on Z_p^p: periods are one of three","Odd-prime Ducci cycles have exactly three period values"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000726,"raw_usage":{"total_tokens":3247,"prompt_tokens":932,"completion_tokens":2315,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":548,"completion_tokens_details":{"reasoning_tokens":2223}},"tokens_in":548,"tokens_out":2315,"duration_ms":15260,"temperature":1.0,"reasoning_tokens":2223,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T05:02:27.767169+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all $5^5=3125$ tuples of $\\mathbb{Z}_5^5$ and compute their periods under $D$. Theorem 4 predicts exactly one tuple with period $1$ (the zero tuple), four with period $4$ (the nonzero constant tuples), and all $3120$ remaining tuples with period $20$. Finding any nonconstant nonzero tuple that returns to itself in fewer than $20$ steps would refute the classification.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 1, which says $\\mathrm{Per}(\\mathbf{u})\\mid P_m(n)$ and bounds lengths, making $P_m(n)$ the maximum period and restricting possible periods to its divisors."},{"cited_title":"The Period of Ducci Cycles on $\\mathbb{Z}_{2^l}$ for Tuples of Length $2^k$","cited_arxiv_id":"2401.17502","evidence_quote":"Supplies the coefficient arithmetic for $a_{r,s}$, including the binomial formula for small $r$ and the consequence that $D^p(\\mathbf{u})=2\\mathbf{u}$ when $n=p$ is prime."},{"cited_title":"Ducci on $\\mathbb{Z}_m^n$ and the Maximum Length for $n$ Odd","cited_arxiv_id":"2403.05319","evidence_quote":"Supplies the companion result that the length (preperiod) is zero in the relevant odd-prime cases, so the equations $D^d(\\mathbf{u})=\\mathbf{u}$ are valid for the tuples considered."},{"cited_title":"The Maximum Length for Ducci Sequences on $\\mathbb{}Z_m^n$ when $n$ is Even","cited_arxiv_id":"2410.18204","evidence_quote":"Supplies the binomial congruence $\\binom{p-1}{s-1}\\equiv(-1)^{s-1}$ mod $p$, used to set up the $D^{p-1}(\\mathbf{u})=\\mathbf{u}$ system that the determinant lemma solves."}],"review_version":1}