{"id":"a3b99dcf-09f6-4748-ad87-982be5016367","arxiv_id":"2607.13442","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Riordan arrays (p1/p2, t p3) over finite fields, column k's eventual periodic block equals A^k C0, where A is a circulant matrix of p3's coefficients; 3-D layers obey an analogous M4^k rule.","lead":"Columns of polynomial-based number tables (Riordan arrays) over finite fields eventually repeat, and the paper shows the repeating block in column k is obtained from the first block by a fixed circulant matrix raised to the k-th power. The same matrix idea links the layers of a 3-D generalization, giving an exact algebraic picture of periodicity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4's period-π claim rests on Lemma 1, whose proof is invalid when the numerator degree exceeds the denominator's—exactly the case for every column k≥1 under the theorem's hypotheses.","rationale":"The paper's central theorems are explicit and the worked examples support the stated formulas when d2 ≤ d1+1. I read the proof of Theorem 4 carefully and it is essentially correct under the stated hypotheses, provided Lemma 1 holds for numerators of degree ≥ d2. However, the proof of Lemma 1 is flawed for that regime: it applies Theorem 8.40 to the full generating function and derives b0 g = p1, which forces deg(p1) < deg(p2). The paper's earlier comment that one may replace p1 by its remainder r modulo p2 is plausible but not formally proved, and it is exactly what is needed to justify the least period for every column k≥1. This is a proof gap, not a counterexample; the theorem is likely true and the gap is addressable. The reader's weakest_assumption also identified this gap, though they bundled it with the degree restriction. I see the degree restriction as a documented boundary condition rather than a hidden flaw, so my agreement is partial. The appropriate verdict remains CONDITIONAL, matching the reader's assessment, so no change is recommended.","tokens_in":20231,"tokens_out":31591,"duration_ms":257720,"concrete_test":"Give a self-contained proof of the reduction implicit in §2.2: for p1 = q p2 + r with deg(r) < d2, show the coefficient sequence of p1/p2 is ultimately periodic with the same least period as r/p2 (e.g., by verifying that its tail equals the purely periodic sequence of r/p2). Then re-derive Theorem 4(i) for k-th columns using this reduction. If the period π follows for all k≥1, the gap is closed; if not, identify the counterexample.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central formula C_k = A^k C_0 (Theorem 4(ii)) and the least-period-π assertion in Theorem 4(i) depend on Lemma 1, which claims that p1/p2 and 1/p2 have the same least period. The proof of Lemma 1 is written as if deg(p1) < deg(p2): it invokes Theorem 8.40 to write p1/p2 = g/f^* with deg(g) < deg(f^*), then concludes b0 g = p1. This equality is impossible when deg(p1) ≥ deg(p2). Under the theorem's own hypothesis d2 ≤ d1+1, every column k≥1 has numerator p1(t)(t p3)^k of degree d1 + k(d3+1) ≥ d2, so for k≥1 the lemma is needed precisely in the case its proof does not handle. The earlier quotient–remainder reduction to r(t)/p2(t) is stated in §2.2 but not proved to preserve the least period; it is the only route to fill the gap. Since Theorem 8 inherits Theorem 4(i), this is a load-bearing gap. It does not appear to be a falsity—Example 5 only shows failure when d2 > d1+1, outside the theorem—but the proof as written is incomplete.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies eventual periodicity in the columns of 2-D Riordan arrays (p1/p2, t p3) over a finite field F_q, and in the layers of 3-D arrays (p1/p2, t p3, p4). The main results are: Theorem 4, stating that, under d2 ≤ d1+1 and appropriate coprimality, every column k ≥ 1 is ultimately periodic with least period π (the least period of 1/p2), and that the periodic block evolves as C_k = A^k C_0 for a circulant matrix A built from the coefficients of p3; Theorem 8, giving an analogous orbit relation M_4^k between layers of a 3-D array; and Theorem 10, asserting that the preperiodic column partial sums form an eventually periodic sequence, with a special family (Corollary 11) for which they vanish. Examples illustrate the degree-condition restriction and show that A- and Z-sequences need not be periodic.","tokens_in":20550,"tokens_out":46681,"duration_ms":390856,"significance":"If