{"id":"09a53a73-221b-406a-a254-433a8dc48025","arxiv_id":"2411.17010","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For numerical semigroups, extremal p-lengths are eventually quasipolynomial with explicit degree, period, and leading coefficient; for arithmetical congruence monoids, p-lengths of x^n grow like Θ(n), Θ(n^(1/2)), or Θ(n^(2/3)) depending on the element.","lead":"This math paper introduces p-length, a family of factorization statistics based on the ℓ_p norms of exponent vectors, and studies how the extremal p-lengths behave for large elements of two families of monoids. It proves clean asymptotic formulas for numerical semigroups and exhibits growth rates from linear to n^(1/2) and n^(2/3) for arithmetical congruence monoids.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.10's proof of quasipolynomiality for ℓ^m_2 rests on an unproved largeness claim: the global integer minimizer of ℓ2 must eventually have nonnegative coordinates. This step is load-bearing because the entire ℓ^m_2 row of Table 1 depends on it.","rationale":"The reader's weakest assumption is exactly the unproved largeness step in Theorem 2.10, and I agree that this is the most load-bearing concern among those flagged. The central claim of the paper includes the eventual quasipolynomiality of ℓ^m_2 for every numerical semigroup; this is a headline contribution in Table 1, not an auxiliary example. The proof gap is real: Proposition 2.9 gives a shift invariant for the global integer minimizer, but the transition from 'integer solution' to 'factorization' (nonnegative coordinates) is asserted in a single 'As such' without a written argument or a uniform threshold. The other issues noted by the reader (Example 2.11's formula, Proposition 3.6's x=40 factorizations) are concrete but more localized: they affect specific examples or proofs in Section 3, and the asymptotic statements may still be true with corrected constructions. Theorem 2.10, by contrast, is a general theorem about all numerical semigroups, so its proof gap is more consequential. That said, the intended argument is plausibly repairable: one can start from any integer solution for each residue class modulo N and apply Proposition 2.9 repeatedly; since each application increases every coordinate by at least 1, after finitely many shifts the minimizer becomes nonnegative, and the finite number of residue classes gives an eventual uniform statement. The concrete test proposed above would verify the conclusion computationally for a representative semigroup, and if it passes, the theorem is likely true and the paper only needs a rigorous proof of the omitted step. Therefore the reader's CONDITIONAL verdict remains appropriate, and no change is needed.","tokens_in":10933,"tokens_out":24728,"duration_ms":210191,"concrete_test":"For a nontrivial numerical semigroup with small generators, say S = ⟨4,7,9⟩ (so N = 4^2+7^2+9^2 = 146), compute L(n) = min{ℓ2(z) : z ∈ Z(n)} exactly for all n up to 3000 using dynamic programming. Then verify that L(n+146) − L(n) = 2n + 146 for all n from 200 to 3000. If any value fails, Theorem 2.10 is false. Also, for each n in that range, independently compute the unique (or a chosen) minimizer of ℓ2 over all integer solutions to 4x+7y+9z = n (via brute force in a bounded box around the real minimizer) and check whether it has nonnegative coordinates and agrees with the dynamic-programming minimizer. This would directly confirm or refute the missing largeness assertion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 2.10 (Section 2), the proof claims: 'As such, one may choose n ∈ S large enough to ensure some z ∈ Z(n) minimizes ℓ2(·) over all integer solutions.' This is the critical step, but it is not justified in the text. Proposition 2.9 shows that if a vector z minimizes ℓ2 among all integer solutions to g·x = n, then z + (g1,...,gk) minimizes among all integer solutions to g·x = n + N (where N = g1^2 + ... + gk^2). This shift increases every coordinate by at least 1, so iterating it eventually produces a