{"id":"4c2864f0-e36e-4821-8c3f-3988c7a45d3c","arxiv_id":"2608.01497","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new mapping from pairs of power series to Sprugnoli arrays yields square roots of aerated Appell arrays and new involutions in the Sprugnoli group.","lead":"This paper constructs a new map that turns pairs of power series into Sprugnoli arrays, and shows that squaring these arrays often produces 'aerated Appell' matrices. This gives explicit square roots for a family of Riordan matrices and a method to build matrices whose square is the identity.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Prop 13's printed square-root formula is missing a factor (1-x^2); as stated it is contradicted by the proof and by g=1,f=0.","rationale":"The intended construction is mathematically sound: computing the proof's first identity with p=g/(1-x), q=g(-x)/(1+x), s=x f(x) gives (G,F1,x)·G = (g(x)g(-x)(1-x^2)+x^2 f^2)/(1-x^2)^2, and the second identity also checks out. So the square-root formula works after restoring the missing factor. The load-bearing problem is a statement-level typo in Proposition 13: the factor (1-x^2) is present in the proof's simplification, in Corollary 14's usage, in Examples 15-18, and in the involution equations, but absent from the proposition's displayed formula. The counterexample g=1,f=0 makes the discrepancy concrete. This is not a deep mathematical flaw, but the theorem as printed is false, so conditional acceptance with correction is appropriate. The reader's concern about the omitted algebra is real but secondary: the algebra is verifiable and correct once the factor is restored, so it does not by itself invalidate the claim.","tokens_in":13174,"tokens_out":14954,"duration_ms":153606,"concrete_test":"Set g=1, f=0 in the definitions and compute M(1,0)^2 two ways: directly from the column generating functions of the Sprugnoli array (1/(1-x), -x, x), and from Proposition 13 as printed. The printed RHS is (1/(1-x^2)^2, x), while direct calculation gives (1/(1-x^2), x), isolating the missing factor. Then symbolically expand the two 'Simplifying this' identities in the proof to confirm they match the corrected numerator.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 13 as printed states M(g,f)^2 = (g(x)g(-x)+x^2 f(x)^2)/(1-x^2)^2, but the proof's own simplification gives (G,F1,x)·G = (g(x)g(-x)(1-x^2)+x^2 f(x)^2)/(1-x^2)^2, and every worked example (Examples 15-18, the involution equations in Section 7) uses the factor (1-x^2). The printed statement is false: for g=1, f=0, M(1,0) is the Sprugnoli array (1/(1-x), -x, x), and direct matrix multiplication gives M(1,0)^2 = (1/(1-x^2), x), not (1/(1-x^2)^2, x). Thus the central constructive formula in the main theorem needs the factor restored. The two 'Simplifying this, we find' jumps are a secondary exposition gap; independently expanding them confirms the corrected identities, so the intended theorem is sound but the displayed formula is not.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a mapping from pairs (g,f) of power series to the bivariate generating function B(x,y)=g(x)/((1-x)(1+yx))+xf(x)/((1-x^2)(1-yx)), whose coefficient matrix M(g,f) is shown to be a Sprugnoli array. The central result (Proposition 13) is that when f is an even power series, M(g,f)^2 is an aerated Appell array; this gives a constructive procedure for square roots of such arrays and, when the square equals the identity, yields involutions in the Sprugnoli group. The paper includes many worked examples and a Riordan-group interpretation of the construction.","tokens_in":13406,"tokens_out":6179,"duration_ms":58896,"significance":"If the corrected formula is used, the paper gives a clean, explicit construction of square roots for a class of aerated Appell arrays, with the square roots landing in the less familiar Sprugnoli group. This is a useful contribution to the Riordan-group toolkit. The main claims are concrete and falsifiable, and the worked examples provide genuine verification. The paper also correctly identifies a source of involutions in the Sprugnoli group. Its main limitation is that the Sprugnoli group machinery is imported from the author's previous preprint [1], and