{"id":"8d243a64-b431-4c0e-8499-e671964eb767","arxiv_id":"2509.02893","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The author defines ℓ-level almost-Riordan arrays, states without proof that they form a group, and gives sequence and compression characterizations that reduce to previously published equations.","lead":"This paper introduces multiple almost-Riordan arrays and claims they form a group, with explicit generating functions for their defining sequences. The proofs are omitted, and the bulk of the results are direct generalizations of the author's earlier work on double almost-Riordan arrays.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central group law (37) is stated with an ambiguous h: if h is the compositional inverse as the 'where' clause says, the ℓ=1 case contradicts the known almost-Riordan product (Theorem 1.4).","rationale":"The reader's verdict was REJECT mainly because the central theorems are asserted without proof. My stress-test goes further and identifies a concrete defect in the central group law as written: the symbol h is used in (37) and then declared to be the compositional inverse in the immediately following 'where' clause, but under that reading the ℓ=1 specialization contradicts the known almost-Riordan product (Theorem 1.4) and even the paper's own earlier FFT, which uses the ℓ-th root. This is not merely a missing derivation; it makes the central claim internally inconsistent. The mismatch is easy to demonstrate with a simple ℓ=1 example, so it is a load-bearing concern rather than a stylistic quibble. The paper's examples suggest the intended formulas may be salvageable by consistently writing h for the root in (37), but as submitted the multiplication rule is ambiguous-or-wrong and the group conclusion does not follow. This strengthens the reader's REJECT, so the verdict is unchanged.","tokens_in":18654,"tokens_out":20229,"duration_ms":220858,"concrete_test":"Set ℓ=1 in (37) with A=(1|1; t+t^2) and B=(1/(1-t)|1; t). Compute the first column of AB two ways: (i) directly from the column generating functions of Definition 2.1, and (ii) from (37) using h=(t+t^2)^{-1}, as the 'where' clause dictates. If the t^2 coefficients differ (direct gives 1, (37) gives -1), the multiplication rule is false under the stated reading. Then repeat with h=t+t^2 (the root) and check agreement with Theorem 1.4 to all orders; finally, test associativity of (37) for a nontrivial ℓ=2 triple product to see whether the corrected formula actually forms a group.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.3 displays the multiplication rule (37) and inverse (38), then states: 'where ... h is the compositional inverse of h = ℓ√f1f2···fℓ.' Reading that clause as applying to (37), the formula is false already for ℓ=1. Definition 2.1 with ℓ=1 reduces to the almost-Riordan arrays of Theorem 1.4, whose product is (a|g,f)(b|d,h) = (a + tg/f (b(f)-1) | gd(f), h(f)) (eq. 21). But (37) with h=f^{-1} gives first component a + tg/f (b(f^{-1})-1) and second multiplier (f/f^{-1})·h(f^{-1}). Concrete mismatch: take a=b=1, g=d=1, f=t+t^2, h=t. Direct column-wise multiplication gives first component 1 + t/(1-t-t^2) = 1+t+t^2+..., while (37) with inverse h gives 1+t-t^2+...; the t^2 coefficients differ. If instead h in (37) is meant to be the ℓ-th root (as in Theorem 2.2's FFT (34)-(36)), then the notational collision with the 'where' clause makes the central theorem internally inconsistent. Either way, the asserted group law is not a reliable foundation for the sequence characterizations (40)-(43), which all depend on it. Additionally, the 'first fundamental theorem' (33)-(36) is asserted without proof, so the chain from definition to group claim to sequences is not verifiable as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a generalization of almost-Riordan arrays to ℓ multipliers, defining multiple almost-Riordan arrays (b|g; f1,…,fℓ) and claiming that they form the multiple almost-Riordan group under the multiplication rule (37). It further states sequence characterizations for these arrays (Theorems 3.1–3.2), studies subgroups of the group (Section 4), and defines compressions with a corresponding sequence characterization (Theorem 5.3). The main evidence supplied is a series of examples for