nonnegative minimizer. However, the proof never states this iteration, never establishes a uniform threshold across residue classes modulo N, and even contains a garbled sentence ('the smallest coordinate ... is strictly larger than the smallest coordinate ...') that obscures the intended argument. Without a rigorous demonstration that for all sufficiently large n ∈ S some minimizer over Z(n) is also the global integer minimizer, the recurrence ℓ^m_2(n+N) − ℓ^m_2(n) = 2n + N, and hence the claimed degree-2 quasipolynomial behavior, is unsupported. This is not a mere typo: it is the hinge on which the quasipolynomiality of ℓ^m_2 rests. If the claim is false, Table 1's ℓ^m_2 entry is wrong for some semigroup; if it is true but unproved, the paper needs a repair in the exposition.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces p-length invariants for factorizations in commutative monoids, defined as the ℓ_p-norm of the exponent vector, and studies their extremal values for large elements of numerical semigroups and arithmetical congruence monoids (ACMs). For numerical semigroups, the paper claims that the minimal and maximal p-length functions are eventually quasipolynomial for p ∈ {0,1,2,∞}, with the degrees, periods, and leading coefficients listed in Table 1; it also claims by example that for p ≥ 3 the minimal p-length need not be eventually quasipolynomial. For ACMs, the paper establishes linear growth of some p-length functions along powers x^n and exhibits two singular ACMs where the maximum 0-length grows as n^{1/2} and n^{2/3}.","tokens_in":11240,"tokens_out":20618,"duration_ms":191101,"significance":"If the results are correct, the paper introduces a genuinely new family of factorization invariants and gives clean, explicit asymptotics for numerical semigroups and for several ACMs. The table of quasipolynomial attributes is a useful contribution, and the ACM examples displaying sublinear growth are striking and well-motivated. The paper also benefits from using explicit external benchmarks for the p=0 and p=1 cases rather than fitting parameters, and many of the Section 2 arguments are simple and checkable. However, several load-bearing steps currently contain gaps or explicit computational errors, so the results cannot be accepted in their present form.","major_comments":[{"comment":"The proof of Theorem 2.10 is incomplete at the step asserting that one may choose n ∈ S large enough so that some z ∈ Z(n) minimizes ℓ2(·) over all integer solutions to g1 x1 + ... + gk xk = n. This is the load-bearing point: it must show that for all sufficiently large n, and in every residue class modulo N = g1^2 + ... + gk^2, a global integer minimizer can be taken with nonnegative coordinates. Proposition 2.9 only propagates an existing integer minimizer from n to n+N; it does not establish the existence of such a minimizer for all large n. The sentence comparing the smallest coordinate of the minimizer for (2.3) with the smallest coordinate of the minimizer for (2.3) is garbled and does not prove a uniform threshold. Without this step, the recurrence ℓ^m_2(n+N) − ℓ^m_2(n) = 2n + N, and hence the degree-2 quasipolynomiality asserted in Table 1, is unsupported. A repair is available, for example by showing that every integer minimizer lies within bounded distance of the real minimizer (n/N)(g1,...,gk), but the proof as written needs a substantial addition.","section":"Theorem 2.10 (Section 2)"},{"comment":"The displayed closed form for the ℓ^m_3(n) minimizer in S = <2,3> is incorrect. Solving the Lagrange multiplier condition for minimizing z1^3 + z2^3 subject to 2z1 + 3z2 = n gives z1 = n(3√6 − 4)/19 (up to the appropriate integer rounding), not (−8 ± n√130)/19. The qualitative conclusion may be salvageable with the corrected formula, but as written the example is false. Moreover, the phrase 'similar computation to the proof of Proposition 2.9' is misleading: over all integer solutions the cubic is unbounded below along the kernel direction (−3,2), so the minimization must be restricted to