the proof of the key proposition contains algebraic simplifications that are only asserted.","major_comments":[{"comment":"The displayed square formula is missing a factor (1-x^2) in the numerator. As printed it reads M(g,f)^2 = (g(x)g(-x)+x^2 f(x)^2)/(1-x^2)^2, but the proof's own simplification gives (g(x)g(-x)(1-x^2)+x^2 f(x)^2)/(1-x^2)^2. The printed statement is false: for g=1, f=0, M(1,0)=(1/(1-x), -x) and its square is (1/(1-x^2), x), not (1/(1-x^2)^2, x). Every later use (Corollary 14, Examples 15–18, Section 7) uses the corrected numerator. Please restore the factor throughout the displayed Proposition.","section":"§6, Proposition 13"},{"comment":"The proof of Proposition 13 relies on two nontrivial algebraic identities, introduced by 'Simplifying this, we find' and 'With these expressions, we find': the formula for (G,F1,x)·G and the identity (G,F1,x)·GF1 = x(G,F1,x)·G. These are load-bearing for the central claim. Please include the intermediate expansions or a concise derivation for both, so the proof can be checked without redoing the computation. An independent expansion shows the identities are correct, so this is a presentation/verification issue rather than a fundamental error.","section":"§6, proof of Proposition 13"}],"minor_comments":[{"comment":"The matrix displays in this example are difficult to read because of missing spacing/line breaks, especially the right-hand matrix. Please typeset them more carefully.","section":"§2, Example 2"},{"comment":"The product and inverse of Sprugnoli arrays are quoted from [1] without proof. Since [1] is an arXiv preprint, a brief statement of the group law or a citation to a published version would improve self-containedness.","section":"§4"},{"comment":"The notation G e(x^2), G o(x^2) is ambiguous. It should be written as (G_e)(x^2) and (G_o)(x^2), with a short clarification that these are the even/odd bisections of G.","section":"§6"},{"comment":"The sentence 'if the element (g, xf)∈R is such that xf is odd' is slightly confusing: since x is odd, xf is odd exactly when f is even. Please phrase it to match the earlier hypothesis directly.","section":"§8"},{"comment":"Reference [1] is to an arXiv preprint. If a journal version or a more stable published source exists, it would be preferable to cite that.","section":"References"}],"recommendation":"minor_revision","confidential_remarks":"The central construction is sound once the missing factor in Proposition 13 is restored. The two algebraic simplifications in the proof should be expanded, but they check out. I would recommend minor revision rather than major revision: the issues are local and the intended theorem is correct. The paper depends heavily on the author's own preprint [1] for the Sprugnoli group framework; the editor may wish to confirm that [1] is available or accepted, and that this submission is sufficiently distinct from it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: this is a narrow but real result in Riordan-array combinatorics. The author defines a map M(g,f) from pairs of power series to Sprugnoli arrays, shows that when f is even the square lands in the aerated Appell subgroup, and uses this to construct explicit square roots and involutions. The novelty is modest—it extends his own Sprugnoli-group machinery from [1]—but the construction is genuinely new and the worked examples match the formulas.\n\nThe main problem is that Proposition 13, as printed, is false. It states M(g,f)^2 = (g(x)g(-x) + x^2 f(x)^2)/(1-x^2)^2. But the proof's own simplification gives (g(x)g(-x)(1-x^2) + x^2 f(x)^2)/(1-x^2)^2, and every example, including the involution equations in Section 7, uses that corrected version. For the simplest test g=1, f=0, the printed formula would give (1/(1-x^2)^2, x), while M(1,0)^2 is (1/(1-x^2), x). So the display in Prop 13 needs the factor (1-x^2) restored in the numerator. That is a load-bearing typo, but it is easily fixed.\n\nThe other soft spot the reader flagged—the two 'Simplifying this, we find' jumps in the proof—is secondary. I expanded them and they do check out; the algebra is routine, just omitted. The author should show the steps or at least note they are mechanical. The reliance on [1] for the Sprugnoli group product is fine for a sequel; readers will need that paper in hand, but that is not a flaw.