{"id":"3c51a7d7-d7d6-40bc-9404-9f943e1a60dc","arxiv_id":"2607.26039","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The claimed N-bonacci upper bound on degree growth under iterated Wronskians is false: closing under all pairwise Wronskians from x^2,x^3 yields degrees 2^n+2.","lead":"The paper claims that repeatedly taking Wronskians of polynomials and feeding outputs back in produces degrees bounded by N-bonacci numbers, but a simple two-polynomial example in one variable already breaks the bound. The failure matters because the claimed bound would have connected polynomial Wronskian algebras to Kirillov's problem on growth.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Span-closure allows combining two largest elements; d=1,k=1,N=2 counterexample grows as 2^n, refuting N-bonacci bound.","rationale":"The central claim is the universal upper bound under the formal span-closure definition. The proof in Appendix A does not control arbitrary recombinations of accumulated elements; it analyzes only a single sliding-window chain. The reader's counterexample exploits exactly this gap and is correct for the smallest nontrivial arity. My check verifies the induction explicitly: the Wronskian of the two largest elements doubles the top degree each round. This is a decisive falsification of both the theorem and the corollary, not merely an edge case. The paper's N-bonacci examples are legitimate for specially chosen re-use protocols but do not establish the claimed universal bound. The reader's verdict of REJECT is therefore sustained; no change to the verdict is needed.","tokens_in":11802,"tokens_out":5803,"duration_ms":47190,"concrete_test":"Analytically verify the d=1,k=1,N=2 counterexample: prove by induction that A_n=span{x^2,...,x^{2^n+2}}. Base n=0: W(x^2,x^3)=x^4. Inductive step: for any a,b ∈ [2,2^n+2], deg W(x^a,x^b)=a+b-1, maximized by a=2^n+1,b=2^n+2, giving degree 2^{n+1}+2. Thus d_n=2^n+2 and d_n/F_n→∞. Optionally run a small SymPy script for n≤8 to confirm the maximum degree follows 2^n+2 rather than Fibonacci.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Appendix A reduces the problem to a single N-window recurrence (a_n = a_{n-1}+...+a_{n-N}-const) by asserting that the fastest growth comes from immediately re-using the newest monomial and dropping the smallest-degree old one. That assertion is not justified by the paper's own span-closure definition: A_{n+1} = span(A_n ∪ B_{n+1}) with B_{n+1} the Wronskians of arbitrary N elements of A_n. Because A_n retains all previously created monomials, one may pick the two (or N) largest available elements at each step, not those in a fixed sliding window. For d=1,k=1,N=2, start A0=span{x^2,x^3}. Since W(x^a,x^b)=(b-a)x^{a+b-1}, induction gives A_n=span{x^2,...,x^{2^n+2}}: the two largest elements x^{2^n+1},x^{2^n+2} yield W=x^{2^{n+1}+2}, and no pair has larger a+b-1. Hence d_n=2^n+2, while 2-bonacci numbers grow as φ^n; d_n/F_n→∞. This contradicts the Theorem and the Corollary's dimension bound dim(A_n)≲(α_N^d)^n=φ^n, since dim(A_n)=2^n+1. The Appendix's recurrence describes only a restricted sliding-window re-use scheme, not the span closure defined in the research problem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies iteration of the complete generalized Wronskian determinant W_k^d on the polynomial algebra R[x^1,...,x^d], viewed as an N-ary bracket with N=binom(d+k,d). The formal research problem defines A_{n+1}=span(A_n ∪ B_{n+1}), where B_{n+1} contains the Wronskians of arbitrary N-tuples of elements of A_n, and d_n is the highest total degree appearing in A_n. The main theorem claims that for any initial arguments, d_n grows asymptotically no faster than the N-bonacci numbers, and a corollary bounds dim(A_n) by (α_N^d)^n < (2^d)^n. The paper also gives examples attaining the bound for d=1 and odd k, discusses the coefficient nonvanishing problem via Skolem–Pisot, and conjectures attainment in general.","tokens_in":12139,"tokens_out":6953,"duration_ms":58028,"significance":"If the theorem were correct, the paper would provide a general growth bound for a natural polynomial analogue of Kirillov's problem and a dimension bound for subspaces generated by iterated Wronskians. The paper contains correct elementary material on N-bonacci characteristic roots and explicit, reproducible monomial examples. However, the central theorem