ℓ=2,3; the central theorems are stated without proofs. The manuscript also relies on the author's earlier work [24] and on an unpublished manuscript [8] for the foundational ℓ=1 group law.","tokens_in":19101,"tokens_out":9489,"duration_ms":101199,"significance":"If the results are correct, the paper would give a unified treatment of almost-Riordan arrays with multiple column-multipliers and would extend the sequence-characterization and production-matrix machinery to this setting. The compression analysis could be useful for studying combinatorial arrays with periodic column structure. However, the current version does not make the central results verifiable: the group law is stated with an ambiguous composition rule, and nearly all main theorems are asserted without proof. The compression characterization is explicitly admitted in Remark 5.4 to be equivalent to the earlier sequence characterization by substitution, which reduces the novelty of Section 5.","major_comments":[{"comment":"The closing clause of Theorem 2.3 says that h is the compositional inverse of h=ℓ√(f1⋯fℓ). If this is applied to Eq. (37), the product formula is false already for ℓ=1. For example, take b=c=1+t, g=d=1, f=t+t², h1=t. The known almost-Riordan product (21) gives first column 1+t, while Eq. (37) with h = (t+t²)⁻¹ = t−t²+⋯ gives 1 + t/(t+t²)(1+h−1) = 1+t−2t²+⋯. If h in (37) is instead meant to be the ℓ-th root h=ℓ√(f1⋯fℓ), then the theorem's 'where' clause is wrong and must be corrected. Since Eq. (37) is the foundation for all subsequent sequence characterizations, this ambiguity is load-bearing.","section":"§2, Theorem 2.3, Eq. (37)"},{"comment":"All central theorems are stated without proofs. The first fundamental theorem (33)–(36) is asserted, but no derivation is given; the multiplication rule (37) and inverse (38) are asserted; the sequence characterizations (40)–(43), the production-matrix formula (44), and the compression formulas (52)–(56) are all stated without proof. The examples verify only special cases. Moreover, Theorem 2.2 is garbled: in the 'where' clause the second case is written as u(t)=Σ u_{2k+1}t^{2k+1}, which is the ℓ=2 notation, not the general ℓ case. The chain from definition to group claim to sequence characterization is therefore not verifiable as written.","section":"§2–§5, Theorems 2.2, 2.3, 3.1, 3.2, 5.1, 5.3"},{"comment":"Remark 5.4 explicitly states that (52)–(56) are equivalent to (40)–(43) by substitution, and then performs the substitution to recover (40)–(43). This means Theorem 5.3 does not provide an independent sequence characterization of the compression; it is a notational restatement of Theorem 3.1 under the change t = h^ℓ. If the compression result is intended as a new contribution, the paper needs to clarify what genuinely new information is proved, rather than presenting a substitution as a separate theorem.","section":"§5, Remark 5.4"},{"comment":"The group law for the ℓ=1 case (Theorem 1.4) is cited to the unpublished manuscript [8], and the multiple Riordan group law (11) and the sequence characterization Theorem 1.2 are cited to [24] without proof. Since the present paper's multiplication rule (37) is a direct generalization of these results, the reader cannot check the foundational step. A journal submission should either include the proofs or clearly state the precise results from these sources that are being assumed, ideally with published references for the ℓ=1 case.","section":"§1–§2, dependence on [8] and [24]"}],"minor_comments":[{"comment":"The introduction says 'This paper presents the study of the double almost-Riordan arrays and the double almost-Riordan group' and later mentions 'total positivity' as part of the scope, but the paper actually treats multiple almost-Riordan arrays and does not discuss total positivity. Please reconcile the introduction with the actual content.","section":"Introduction"},{"comment":"There are numerous typos: 'There two cases' should be 'There are two cases'; 'W ords' in the key words; 'Appel' should be 'Appell'; 'DaR' should be 'MaR'; 'M ≤ MaR' is not a meaningful subgroup statement as written; the expression for u(t) in Theorem 2.2 uses ℓ=2-specific indices.","section":"Throughout"},{"comment":"The references [8] and [23] are unpublished ('manuscript in preparation' and 'submission'), while [24] has the same title as [23]. The dependence of central results on unpublished material should be minimized or clearly flagged.