the nonnegative solution set Z(n).","section":"Example 2.11 (Section 2)"},{"comment":"For x = 40 = 2^3·5, the two displayed factorizations of (2^3·5)^n contain only n total factors of 2, not 3n; they factor 10^n rather than 40^n. The exponent on the atom 2^2 should be (3n−k−1)/2 in the first case and (3n−k−2)/2 in the second. In addition, the second construction has final 5-exponent n−k^2−1, which is negative at the endpoint n = k^2 of the claimed interval. The Θ(n^{1/2}) claim may be true, but the proof as written is invalid and needs correction.","section":"Proposition 3.6 (Section 3)"},{"comment":"The lower-bound construction for x = 70 contains negative exponents: the factor (2·5^{2i+1}7^{2a−2i−1}) has exponent 2a−2i−1 = −1 when i = a. The subsequent assertion that 'we may choose c appropriately so that ... for some n ≤ k^3' is not justified. Since this construction is the entire lower bound for the Θ(n^{2/3}) claim, the proof must be rewritten with a valid choice of exponents and an explicit verification that c is a nonnegative integer and that the resulting n lies in the claimed range.","section":"Proposition 3.7 (Section 3)"}],"minor_comments":[{"comment":"The definition of ℓ0(z) via z_i^0 is ambiguous because 0^0 is undefined; the authors should define ℓ0(z) as the number of nonzero coordinates of z.","section":"Section 1 and Section 2"},{"comment":"The proof assumes that g2 is the smallest among g2,...,gk; this should be stated explicitly, for instance by assuming the generators are ordered g1 < g2 < ... < gk.","section":"Lemma 2.7"},{"comment":"The line 'ℓm∞(n+ai) = l∞(qg) = q' contains a typo; it should be 'ℓm∞(qg) = q'.","section":"Theorem 2.6 proof"},{"comment":"The chain 'a(Q′+R′) ≤ a(Q+R+Q′+R′) = 1/n a(b+c) = (P+P′)(b+c)' appears to contain a typo; it should read '= a n (b+c) = (P+P′)(b+c)'. Also, the derived inequality 2P′ < P actually yields m < k, so the stated m ≤ 2k is weaker than what the argument gives.","section":"Theorem 3.5 proof"}],"recommendation":"major_revision","confidential_remarks":"Dear Editor, the paper introduces a promising invariant and contains several plausible and checkable results for numerical semigroups. However, the proof gap in Theorem 2.10 and the concrete computational errors in Example 2.11, Proposition 3.6, and Proposition 3.7 are load-bearing and currently prevent acceptance. I found no reason to doubt the authors' intent, and the issues appear repairable within the scope of the manuscript, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The p-length invariant is a good idea, and the numerical semigroup part mostly delivers what it promises. The paper introduces ℓ_p-extremal factorization lengths, proves eventual quasipolynomiality with explicit degree, period, and leading coefficient for p = 0, 1, 2, ∞, and shows ℓ^m_3 is not eventually quasipolynomial via an example. That last result is the most interesting thing here: the minimal cube-sum problem on ⟨2,3⟩ has irrational slopes, and the paper’s claim that this kills quasipolynomiality is exactly the kind of subtlety the field needs. I also like the ACM growth-rate results; the contrast between regular and singular ACMs is real.\n\nThe central theorems in Section 2 (2.4, 2.6, 2.8) look right. The arguments are short and mostly self-contained, and the reliance on Apéry sets is clean. The reader’s spot checks didn’t turn up issues there, and I agree.\n\nThe soft spots are real but localized. Theorem 2.10’s proof has a load-bearing gap: it asserts that for large enough n ∈ S, some z ∈ Z(n) minimizes ℓ2 over all integer solutions. That is not justified in the text. The stress-test note is right: Proposition 2.9 gives a shift that raises every coordinate, so an iteration argument is plausible, but the paper needs to state it and prove a uniform threshold across residue classes. The garbled sentence about smallest coordinates doesn’t help. This gap underlies the entire ℓ^m_2 row of Table 1, so it needs fixing.