is false under the paper's own span-closure definition, and the dimension corollary fails with it. The counterexample is simple, within the stated research problem, and does not depend on subtle interpretations. The paper's contribution therefore reduces to a set of examples and a proof for a restricted sliding-window iteration scheme, not the theorem advertised in the abstract and conclusion.","major_comments":[{"comment":"The theorem is false as stated. Take d=1,k=1,N=2 and A0=span{x^2,x^3}. Since W(x^a,x^b)=(b-a)x^{a+b-1}, induction gives A_n=span{x^2,...,x^{2^n+2}}: pairing the two largest monomials x^{2^n+1} and x^{2^n+2} produces x^{2^{n+1}+2}, and no pair gives a larger exponent. Hence d_n=2^n+2. The 2-bonacci numbers F_n grow like φ^n, so d_n/F_n→∞, contradicting the Theorem's assertion lim d_n/F_n<∞. This is a direct counterexample to the central claim.","section":"Research problem and Theorem, p.5; Appendix A"},{"comment":"For the same initial data, A_n contains exactly one monomial of each degree from 2 to 2^n+2, so dim(A_n)=2^n+1. The corollary claims dim(A_n)≲(α_2^d)^n=φ^n, which is impossible because (2^n+1)/φ^n→∞. Thus the dimension bound fails together with the theorem.","section":"Corollary, Eq. (5), p.5"},{"comment":"The proof reduces d_n to a single N-window recurrence a_n=a_{n-1}+...+a_{n-N}-kN/(d+1) by asserting that fastest growth requires immediate re-use of the newest monomial and dropping the smallest-degree old one. This assertion is not justified by the research problem's definition A_{n+1}=span(A_n∪B_{n+1}) with B_{n+1} consisting of Wronskians of arbitrary N elements of A_n. The counterexample above uses exactly this freedom by pairing the two largest elements at each step. The Appendix therefore proves at most an upper bound for the sliding-window scheme m_n:=W(m_{n-N},...,m_{n-1}) used in Examples 3–4 and the Conjecture, not for the stated theorem.","section":"Appendix A"}],"minor_comments":[{"comment":"The title contains OCR-type typos ('ITERA TE', 'OVERR'); please proofread the manuscript carefully.","section":"Title and running head"},{"comment":"The paper repeatedly invokes [2, Theorem 7] for the structure of Wronskians of monomials and for generalized Vandermonde determinants, but [2] is an unpublished preprint and the formula is not stated. A self-contained statement of this lemma would improve readability and verifiability.","section":"Appendix A and Example proofs"},{"comment":"The coefficient recurrence c_n=-c_{n-1}-c_{n-2}+c_{n-3} is asserted without derivation, and the statement that coefficients are nonzero 'up to n≈10,000' is experimental evidence, not a proof of nonvanishing. The distinction should be made explicit in the text.","section":"Example 4"}],"recommendation":"reject","confidential_remarks":"The stress-test counterexample is decisive and lands exactly on the theorem as stated in the Research problem. The paper's main novelty is the false bounding theorem; the examples and N-bonacci root analysis are correct but do not support the advertised conclusion. I do not see a way to repair the central claim within the manuscript's scope — at most the paper could be reframed as a study of the sliding-window iteration scheme, which would be a substantially weaker contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe central theorem is false under the paper's own definition. The research problem closes A_{n+1} = span(A_n ∪ all Wronskians of N elements of A_n), but Appendix A proves a bound for a different process: a single sliding window where each new output is immediately re-used and the oldest argument is dropped. That sketch misses that you can combine any two large elements already present. For d=1, k=1, N=2, start with x^2 and x^3. By induction A_n contains every monomial of degree from 2 to 2^n + 2: the two largest give W(x^{d_n}, x^{d_n-1}) = x^{2d_n-2}, so d_{n+1} = 2d_n - 2 and hence d_n = 2^n + 2. Fibonacci numbers grow like φ^n, so d_n/F_n → ∞, contradicting both the theorem and the dimension corollary.\n\nThat said, the paper is not empty. Examples 1–3 correctly exhibit N-bonacci degree growth for the restricted sliding-window iteration scheme; that is a useful observation. Appendix B re-proves the known growth rate of N-bonacci numbers with Rouché's theorem and a geometric bound; the comparison with Wolfram's constant is a small but clean contribution. The computer experiments and the link to the Skolem–Pisot problem are also worth a look.