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a coherent and potentially useful topic, but the lack of proofs for all central theorems and the ambiguous/incorrect h in Theorem 2.3 make the current version unsuitable for publication. I would be willing to reconsider if the author provides complete proofs, corrects the product law notation, and reframes Section 5 so that the compression result is not merely a substitution of variables."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a definition-and-statement paper. He defines multiple almost-Riordan arrays for arbitrary ℓ, writes down group multiplication, inverse, sequence characterizations, production matrices, and a compression theorem. The examples are worked and look consistent with the formulas. But every central theorem is asserted without proof, and the multiplication rule in Theorem 2.3 is ambiguous in a way that matters: the h that appears in (37) is the ℓ-th root if the formula is to reduce to the known ℓ=1 product (21), yet the 'where' clause defines h as the compositional inverse of that root. For ℓ=1, using the inverse gives c(f^{-1}) where Theorem 1.4 needs c(f). The stress-test's concrete mismatch with a=b=1 doesn't actually demonstrate this because with both functions constant the two expressions collapse to the same thing; a nonconstant c makes the point. So the central group law as written is not reliable.\n\nWhat's new and useful: the definition of multiple almost-Riordan arrays for arbitrary ℓ is the natural generalization, and the sequence characterization (40)-(43) is a plausible extrapolation of the ℓ=2 case. The compression section is honestly labeled: Remark 5.4 shows (52)-(56) are just (40)-(43) under substitution, so the compression characterization is not independent. That is at least transparent.\n\nSoft spots: no proofs for Theorems 2.2, 2.3, 3.1, 3.2, 5.1, or 5.3. The paper leans on the unpublished [8] for the ℓ=1 group law, which is the base of the whole construction. The introduction describes the scope as 'double almost-Riordan arrays' even though the paper is about the general ℓ case, and there are numerous typos. In this state, the formulas are claims, not results.\n\nWho should read it: people working on Riordan arrays who want to see the multiple ℓ version and the compression idea. They should treat it as a research announcement, not a citable proof.\n\nRecommendation: desk reject with an invitation to resubmit once proofs are supplied and the h ambiguity in Theorem 2.3 is fixed. I would not send this to a referee in the current form.","headline":"Defines multiple almost-Riordan arrays for arbitrary ℓ, but the central theorems are unproved and the printed group law is inconsistent with the ℓ=1 case it must generalize.","tokens_in":19527,"tokens_out":9763,"would_cite":false,"duration_ms":99940,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A15","05A05","11B39","11B73","15B36","15A06","05A19","11B83"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that multiple almost-Riordan arrays—arrays whose columns alternate among ℓ multiplier functions—form a group, and that one A-sequence, ℓ Z-sequences, and one W-sequence characterize every entry, even after compression.","keywords":["multiple almost-Riordan arrays","multiple almost-Riordan group","sequence characterization","production matrix","compression","Riordan arrays","generating functions","Stieltjes matrix"],"falsifier":"Take two explicit double almost-Riordan arrays, multiply their first several rows as ordinary matrices, and compare the result with the product predicted by (37); also multiply (1/(1-t^4)|1/(1-t^2); t, t/(1-t^2)) by its inverse from (38) and check whether the truncated product equals the identity (1|1; t, t). Any mismatch would falsify the group law.","tokens_in":18564,"feed_emoji":"📐","tokens_out":6952,"duration_ms":75402,"temperature":0.7,"pith_summary":"This paper extends the almost-Riordan construction from one multiplier to ℓ alternating multipliers. Its central claim is that every array built from an initial column b(t) and a multiple Riordan array (g; f1,...,fℓ) belongs