\n\nExample 2.11’s displayed minimizer formula is also wrong—Lagrange multipliers on z1^3 + z2^3 with 2z1 + 3z2 = n give c1 = n(3√6 − 4)/19, not (−8 ± n√130)/19. The conclusion that ℓ^m_3 is not eventually quasipolynomial probably survives with the corrected formula, but as printed it’s an error. Proposition 3.6’s x = 40 construction has a 2-adic imbalance: 40^n = 2^{3n}5^n, and the two displayed factorizations don’t have matching 2-exponents. That one I didn’t repair in my head; it may need a different construction.\n\nNone of this looks like a broken core. The paper is honest about which results are imported and which are new, and the numerical semigroup proofs are not circular. It’s a solid contribution that needs a careful revision.\n\nI’d send this to a serious referee. A referee can ask for a real largeness lemma for Theorem 2.10 and corrected examples. For anyone working on factorization invariants or quasipolynomial phenomena, this is worth reading, but wait for the revision.","headline":"Genuinely new p-length invariant with mostly clean asymptotics; the ℓ^m_2 proof has a real gap and the examples need repair, but this deserves refereeing.","tokens_in":11936,"tokens_out":2968,"would_cite":true,"duration_ms":28045,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20M14","20M13"],"pacs":[],"model":"deepseek-v4-flash","headline":"Extremal p-lengths of factorizations in numerical semigroups are eventually quasipolynomial for p = 0, 1, 2, and ∞, with explicit periods and leading coefficients.","keywords":["p-length","factorization length","numerical semigroup","quasipolynomial","arithmetical congruence monoid","Apéry set","ℓ_p-norm","asymptotic growth"],"falsifier":"Take the numerical semigroup S = <2,3> and compute, for each n up to a large bound (say $10^{5}$), the unique integer solution (x,y) to 2x + 3y = n minimizing $x^{2}$ + $y^{2}$. If there are infinitely many n with the minimizing solution having x < 0 or y < 0, or if the second difference of the true minimum ℓ^m_2(n) ever fails to equal 2($2^{2}$ + $3^{2}$) = 26 for large n, then the claimed quasipolynomiality with period 13 is false. For a semigroup with three generators, the analogous check is whether the integer minimizer of $x1^{2}$ + $x2^{2}$ + $x3^{2}$ subject to g1x1 + g2x2 + g3x3 = n eventually has all coordinates nonnegative.","tokens_in":10663,"feed_emoji":"📈","tokens_out":13943,"duration_ms":103148,"temperature":0.7,"pith_summary":"This paper introduces p-length, a generalization of the classical factorization length that replaces the sum of exponents (the 1-norm) with the ℓ_p-norm of the exponent vector. It establishes that for every numerical semigroup S = ⟨g1,...,gk⟩, the minimum and maximum p-lengths of factorizations of n are eventually quasipolynomial functions of n for p ∈ {0,1,2,∞}, and it computes the degree, period, and leading coefficient of each. For example, the maximum ∞-length satisfies ℓ^M_∞(n) = (n - a_i)/g1 for all sufficiently large n, where a_i is the Apéry element of S congruent to n modulo g1. For arithmetical congruence monoids, the paper shows that in the singular monoid M_{4,6}, the minimal 1- and ∞-lengths of x^n grow linearly for every x = 2^a 5^b 7^c, whereas the maximum 0-length grows like Θ($n^{{1/2}}$) for x = 28 and 40 and Θ($n^{{2/3}}$) for x = 70. The result matters because it gives a complete asymptotic picture for a new invariant of factorization in the two most-studied monoid families, and it exposes a boundary: for p ≥ 3 the minimum p-length of a numerical semigroup need not be eventually quasipolynomial.","feed_headline":"p-lengths of factorizations are eventually quasipolynomial","feed_subtitle":"For numerical semigroups, min and max p-lengths have explicit periods and leading coefficients for p = 0, 1, 2, ∞.","key_machinery":"The machinery is the pair of extremal functions ℓ^m_p(n) and ℓ^M_p(n) on the factorization set Z(n) = {z ∈ Z≥0^k : z1g1 + ··· + zkgk = n}, together with three structural identities that turn the discrete optimization into a quasipolynomial: the Apéry-set formulas ℓ^M_∞(n) = (n - a_i)/g1 and ℓ^m_∞(n) = (n + a_i)/g, where a_i is