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REVIEW 2 major objections 3 minor 20 references

Elasticity in Apery sets

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For every rational number $r>1$, there is a numerical semigroup whose Apéry set has elasticity set exactly $\{1,r\}$.

desk verdict New invariants and several sharp constructions, but Theorem 2.2 fails at b=1, so the 'all two-element sets realizable' claim is unproven for integer elasticities. read the letter →

arxiv 1908.06448 v1 pith:7GAWEBXJ submitted 2019-08-18 math.AC

Jackson Autry , Tara Gomes , Christopher O'Neill , Vadim Ponomarenko This is my paper · ORCID
classification math.AC MSC 20M14
keywords numericalsemigroupApérysetelasticityhalf-factorialembeddingdimensionmeanfactorization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Numerical semigroups are additive subsemigroups of the nonnegative integers with finite complement, and their Apéry set records the smallest representative of each residue class modulo the multiplicity. This paper asks how much non-unique factorization can occur among those representatives and introduces two invariants: the Apéry half-factorial fraction (AHFF) and the mean Apéry elasticity (MAE). Its main construction shows that for every rational $r>1$, there is a numerical semigroup whose Apéry set has elasticity set exactly $\{1,r\}$, with a single non-half-factorial element. The same tools show that AHFF can be arbitrarily close to 0 and MAE arbitrarily large even when the semigroup has only three atoms, and that Apéry half-factorial semigroups are asymptotically generic for fixed multiplicity but not among all semigroups of a given genus.

What carries the argument

The Apéry set $\operatorname{Ap}(S)=\{n\in S : n-m\notin S\}$, where $m$ is the multiplicity, is the minimal complete set of residues of $S$ modulo $m$. Factorization structure is read through the Apéry poset, ordered by $x\preceq y$ when $y-x\in S$; a chain from $0$ to $n$ corresponds to an ordered factorization of $n$, so the presence of two chain lengths to the same element detects non-half-factorality. The load-bearing construction is the family $S=\langle a+b,pa,pb\rangle$, whose Apéry set is explicitly listed and has all elements uniquely factorable except $pab$; the two factorizations $pab=a(pb)=b(pa)$ supply the elasticity ratio $a/b$.

What would settle it

Compute $R(\operatorname{Ap}(S))$ for the family $S=\langle a+1,p,ap\rangle$ with $p$ prime and $a+1<p$; if, as the listed Apéry set suggests, every element is uniquely factorable, then $R(\operatorname{Ap}(S))=\{1\}$ for each $a$, which would falsify Corollary 2.3 for integer $r=a$ unless a new construction is supplied.

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Extended reading notes

Core claim

The central discovery is that the two-element elasticity sets of Apéry sets are exactly the sets $\{1,r\}$ with rational $r>1$. The witness family is $S=\langle a+b, pa, pb\rangle$ for coprime $a>b\geq 1$, with $p$ prime, $p\nmid(a+b)$, and $a+b<pb$; its Apéry set is $\{0,pb,2pb,\ldots,(a-1)pb,pa,2pa,\ldots,(b-1)pa,pab\}$, and every element has unique factorization except $pab$, which is $a$ copies of $pb$ and $b$ copies of $pa$ and therefore has elasticity $a/b$. The paper also proves that the Apéry half-factorial fraction can be made arbitrarily close to 0 inside embedding dimension 3, that mean Apéry elasticity can be made arbitrarily large in embedding dimension 3, and that among numerical semigroups with fixed multiplicity the proportion that are Apéry half-factorial approaches 1 as genus grows, while across all genera the limiting proportion lies strictly between 0 and 1.

Load-bearing premise

The two-element classification rests on the construction in Theorem 2.2 treating both $pa$ and $pb$ as atoms of $S=\langle a+b,pa,pb\rangle$; that treatment fails when $b=1$, because $pa$ is then a sum of $a$ copies of $p$, so the proof as written does not establish integer ratios $r=a$.