\n\nSo: the central argument survives, but only after correcting the displayed formula. This is exactly the kind of thing a referee should catch. The paper is for specialists in Riordan arrays and generating-function matrix groups; it will not matter beyond that subfield. With the correction, it is a solid, citable structural note.\n\nRecommendation: send it to peer review. The intended theorem appears sound, the examples are consistent with the corrected formula, and the fix is a one-line change plus a request to expand the skipped algebra. A serious referee would be doing the author a favor.","headline":"The intended square-root theorem is sound and the examples check out, but Proposition 13 as printed is missing a factor (1-x^2) in the numerator, so the paper needs a one-line correction before the central formula can be trusted as stated.","tokens_in":13854,"tokens_out":1599,"would_cite":false,"duration_ms":18119,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15B36","05A15","11B83","11C20","15A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"A map from power-series pairs to Sprugnoli arrays yields square roots of aerated Appell arrays and new involutions in the Sprugnoli group.","keywords":["Riordan array","Appell subgroup","Sprugnoli group","generating function","square root","involution","aerated array"],"falsifier":"Symbolically expand the two identities (G,F1,x)·G = (g(x)g(-x)(1-$x^{2}$)+$x^{2}$ f(x)^2)/(1-$x^{2}$)^2 and (G,F1,x)·GF1 = x(G,F1,x)·G for a generic even f(x) and generic g(x); any counterexample from the expansions would invalidate Proposition 13. Alternatively, compute a finite truncation of M(g,f)^2 for a concrete (g,f) with even f by direct matrix multiplication and compare to the Appell formula; a mismatch at any entry settles it.","tokens_in":13058,"feed_emoji":"🧮","tokens_out":6370,"duration_ms":47188,"temperature":0.7,"pith_summary":"The paper introduces a mapping from pairs of power series (g,f) to Sprugnoli arrays, a family of infinite lower-triangular matrices, and proves that when f is an even power series, the square of the mapped matrix is an aerated Appell array. This gives a constructive method for finding square roots of certain aerated Appell arrays in the Riordan group: any such array expressible in the closed form (g(x)g(-x)+$x^{2}$ f(x)^2)/(1-$x^{2}$)^2 has a Sprugnoli array as a square root. Because the identity matrix is itself an aerated Appell array, the same mechanism produces a family of involutions inside the Sprugnoli group. The result builds a bridge between the Sprugnoli and Riordan groups, providing a new toolbox for root problems in matrix groups.","feed_headline":"Even power series yield Sprugnoli square roots of Appell arrays","feed_subtitle":"Power-series pairs map to matrices whose squares are aerated Appell arrays, yielding square roots and involutions.","key_machinery":"The load-bearing object is the mapping M(g,f), defined by the bivariate generating function B(x,y) = g(x)/((1-x)(1+yx)) + x f(x)/((1-$x^{2}$)(1-yx)). It sends pairs of power series to Sprugnoli arrays, and the key identity for its square reduces the squared generating function to a single Appell form. The Sprugnoli group product and the fundamental theorem for Sprugnoli arrays are the tools that carry the calculation.","core_discovery":"The central discovery is Proposition 13: when f(x) is an even power series, the square of the Sprugnoli array M(g,f) equals ((g(x)g(-x)+$x^{2}$ f(x)^2))/(1-$x^{2}$)^2, which is an aerated element of the Appell subgroup of the Riordan group. Consequently, every aerated Appell array that can be written in this form has a square root that is a Sprugnoli array M(g,f). If the right-hand side is the constant 1, then M(g,f) is an involution in the Sprugnoli group. The proof uses the Sprugnoli group product rule and the fundamental theorem of Sprugnoli arrays to compute the square componentwise.","pith_inferences":["The method likely extends beyond aerated Appell arrays: other Riordan arrays might have Sprugnoli square roots, a possibility the paper leaves open.","The algebraic