established, the paper would give a clean, explicit description of column periodicity in terms of circulant matrices, extending earlier work on Riordan arrays over finite fields. The concrete finite-field examples and the explicit matrix formulas are valuable and make the intended statements easy to test. The main ideas are promising. However, two load-bearing proof gaps—one in Lemma 1 and one in the generating-function argument for Theorem 10—mean that the central claims are not fully justified as written. The gaps appear fixable, but they are not merely cosmetic.","major_comments":[{"comment":"The proof of Lemma 1 silently treats only the case deg(p1) < deg(p2). The step 'b0 g(t) = p1(t)' after invoking Theorem 8.40 requires deg(g) < deg(f*) = deg(p2), which forces deg(p1) < deg(p2). The quotient-remainder reduction r(t)/p2(t) is mentioned in §2.2, but it is not used in the proof. This is load-bearing: under Theorem 4's hypothesis d2 ≤ d1+1, every column k≥1 has numerator p1(t)(t p3(t))^k of degree d1+k(d3+1) ≥ d2, so Lemma 1 is needed exactly in the case its proof omits. Please add an explicit argument that p1/p2 = q + r/p2 has the same least period as r/p2 and hence as 1/p2. The gap is fixable, and Example 5 does not contradict the statement, but the proof as written is incomplete.","section":"Section 2.2, Lemma 1"},{"comment":"The equality S(x) = [t^D] R1(t) Σ (p3/t^{d3})^k x^k = [t^D] t^{d3} R1(t)/(t^{d3}-p3(t)x) requires a precise choice of expansion. The subsequent argument expands 1/Q in the t-adic f.p.s. topology because Q(0,x) ≠ 0, but this expansion does not in general compute the original S(x). For example, over F_2 take p1=1+t, p2=1+t+t^2, p3=1+t, so d1=1, d2=2, d3=1, D=-1; then R1=(1+t)/(1+t^3) and S[k] = [t^{-1}] R1(t)((1+t)/t)^k gives S(x)=x+x^2+x^4+x^5+... = (x+x^2)/(1-x^3). The t-adic coefficient [t^{-1}] of H(t,x)=t R1(t)/(t-(1+t)x), however, is 0. Thus the proof as written proves rationality of a different quantity. The theorem may be true, but it needs a justified diagonal/residue or constant-term argument that works with the x-adic/Laurent-in-t topology rather than the t-adic expansion of 1/Q.","section":"Section 5, Theorem 10 proof, Eq. (5.6)"},{"comment":"The displayed index in (4.11) is missing a factor d4. As written, the left side is [t^{(kd4+d1+1)-d2+m}] p1 p4^k/p2, but substituting p4^k = p4^{k-1} p4 gives a sum over z_j [t^{(kd4+d1+1)-d2+m-j}], and since kd4 = (k-1)d4 + d4, the index should be ((k-1)d4+d1+1)-d2 + d4 + m - j, not ((k-1)d4+d1+1)-d2 + m - j. The subsequent dot-product argument uses the corrected version, but the displayed equality is false as written and should be fixed.","section":"Section 4, Theorem 8 proof, Eq. (4.11)"}],"minor_comments":[{"comment":"The last component of the vector in (3.4) is written as a sum over c_{d3+1-π-iπ}; the notation should be made unambiguous, e.g. c_{d3-(π-1)-iπ}, with the floor taken over the whole numerator. As typeset, the expression is hard to read.","section":"Section 3.2, Theorem 4, Case 2"},{"comment":"Reference [10] spells the first author's name as 'Lidle'; it should be 'Lidl'.","section":"References"},{"comment":"In the proof, the notation 'n+1 = deg(q(t)) + 1' is clear, but the sentence before it refers to the (n+1)-st term without specifying zero-based indexing. A short clarification would help.","section":"Section 2.2, Lemma 2"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the scope of the journal and the intended results are plausible and well motivated. The main obstacles are the two proof gaps noted: Lemma 1's missing degree case and Theorem 10's invalid t-adic coefficient extraction. Both are fixable with additional arguments, but they are load-bearing for the paper's central claims, so the current version should not be accepted as is."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: Theorems 4, 8, and 10 deliver what the abstract promises—a circulant-matrix description of periodic blocks for (p1/p2, t p3) arrays over F_q and the analogous orbit transfer for 3-D arrays. The generalization from the third author's earlier complex-coefficient work is real: the proofs are constructive, the examples are checked by hand, and Example 5 honestly shows the formula fails outside the degree restriction. That is good behavior.