Editorial extensions

If this is right

  • All two-element elasticity sets of Apéry sets are classified: they are exactly $\{1,r\}$ for rational $r>1$.
  • There exist three-atom numerical semigroups whose Apéry half-factorial fraction is arbitrarily close to 0.
  • Mean Apéry elasticity is unbounded over three-atom numerical semigroups, so the Apéry set can be very far from half-factorial on average.
  • For fixed multiplicity, almost every sufficiently large-genus numerical semigroup is Apéry half-factorial.
  • Among all numerical semigroups of a given genus, the proportion that are Apéry half-factorial tends to a limit strictly between 0 and 1.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof of the two-element classification does not cover $b=1$: in $S=\langle a+1,p,pa\rangle$, the element $pa$ is not an atom, so the claimed uniqueness of factorization for all Apéry elements except $pab$ needs a separate argument; integer ratios would require a different witness if the classification is to hold for them.
  • The same construction suggests possible next families with three or more non-half-factorial Apéry elements, which would realize elasticity sets such as $\{1,r,s\}$; the paper does not resolve this.
  • The fixed-multiplicity versus fixed-genus contrast implies the asymptotic density of Apéry half-factorial semigroups depends strongly on which parameter is held fixed, so any census of factorization-theoretic behavior should report multiplicity and genus separately.
  • Mean Apéry elasticity gives a single-number summary of factorization complexity; characterizing which rationals occur as $MAE(S)$ would be a natural continuation of Theorem 3.1.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. This paper introduces and studies factorization invariants of numerical semigroups restricted to Apéry sets. The authors define R(Ap(S)) (the set of elasticities of Apéry elements), the Apéry half-factorial fraction (AHFF), and the mean Apéry elasticity (MAE). The main contributions are: a characterization of Apéry half-factoriality via graded Apéry posets (Theorem 1.1); a construction of semigroups with R(Ap(S))={1,a/b} (Theorem 2.2), used to claim that every two-element subset {1,r} of the rationals at least 1 is realizable (Corollary 2.3); families demonstrating that AHFF can be arbitrarily close to 0 (Theorem 2.4) and that MAE can be arbitrarily large (Theorem 3.1); an infinite AHF family with three generators (Theorem 2.5); and asymptotic density results for AHF semigroups (Theorems 4.1 and 4.2).

Significance. The two-element classification, if established, is a clean and appealing result, and the AHFF and MAE invariants are natural objects that could stimulate further work. The proofs are direct and mostly self-contained, and external results [1] and [20] are invoked for background and asymptotic comparison rather than fitted parameters. The main gap is the b=1 boundary in Theorem 2.2, which removes all integer elasticities from the two-element realizability claim; this is a substantive issue for the paper's central assertion.

major comments (2)
  1. [Section 2, Theorem 2.2 and Corollary 2.3] The construction in Theorem 2.2 fails for b=1. When b=1, the semigroup S=<a+1, pa, p> is actually <a+1, p> because pa is a multiple of p; hence pa is not an atom. The Apéry set is {0, p, 2p, ..., ap}, and every element has unique factorization, so R(Ap(S))={1}, not {1,a}. For example, a=2, b=1, p=5 gives S=<3,5>, Ap(S)={0,5,10}, and R(Ap(S))={1}. Since an integer r has reduced form r=a/1, Corollary 2.3's integer cases are exactly the unproved boundary of this theorem. The assertion that all two-element subsets of the rationals at least 1 are realizable is therefore unsupported for integer r. Please either restrict Theorem 2.2 to b≥2 and Corollary 2.3 to non-integers, or supply a separate construction that covers integer r.
  2. [Section 3, Theorem 3.1] The displayed equality for Ap(S) in Theorem 3.1 is incorrect for most odd primes q. For q=3, with m=20 and S=<20,2p,3p> (p prime, p>10), the Apéry set is {0,2p,3p,4p,...,19p,21p}; in particular 21p is present and 22p is not an Apéry element. The paper's expression {0,2p,4p,...,(q-1)p, qp, (q+1)p,..., (9q+17)/2 p} gives, after removing (4q+8)p, the set {0,2p,3p,...,22p}\{20p}, which includes 22p and omits 21p. The lower-bound argument itself only needs T⊆Ap(S), so the asymptotic conclusion may be salvageable, but the stated theorem needs a corrected Apéry-set description.
minor comments (3)
  1. [Section 4, Theorem 4.2] The text says each constructed semigroup has "genus m+|T|", but the subsequent summation uses |T|=g-(m-1), which corresponds to genus (m-1)+|T|. The stated genus is off by one and should be corrected.
  2. [Section 2, Theorem 2.5] The proof of Theorem 2.5 merely asserts the Apéry set without demonstration. It would be helpful to state explicitly that a(n^2+n)+b(2n^2+1) with 0≤a,b≤n-1 forms a complete residue system modulo n^2 and that no Apéry element can contain n^2 as a summand, since this is what makes the half-factorial conclusion transparent.
  3. [Section 1, Theorem 1.1] The proof speaks of a "chain" as a set of mutually comparable elements, but factorizations correspond to saturated chains in the Hasse diagram. Please clarify that the correspondence is between saturated chains and ordered factorizations into atoms.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the flagged b=1 issue in Theorem 2.2 is a correctness gap, not a circular derivation.