identities asserted 'by simplification' in Proposition 13 could be verified by direct symbolic expansion; if verified, the construction is fully explicit and parameterizes a family of involutions.","The map (g, xf) -> M(g,f) may have additional structural properties, such as compatibility with products or preservation of combinatorial statistics, that could deepen the connection between the Riordan and Sprugnoli groups.","A computational test on other simple aerated Appell arrays (e.g., 1/(1+x^2) or 1/(1-x^2-x^4)) would reveal which admit rational or polynomial choices for g and f, suggesting a classification of representable square roots."],"forward_implications":["Every aerated Appell array expressible as (g(x)g(-x)+x^2 f(x)^2)/(1-x^2)^2 with f even has an explicit Sprugnoli square root M(g,f).","If that expression equals 1, M(g,f) is an involution in the Sprugnoli group, generating new examples such as ((1+rx)/(1-x), -x(1+x+(r-1)x^2)/(1+rx), x)^2 = I.","The mapping can be viewed as a map from the Riordan group to the Sprugnoli group via (g, xf) -> ((g/(1-x)+xf/(1-x^2), ...), and when xf is odd the image squares into the aerated Appell subgroup.","This gives an explicit tool for constructing square roots of aerated Appell arrays, complementing earlier constructions of square roots for Bell matrices."],"supporting_citations":[{"why":"Supplies the Sprugnoli group product and inverse rules used to square M(g,f).","marker":"[1]"},{"why":"Provides the Riordan array background and the fundamental theorem used in the proof.","marker":"[2]"},{"why":"Sets up the problem of square roots of Bell matrices that this paper's square-root construction mirrors.","marker":"[3]"},{"why":"Background reference for the Riordan group and its Appell subgroup.","marker":"[4]"},{"why":"Source for the Riordan group product and the Appell subgroup structure.","marker":"[5]"}],"fun_headline_variants":["Even power series unlock square roots in Appell group","Aerated Appell arrays get Sprugnoli square roots","Even power series produce Sprugnoli square roots","Square roots and involutions from even power series"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The proof of Proposition 13 assumes two algebraic simplifications for the components of M(g,f)^2 that are stated without expansion; if either identity fails, the square-root formula and the subsequent involution constructions collapse. It also assumes the Sprugnoli group product rules of [1] without re-derivation.","fun_headline_variants_meta":{"raw":{"variants":["Even power series unlock square roots in Appell group","Aerated Appell arrays get Sprugnoli square roots","Even power series produce Sprugnoli square roots","Square roots and involutions from even power series"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000728,"raw_usage":{"total_tokens":3036,"prompt_tokens":623,"completion_tokens":2413,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":367,"completion_tokens_details":{"reasoning_tokens":2348}},"tokens_in":367,"tokens_out":2413,"duration_ms":17840,"temperature":1.0,"reasoning_tokens":2348,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T00:04:29.775520+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Symbolically expand the two identities (G,F1,x)·G = (g(x)g(-x)(1-$x^{2}$)+$x^{2}$ f(x)^2)/(1-$x^{2}$)^2 and (G,F1,x)·GF1 = x(G,F1,x)·G for a generic even f(x) and generic g(x); any counterexample from the expansions would invalidate Proposition 13. Alternatively, compute a finite truncation of M(g,f)^2 for a concrete (g,f) with even f by direct matrix multiplication and compare to the Appell formula; a mismatch at any entry settles it.","supporting_citations":[{"cited_title":"Merlini, On the square root of a Bell matrix,Results in Mathematics,76(2021), Article 46, 18 pages","cited_arxiv_id":null,"evidence_quote":"Sets up the problem of square roots of Bell matrices that this paper's square-root construction mirrors."},{"cited_title":"Shapiro, R","cited_arxiv_id":null,"evidence_quote":"Background reference for the Riordan group and its Appell subgroup."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source for the Riordan group product and the Appell subgroup structure."}],"review_version":1}