\n\nThe soft spot is Lemma 1. Its proof writes p1/p2 = b0 g / p2 with deg g < deg p2, then concludes b0 g = p1. That equality cannot hold when deg p1 >= deg p2. Under the theorem's own hypothesis d2 ≤ d1+1, every column k≥1 has numerator degree d1 + k(d3+1) ≥ d2, so the period-π claim in Theorem 4(i) depends on exactly the case the proof skips. The earlier quotient–remainder reduction to r/p2 is stated but not proved to preserve the least period. This is a genuine gap, not a false result: the fix is routine—divide p1 by p2, the polynomial quotient only shifts the preperiod, so the tail has the same least period as r/p2, and Lemma 1 goes through. But as written, the proof is incomplete. I'd ask the authors to patch it before publication.\n\nThe other claimed results look sound. Theorem 8 inherits the circulant argument and the commutativity of circulants; Theorem 10's rational-function argument is plausible, and Note 2 is appropriately modest about where the degree restriction is used. The citation pattern is fine: prior work by the third author is cited, and the new contributions are distinguished. No sign of fitting or circular dependence.\n\nWho is this for? People working on Riordan arrays, finite-field recurrences, and circulant matrices will get value. It is not a major breakthrough, but it is a clean, useful generalization with worked examples. I would send it to peer review. The referee should verify the repaired Lemma 1 and the indexing in Theorem 4's case π < d3+1, but I would not desk-reject.","headline":"New and mostly correct transfer-matrix results for Riordan arrays over finite fields, but the proof of Lemma 1 misses the degree case the main theorem needs; still worth refereeing.","tokens_in":21051,"tokens_out":3361,"would_cite":true,"duration_ms":31968,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A15","11B50","11B83","15B33"],"pacs":[],"model":"deepseek-v4-flash","headline":"Over a finite field, every column of a polynomial Riordan array is eventually periodic, and the periodic blocks are powers of one circulant matrix applied to the zeroth column.","keywords":["Riordan arrays","finite fields","periodicity","linear recurring sequences","circulant matrices","partial sums","formal power series","3-D Riordan arrays"],"falsifier":"Compute column 1 of the Riordan array ((1+t)/(1+t+t^2), t(1+t)) over F_2 directly: its periodic block, extracted from the coefficient table, must equal circ(1,1,0) times the zeroth-column block. If the product differs from the directly extracted block, Theorem 4(ii) fails; if it matches, the identity survives the simplest possible check.","tokens_in":20131,"feed_emoji":"🔄","tokens_out":7649,"duration_ms":69134,"temperature":0.7,"pith_summary":"This paper proves a structural description of the long-run behavior of Riordan arrays over finite fields. For an array built from a rational generating function p1/p2 and a polynomial factor t p3, every column is eventually periodic, and all columns share the same least period — the period of the coefficients of 1/p2. The repeating block in the k-th column is exactly A^k times the block in column 0, where A is a circulant matrix assembled from the coefficients of p3. For the 3-D analogue, the orbit of periodic blocks in layer k is M^k times the orbit in layer 0, with M built from p4. The same tools show that the preperiodic column partial sums are themselves periodic, and a specific family has all such sums equal to zero.","feed_headline":"One circulant matrix governs every column's repeating block","feed_subtitle":"In a Riordan array over a finite field, column k's periodic block is A^k times column 0's.","key_machinery":"The driving object is the circulant matrix generated by the coefficient vector of the polynomial in the second argument: A = circ(c_{d3},...,c_0) padded to size π, with coefficients folded modulo π when π < d3+1. Because multiplication by t p3 shifts and convolves coefficients, the periodic block of the next column is the product of A with the current block; induction on k gives C_k = A^k C_0. The period π itself comes from the standard fact that coefficients of a rational function over a finite field form a linear recurring sequence whose least period is the order of its minimal polynomial, which the paper shows is the reciprocal of p2 up to a scalar.","core_discovery":"On its own terms, the central claim is Theorem 4: under the hypotheses p_i(0)≠0, p2 coprime to p1 and