full rationale

The paper's central results are direct constructions and computations, not fits or renamed inputs. Theorem 2.2 explicitly builds S = <a+b, pa, pb> and computes Ap(S) and the elasticity of pab, with the elasticity value a/b emerging from a direct factorization-length comparison rather than from any assumed conclusion. Corollary 2.3 merely applies Theorem 2.2 to reduced fractions, so it inherits the theorem's content without importing any circular premise. The paper uses self-citations only for background and motivation, and the load-bearing external citations in Section 4 ([1] and [20]) are independent published results used as asymptotic inputs, not as substitutes for the paper's own arguments. The reviewer's noted failure of Theorem 2.2 at b=1 is a genuine mathematical gap in the proof, since pa is not an atom when b=1 and the claimed Ap(S) set has only unique factorizations, but this is a correctness concern about the boundary case, not a case of the paper deriving X from a definition of X or calling a fitted parameter a prediction. Accordingly, no circularity step can be exhibited, and the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces new mathematical definitions (AHF, AHFF, R(Ap), MAE) but these are definitions, not empirical postulates; they carry no independent falsifiable handle outside the paper.

assumptions (3)
  • standard math Apery set contains exactly one minimal element in each residue class modulo m(S); its size is m(S).
    Used throughout, e.g., in identifying Ap(S) in Theorems 2.2, 2.4, 2.5.
  • domain assumption Zhai's theorem that n_g/f_g tends to a positive constant as g grows.
    Invoked in the proof of Theorem 4.2 to convert Fibonacci lower bounds into a positive proportion.
  • domain assumption Kunz polytope result that for fixed multiplicity m, almost all semigroups of genus g have maximal embedding dimension.
    Used in the proof of Theorem 4.1.

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Pith. "Pith review of Elasticity in Apery sets." pith.science (2026). https://pith.science/paper/7GAWEBXJ

@misc{pith2026190806448,
  author       = {Pith},
  title        = {Pith review of: Elasticity in Apery sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7GAWEBXJ}},
  note         = {Machine review of arXiv:1908.06448}
}
abstract

A numerical semigroup $S$ is an additive subsemigroup of the non-negative integers, containing zero, with finite complement. Its multiplicity $m$ is its smallest nonzero element. The Apery set of $S$ is the set $\text{Ap}(S) = \{n \in S : n-m \notin S\}$. Fixing a numerical semigroup, we ask how many elements of its Apery set have nonunique factorization, and define several new invariants.

Figures

Figures reproduced from arXiv: 1908.06448 by the authors.

Figure 1
Figure 1. Examples of Ap´ery posets. combination of atoms in every possible way. The elasticity of x, denoted ρ(x), is the the largest number of atoms that can be used, divided by the smallest number. Clearly ρ(x) ≥ 1; if equality holds, we say x is half-factorial. Conventionally we say that the unit 0 is half-factorial. In this note, we consider the elasticity function restricted to elements of Ap(S). We write ρ(Ap(S)) for t… view at source ↗

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