the element of Ap(S;g1) or Ap(S;g) in the appropriate residue class; the translation property for the Euclidean norm, that adding (g1,...,gk) to a minimizer preserves minimality, which yields ℓ^m_2(n+N) - ℓ^m_2(n) = 2n + N; and the one-variable reduction for p ≥ 2 that the maximum p-length is achieved by the factorization with maximal first coordinate. For ACMs the load-bearing tool is a classification of the atoms of M_{4,6} (an atom divides 2^a 5^b 7^c only in restricted forms), which permits counting distinct atoms in factorizations of x^n.","core_discovery":"The central discovery is that extremal p-lengths are asymptotically rigid for numerical semigroups: for each p ∈ {0, 1, 2, ∞}, both the minimal and the maximal p-length functions ℓ^m_p(n) and ℓ^M_p(n) are eventually quasipolynomial, meaning that for all n beyond a finite threshold they equal a polynomial in n whose coefficients depend periodically on n. The paper determines the full set of attributes: ℓ^M_p has degree p, period g1, and leading coefficient 1/g1^p for p ≥ 1 (degree 0 and period 1 for p = 0), while ℓ^m_1 has degree 1, period gk, and leading coefficient 1/gk, ℓ^m_2 has degree 2, period N = $g1^{2}$ + ··· + $gk^{2}$, and leading coefficient 1/N, and ℓ^m_∞ has degree 1, period g = g1 + ··· + gk, and leading coefficient 1/g. The proof for p = ∞ uses Apéry sets to determine the constant term, and the proof for p = 2 uses a translation identity that reduces the problem to second differences. For arithmetical congruence monoids, the paper does not attempt quasipolynomial formulas but instead establishes explicit growth rates for the singular monoid M_{4,6}.","pith_inferences":["The unproved largeness step in Theorem 2.10 could likely be supplied by a lattice-geometry argument: the Euclidean minimizer over all integer solutions converges to the ray in direction (g1,...,gk) as n grows, so for all n beyond a bound depending only on the generator set, the minimizer lies in the nonnegative orthant; if that holds, the second-difference proof of quasipolynomiality is complete.","The special role of p = 2 may be explained by the fact that the ℓ_2-norm is the only ℓ_p-norm whose sublevel sets are rational ellipsoids, making the translation identity (z + (g1,...,gk)) preserve the norm difference in a linear way; p ≥ 3 norms lack such an identity, which is consistent with the paper's example that ℓ^m_3 for S = <2,3> involves a floor of an irrational multiple.","The growth exponents 1/2 and 2/3 for ℓ^M_0(x^n) in M_{4,6} are reminiscent of the divisor function's behavior in short intervals, suggesting a possible connection between factorization counting in singular ACMs and analytic number theory; examining other singular ACMs might yield other rational exponents.","The good/evil atom dichotomy used for M_{4,6} could be adapted to other finite-atomic ACMs to obtain similar Θ(n^α) bounds, potentially answering Question 3.8 for a larger class."],"forward_implications":["For any numerical semigroup, the value of the extremal p-length for every sufficiently large n is given by an explicit quasipolynomial, so the invariant is effectively computable rather than merely asymptotic.","The period and leading coefficient of each p-length function constitute new arithmetic invariants of the semigroup, such as N = g1^2 + ··· + gk^2 for the minimum 2-length and the Apéry elements for the maximum ∞-length.","The p = ∞ formulas tie extremal factorization lengths directly to the semigroup's Apéry structure, giving a clean geometric meaning to the periodic constant terms.","The ACM results show that the maximum number of distinct atoms in a factorization of x^n can grow like n^{1/2} or n^{2/3} depending on the base element, so the growth rate is a discriminating invariant of the base element even within a single monoid.","The Θ(n) lower bound for minimal 1- and ∞-lengths in M_{4,6} confirms that singular ACMs can mimic the linear growth of regular ACMs despite their unusual factorization behavior."],"supporting_citations":[{"why":"Supplies the sparse-solutions