\n\nThe soft spot is exactly the gap between the stated span-closure problem and the proof. The proof's assertion that not using the newest argument immediately cannot be faster is simply wrong: in the counterexample, at n=3 the fastest output comes from pairing two older monomials, not from the most recent one. This is not a minor fix; the entire upper-bound claim depends on restricting to the sliding-window scheme. If the paper were reframed as a theorem about that scheme, the examples would support it, but the universal bound for the span closure is refuted.\n\nWho is this for? Readers interested in Wronskian N-ary brackets and Kirillov's problem might want to see the examples, but they should not trust the main theorem. It deserves a serious referee—not because the result is plausible, but because the error is subtle enough to slip past and the examples are genuinely interesting. I would recommend peer review with a request for major revision: either repair the proof by addressing branching, or honestly restrict the claim to single-chain iteration.","headline":"Main theorem is false: span-closure allows combining any two accumulated monomials, giving 2^n degree growth for d=1,k=1,N=2, not Fibonacci.","tokens_in":12645,"tokens_out":5716,"would_cite":false,"duration_ms":44112,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E15","11B39","15A15","05A10","05A16","17B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any starting polynomials, iterating the generalized Wronskian grows the highest degree no faster than the N-bonacci numbers.","keywords":["Differential polynomial","N-ary Lie bracket","multivariate Wronskian determinant","N-bonacci numbers","asymptotic degree growth","monomial degree recurrence","dimension growth","zero-detection in linear recurrences"],"falsifier":"Take d=1, k=1, N=2, start with x^2 and x^3, and at each step compute the Wronskian of every pair among the accumulated polynomials. Record the maximum total degree. If the maximum degree ever exceeds the nth Fibonacci number or shows an asymptotic exponent greater than φ≈1.618, the theorem's bound collapses; if it stays below the bound, the single-chain reduction is supported.","tokens_in":11659,"feed_emoji":"📈","tokens_out":8855,"duration_ms":82663,"temperature":0.7,"pith_summary":"The paper considers an iterative process on polynomials in d variables: take N polynomials, compute their complete generalized Wronskian determinant (a determinant built from partial derivatives up to order k), treat the result as a new polynomial, and keep re-using the newest polynomials as inputs. The question is how fast the highest total degree can grow. The main theorem gives a universal ceiling: after n iterations, the highest total degree—and the highest degree in each separate variable—grows asymptotically no faster than the nth N-bonacci number, the Fibonacci-like sequence in which each term is the sum of the previous N terms. This matters because degree growth bounds how much new polynomial space the iteration can span, connecting the problem to a classical question about growth of Lie algebras generated by iterated brackets. The ceiling is exact for one-variable polynomials with odd-order Wronskians, and the authors conjecture—with extensive computer evidence—that it is exact in every dimension and differential order.","feed_headline":"Iterated Wronskians grow no faster than N-bonacci","feed_subtitle":"A universal ceiling on polynomial-degree growth from repeated Wronskians, with sharp cases in one variable.","key_machinery":"The engine of the proof is the complete generalized Wronskian determinant W^k_d of differential order k over R^d, viewed as an N-ary bracket with N = binom(d+k,d). For monomial inputs, the output is again a monomial; its degree in each variable is the sum of the input degrees in that variable minus Nk/(d+1), and its coefficient is a determinant built from the exponents. This turns degree evolution into a linear recurrence with constant coefficients. The corresponding N-bonacci recurrence has characteristic polynomial λ^N − λ^{N−1} − ⋯ − λ − 1, whose unique root α_N above 1 lies between the golden ratio and 2, while all other roots lie inside the unit disk. That separation of roots converts t","core_discovery":"The central claim is