p3, and d2 ≤ d1+1, the k-th column (k≥1) of the Riordan array (p1/p2, t p3) over F_q is ultimately periodic with least period π, the least period of the coefficient sequence of 1/p2; its periodic block is C_k = A^k C_0, with A a π×π circulant matrix generated by the coefficients of p3. Theorem 8 extends the picture to the 3-D array (p1/p2, t p3, p4): the orbit of periodic blocks in the k-th layer is M_4^k applied to the orbit in layer 0, M_4 a circulant built from p4. Theorem 10 adds that the sequence of preperiodic column partial sums is eventually periodic, and Corollary 11 exhibits a fami","pith_inferences":["If the degree condition d2 ≤ d1+1 is dropped, Example 5 suggests the block law becomes C_k = P^{φ(k)} A^k C_0 with P a cyclic shift and φ(k) piecewise linear; one could test whether such a modified law holds generally outside the theorem's range.","The claim that every high-index column still has period π leans on an implicit quotient-remainder step in Lemma 1 for numerators of degree exceeding d2; making that step explicit, or finding a counterexample, would settle whether the period-π assertion survives for all k.","Because A and M_4 are circulant, the block-evolution law is essentially a linear recurrence on blocks; one might extend the same circulant-block analysis to other triangular arrays whose generating functions are rational, not just polynomial Riordan pairs.","The vanishing partial-sums family suggests a mod-p combinatorial identity: for even p3, the first linear-length window of each column sums to zero; a combinatorial proof could illuminate why the preperiodic sums cancel."],"forward_implications":["Each column k≥0 of the 2-D array (p1/p2, t p3) is ultimately periodic with the same least period π, independent of the column index, whenever d2 ≤ d1+1 and p2 is coprime to p1 and p3.","The periodic block in column k is C_k = A^k C_0, so the entire collection of column blocks is the orbit of C_0 under A; since A has finite order, only finitely many distinct periodic blocks occur.","In the 3-D array, the periodic-block orbit of layer k is M_4^k times the orbit of layer 0, so layer-to-layer evolution is governed entirely by the coefficients of p4.","The preperiodic column partial sums S[k] form an eventually periodic sequence, and for the family with p1=1, p2=1+t, and even p3 whose coefficient sum is nonzero mod p, all S[k] vanish.","Consequently, for any concrete array of this type, the full periodic structure of all columns can be written down from C_0 and A alone, without computing column by column."],"fun_headline_variants":["A circulant matrix A makes columns repeat as A^k times column zero","Riordan arrays: column k's periodic block is A^k C_0","One circulant matrix governs every column's repeating block","Column periodicity in finite-field Riordan arrays: A^k does it","The power of A: how Riordan columns get their periodic blocks"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole description rests on the degree inequality d2 ≤ d1+1 and the coprime hypotheses: without them the clean identity C_k = A^k C_0 can fail (Example 5), and the proof that every column has period π depends on an implicit quotient-remainder reduction as numerator degree grows.","fun_headline_variants_meta":{"raw":{"variants":["A circulant matrix A makes columns repeat as A^k times column zero","Riordan arrays: column k's periodic block is A^k C_0","One circulant matrix governs every column's repeating block","Column periodicity in finite-field Riordan arrays: A^k does it","The power of A: how Riordan columns get their periodic blocks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000487,"raw_usage":{"total_tokens":2263,"prompt_tokens":795,"completion_tokens":1468,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":1376}},"tokens_in":539,"tokens_out":1468,"duration_ms":10893,"temperature":1.0,"reasoning_tokens":1376,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T05:11:12.283699+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute column 1 of the Riordan array ((1+t)/(1+t+t^2), t(1+t)) over F_2 directly: its periodic block, extracted from the coefficient table, must equal circ(1,1,0) times the zeroth-column block. If the product differs from the directly extracted block, Theorem 4(ii) fails; if it matches, the identity survives the simplest possible check.","supporting_citations":[],"review_version":1}