result that underpins the periodic ℓ^m_0(n) statement in Theorem 2.1(c).","marker":"[2]"},{"why":"Provides the eventual quasilinearity of ℓ^m_1 and ℓ^M_1 used as the p = 1 base cases.","marker":"[7]"},{"why":"Gives the Apéry set structure and the definition of Ap(S;n) that drives the p = ∞ formulas.","marker":"[4]"},{"why":"Gives the full characterization of atoms of M_{4,6} that underlies all the ACM growth-rate proofs.","marker":"[6]"},{"why":"Provides the Krull property and general ACM facts that justify the Θ(1) and linear-growth statements for regular and bifurcus ACMs.","marker":"[5]"},{"why":"Gives the Frobenius-number bounds used to prove that ℓ^M_0(n) equals k for all large n in Theorem 2.1(d).","marker":"[16]"}],"fun_headline_variants":["p-lengths eventually follow quasipolynomials","Extremal p-lengths: quasipolynomial for numerical semigroups","Explicit periods for p-length extrema when p=0,1,2,∞","Asymptotic rigidity: p-lengths become quasipolynomial","Growth rates for p-lengths in arithmetical congruence monoids"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that the minimum 2-length is eventually quasipolynomial assumes, without proof, that for all sufficiently large n there is a factorization of n that minimizes the Euclidean norm over all integer solutions to the same linear equation; if that largeness condition ever fails, the quasipolynomiality of ℓ^m_2 rests on nothing.","fun_headline_variants_meta":{"raw":{"variants":["p-lengths eventually follow quasipolynomials","Extremal p-lengths: quasipolynomial for numerical semigroups","Explicit periods for p-length extrema when p=0,1,2,∞","Asymptotic rigidity: p-lengths become quasipolynomial","Growth rates for p-lengths in arithmetical congruence monoids"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000955,"raw_usage":{"total_tokens":4127,"prompt_tokens":1058,"completion_tokens":3069,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":674,"completion_tokens_details":{"reasoning_tokens":2972}},"tokens_in":674,"tokens_out":3069,"duration_ms":20892,"temperature":1.0,"reasoning_tokens":2972,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:40:46.963293+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the numerical semigroup S = <2,3> and compute, for each n up to a large bound (say $10^{5}$), the unique integer solution (x,y) to 2x + 3y = n minimizing $x^{2}$ + $y^{2}$. If there are infinitely many n with the minimizing solution having x < 0 or y < 0, or if the second difference of the true minimum ℓ^m_2(n) ever fails to equal 2($2^{2}$ + $3^{2}$) = 26 for large n, then the claimed quasipolynomiality with period 13 is false. For a semigroup with three generators, the analogous check is whether the integer minimizer of $x1^{2}$ + $x2^{2}$ + $x3^{2}$ subject to g1x1 + g2x2 + g3x3 = n eventually has all coordinates nonnegative.","supporting_citations":[{"cited_title":"Aliev, J","cited_arxiv_id":null,"evidence_quote":"Supplies the sparse-solutions result that underpins the periodic ℓ^m_0(n) statement in Theorem 2.1(c)."},{"cited_title":"Barron, C","cited_arxiv_id":null,"evidence_quote":"Provides the eventual quasilinearity of ℓ^m_1 and ℓ^M_1 used as the p = 1 base cases."},{"cited_title":"Assi and P","cited_arxiv_id":null,"evidence_quote":"Gives the Apéry set structure and the definition of Ap(S;n) that drives the p = ∞ formulas."},{"cited_title":"Banister, J","cited_arxiv_id":null,"evidence_quote":"Gives the full characterization of atoms of M_{4,6} that underlies all the ACM growth-rate proofs."},{"cited_title":"Baginski and S","cited_arxiv_id":null,"evidence_quote":"Provides the Krull property and general ACM facts that justify the Θ(1) and linear-growth statements for regular and bifurcus ACMs."},{"cited_title":"Ram ´ ırez Alfons ´ ın,The Diophantine Frobenius problem, Oxford Lecture Series in Math- ematics and its Applications, 30","cited_arxiv_id":null,"evidence_quote":"Gives the Frobenius-number bounds used to prove that ℓ^M_0(n) equals k for all large n in Theorem 2.1(d)."}],"review_version":1}