that iterating the complete generalized Wronskian W^k_d as an N-ary bracket on R[x^1,...,x^d], where N = binom(d+k,d), cannot make degrees explode faster than the N-bonacci numbers. For any initial N-tuple of polynomials, the sequence of highest total degrees d^{(N)}_n satisfies lim_{n→∞} d^{(N)}_n / F^{(N)}_n < ∞, and the same holds for the highest degree in each coordinate. The argument reduces to monomial seeds: the Wronskian of monomials is again a monomial, and its degree in each variable is the sum of the input degrees minus the constant Nk/(d+1). Along the fastest single chain of re-use, this yields a linear recurrence that, after homogenization, is exactly the N-","pith_inferences":["The proof as written bounds a single chain of reuse (newest output always enters immediately, oldest drops out). If the problem is read literally as allowing any N polynomials from the whole current span at each step, branching could in principle accelerate growth; a direct small-scale test would show whether the ceiling still holds under that wider reading.","The constant term Nk/(d+1) in the per-variable degree recurrence suggests the total-degree bound may be looser in higher dimensions; tracking exact constants could yield a sharper effective bound for specific seeds.","Because coefficient nonvanishing is a zero-detection problem for linear recurrences, the conjecture implies that certain constructed integer sequences never hit zero; proving that for even one nontrivial seed would be a concrete positive case of a known hard decision problem."],"forward_implications":["For any starting polynomials, after n iterations the highest total degree is eventually no larger than the nth N-bonacci number; the same ceiling applies to the highest degree in each variable.","The dimension of the generated polynomial subspaces grows at most like (α_N^d)^n, which is strictly below the naive (2^d)^n bound.","In one variable with odd differential order k, the ceiling is attained: there exist monomial seeds whose degrees are the N-bonacci numbers up to a shift and whose coefficients never vanish, so the bound is sharp there.","If the paper's conjecture is correct, the N-bonacci ceiling is asymptotically sharp in every dimension and differential order, not just on the line.","Whether a particular seed choice produces nonzero coefficients forever is a zero-detection problem for an integer linear recurrence; the paper reports many cases where coefficients stay nonzero for about 10,000 iterations."],"fun_headline_variants":["Wronskian iteration: degree growth capped by N-bonacci","N-bonacci numbers bound degree growth in iterated Wronskians","Iterating Wronskians: polynomial degrees hit N-bonacci ceiling","Repeated Wronskians: degree growth no faster than N-bonacci"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The bound rests on the assumption that the fastest growth comes from a single window of length N—using the newest output immediately and dropping the oldest argument—so that degrees obey one N-term recurrence; if choosing arbitrary N-tuples from the whole current span can create branched, faster growth, that recurrence argument would not cover the stated problem.","fun_headline_variants_meta":{"raw":{"variants":["Wronskian iteration: degree growth capped by N-bonacci","N-bonacci numbers bound degree growth in iterated Wronskians","Iterating Wronskians: polynomial degrees hit N-bonacci ceiling","Repeated Wronskians: degree growth no faster than N-bonacci"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000254,"raw_usage":{"total_tokens":1459,"prompt_tokens":852,"completion_tokens":607,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":596,"completion_tokens_details":{"reasoning_tokens":528}},"tokens_in":596,"tokens_out":607,"duration_ms":5348,"temperature":1.0,"reasoning_tokens":528,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T00:48:11.216151+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take d=1, k=1, N=2, start with x^2 and x^3, and at each step compute the Wronskian of every pair among the accumulated polynomials. Record the maximum total degree. If the maximum degree ever exceeds the nth Fibonacci number or shows an asymptotic exponent greater than φ≈1.618, the theorem's bound collapses; if it stays below the bound, the single-chain reduction is supported.","supporting_citations":[],"review_version":1}