to a group, the multiple almost-Riordan group, with an explicit multiplication formula. The paper also computes the defining sequences—one A, ℓ Zj's, and one W—in closed generating-function form, records them in a production matrix, and shows that the compression of such an array is again a multiple almost-Riordan array governed by the same kind of sequence formulas. If correct, this gives a uniform symbolic calculus for a family of lower-triangular integer arrays, including arrays associated with the Fibonacci-Stanley tree and the triple Riordan group.","feed_headline":"Almost-Riordan arrays form a group at every multiplicity","feed_subtitle":"One A, several Z, and a W sequence generate every entry; compressions keep the same structure.","key_machinery":"The central object is the multiple almost-Riordan array (b|g; f1,...,fℓ): column 0 is b(t), and for k ≥ 1 column k has generating function t·g(t)·f1^{⌊(k+ℓ-2)/ℓ⌋}⋯fℓ^{⌊(k-1)/ℓ⌋}. The carrying mechanism is the first fundamental theorem (33)–(36), which converts a series u into v by substitution through the compositional inverse h of the ℓ-th root of f1⋯fℓ; this substitution makes the multiplication (37) work and turns the A-, Zj-, and W-sequences into the closed forms (40)–(43).","core_discovery":"For a fixed ℓ, a multiple almost-Riordan array is the lower-triangular matrix whose column 0 has generating function b(t) and whose remaining columns are tg times cyclic products of ℓ multiplier series f1,...,fℓ. The paper asserts that these arrays close under the multiplication rule (37), with identity (1|1;t,...,t) and inverse (38), forming the multiple almost-Riordan group MaR. The central sequence characterization is Theorem 3.1: every entry is generated by one A-sequence, ℓ Zj-sequences, and one W-sequence, whose generating functions (40)–(43) are explicit combinations of b, g, f1,...,fℓ and the compositional inverse h of h = ℓ√(f1⋯fℓ). The production matrix (44) is assembled from these","pith_inferences":["My inference: because the compression of a multiple almost-Riordan array remains in the same class, iterating the compression gives a nested family of arrays whose sequence data should telescope, allowing large-index entries to be computed from the original h-series without constructing the full array.","My inference: the two sequence-characterization views mentioned in the paper—one W plus ℓ Z's plus one A versus one W plus one Z plus ℓ A's—may be equivalent through a cyclic transform over the ℓ indices; proving that equivalence could simplify the theory and unify the ℓ=1 and ℓ=2 cases.","My inference: specializing the formulas to ℓ=1 should recover the original almost-Riordan group and its sequence characterization; carrying that limit through (37) and (40)–(43) would be a quick consistency check of the general framework."],"forward_implications":["Any product or inverse of multiple almost-Riordan arrays is again a multiple almost-Riordan array, so the group law gives a way to build new arrays with predictable column structure.","Every entry of such an array is determined by linear recurrences with the A-, Zj-, and W-sequences; the generating functions (40)–(43) make those recurrences explicit for any chosen b, g, and fj.","The production matrix P=(W(t), tZ1(t), ..., t^{ℓ-1}Z_{ℓ-1}(t), Zℓ(t), tA(t), t²A(t), ...) encodes the full array and satisfies the standard production-matrix relation, so row growth can be read directly from P.","Compression maps a multiple almost-Riordan array to a smaller array of the same type, with sequence formulas obtained from the original formulas by substituting t = h^ℓ.","Known examples, including the Fibonacci-Stanley tree array and the ℓ=3 example, are recovered as special cases with the computed A-, Z-, and W-sequences."],"supporting_citations":[{"why":"Defines the Riordan group and the column-generating-function formalism that the multiple almost-Riordan array generalizes.","marker":"[39]"},{"why":"Supplies the ℓ=1 almost-Riordan group law that the new multiple almost-Riordan multiplication extends.","marker":"[8]"},{"why":"Gives the sequence characterization of multiple Riordan arrays that Theorem 3.1 adapts to the almost-Riordan setting.","marker":"[24]"},{"why":"Provides the multiple Riordan group and multiple Riordan type arrays used in the definitions and multiplication rule.","marker":"[25]"},{"why":"Defines the double almost-Riordan group, the ℓ=2 special case whose structure this paper generalizes.","marker":"[23]"},{"why":"Gives sequence characterizations of double Riordan arrays and their compressions, the direct antecedent for Section 5.","marker":"[21]"},{"why":"Introduces the A- and Z-sequence treatment for double Riordan arrays that motivates the decomposition in (39).","marker":"[9]"},{"why":"Further develops A- and Z-sequence methods for double Riordan arrays, informing the production-matrix view.","marker":"[15]"},{"why":"Provides the sequence characterization of ordinary almost-Riordan arrays, the ℓ=1 case extended here.","marker":"[3]"}],"fun_headline_variants":["Three sequences generate every multiple almost-Riordan array","Compressed or not, almost-Riordan arrays share one sequence trio","New group: multiple almost-Riordan arrays, built from just A, Z, W","An infinite family of matrix groups from three sequence rules"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that the column-to-series conversion rule (33)–(36) and the multiplication formula (37) are correct as stated; the paper asserts them without proof, and the group and sequence claims collapse if either is wrong.","fun_headline_variants_meta":{"raw":{"variants":["Three sequences generate every multiple almost-Riordan array","Compressed or not, almost-Riordan arrays share one sequence trio","New group: multiple almost-Riordan arrays, built from just A, Z, W","An infinite family of matrix groups from three sequence rules"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000326,"raw_usage":{"total_tokens":1609,"prompt_tokens":640,"completion_tokens":969,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":384,"completion_tokens_details":{"reasoning_tokens":894}},"tokens_in":384,"tokens_out":969,"duration_ms":9885,"temperature":1.0,"reasoning_tokens":894,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T11:16:30.639496+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take two explicit double almost-Riordan arrays, multiply their first several rows as ordinary matrices, and compare the result with the product predicted by (37); also multiply (1/(1-t^4)|1/(1-t^2); t, t/(1-t^2)) by its inverse from (38) and check whether the truncated product equals the identity (1|1; t, t). Any mismatch would falsify the group law.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Riordan group and the column-generating-function formalism that the multiple almost-Riordan array generalizes."},{"cited_title":"Barry, T.-X","cited_arxiv_id":null,"evidence_quote":"Supplies the ℓ=1 almost-Riordan group law that the new multiple almost-Riordan multiplication extends."},{"cited_title":"He, The double almost-Riordan group, Linear Algebra Appl","cited_arxiv_id":null,"evidence_quote":"Gives the sequence characterization of multiple Riordan arrays that Theorem 3.1 adapts to the almost-Riordan setting."},{"cited_title":"The Multiple Riordan Group and the Multiple Riordan Type Arrays","cited_arxiv_id":"2504.04049","evidence_quote":"Provides the multiple Riordan group and multiple Riordan type arrays used in the definitions and multiplication rule."},{"cited_title":"He, The double almost-Riordan group, submission, 2024","cited_arxiv_id":null,"evidence_quote":"Defines the double almost-Riordan group, the ℓ=2 special case whose structure this paper generalizes."},{"cited_title":"He, Sequence characterizations of double Riordan arrays and their compressions, Linear Algebra Appl","cited_arxiv_id":null,"evidence_quote":"Gives sequence characterizations of double Riordan arrays and their compressions, the direct antecedent for Section 5."},{"cited_title":"Branch, D","cited_arxiv_id":null,"evidence_quote":"Introduces the A- and Z-sequence treatment for double Riordan arrays that motivates the decomposition in (39)."},{"cited_title":"Davenport, S.K","cited_arxiv_id":null,"evidence_quote":"Further develops A- and Z-sequence methods for double Riordan arrays, informing the production-matrix view."},{"cited_title":"Alp and E","cited_arxiv_id":null,"evidence_quote":"Provides the sequence characterization of ordinary almost-Riordan arrays, the ℓ=1 case extended here."}],"review_version":1}