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REVIEW 4 major objections 3 minor 20 references

On primality and atomicity of numerical power monoids

T0 review · 4 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The size of a uniformly random atom with maximum at most n has the same moments, asymptotically, as the number of heads in n fair coin flips.

desk verdict Primality result is solid and clean; the atomic-density side has a genuine proof gap in the main asymptotic, so the paper needs major revision before the moment theorem can be trusted. read the letter →

arxiv 2412.05857 v1 pith:EAIBTDL4 submitted 2024-12-08 math.CO

Anay Aggarwal , Felix Gotti , Susie Lu This is my paper · ORCID
classification math.CO MSC 13A0511Y0506F05
keywords powermonoidrestrictedatomicnumericaldensityprimalelementbinomialmoments
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

The paper establishes two groups of results about power monoids built from finite subsets of numerical monoids under the sumset operation. First, it proves that restricted numerical power monoids contain no primal elements — the algebraic analogue of primes introduced by Cohn — and that among unrestricted numerical power monoids the only one containing a primal element is $\mathcal{P}_{\text{fin}}(\mathbb{N}_0)$, whose unique primal element is $\{1\}$. Second, it gives a precise asymptotic description of the atomic density of $\mathcal{P}_{\text{fin},0}(\mathbb{N}_0)$ by partitioning its atoms into blocks $\mathcal{A}_{n,k}$ of sets of size $k$ with maximum at most $n$. The paper shows that the number of atoms in each block is almost the number of all sets of that size, so the block counts form an almost unimodal sequence, and proves that for every fixed $r$ the $r$-th moment of the size of a uniformly random bounded atom matches the $r$-th moment of a $\text{Bin}(n, 1/2)$ random variable asymptotically. A sympathetic reader would care because this turns a purely algebraic counting question into a probabilistic limit law: the sizes of atoms with bounded maximum behave, in every moment, like the number of heads in $n$ fair coin flips.

What carries the argument

The argument runs on the block decomposition $\mathcal{A}_{n,k}$ and the counts $\alpha_{n,k} = |\mathcal{A}_{n,k}|$. The central identity is the asymptotic expansion $\alpha_{n,\epsilon n} = \binom{n}{\epsilon n - 1}(1 - \gamma_\epsilon(n))$ with $\gamma_\epsilon(n) = O(n^{-1})$, proved by bounding the number of sets $A = B + C$ that decompose with $\min(|B|,|C|) \ge 2$. The bound uses Shitov's coefficient-counting of pairs $(B,C)$ via the coefficients of $(1+x)^{2n}$, together with a Chebyshev concentration lemma that controls the number of random subsets of size $\epsilon n$ admitting a factorization with a small summand. For the moment theorem, the paper writes $\alpha_{n,k} = \binom{n}{k-1} - \beta_{n,k}$, splits the weighted error sum into the ranges $k < \epsilon n$, $\epsilon n \le k \le (1-\epsilon)n$, and $k > (1-\epsilon)n$, and shows each range contributes negligibly: the middle range by the claimed uniform decay of $\gamma_\epsilon(n)$, the tails by Hoeffding's inequality applied to the binomial moments. For the primal-element part, the key device is Lemma 3.4, which characterizes when $\{0,a\}$ divides a set $S$, and a descent argument based on representing intervals as sums of smaller intervals.

What would settle it

For $n$ up to a few thousand, exhaustively list all subsets $A = \{0\} \cup B$ with $B \subseteq \{1,\dots,n\}$ and test atomicity by checking no factorization $A = C + D$ with $\min(|C|,|D|) \ge 2$; then compare the empirical moments of $|A|$ against those of $\text{Bin}(n,1/2)$. If for some fixed $r$ the ratio $\mathbb{E}(X_n^r)/\mathbb{E}(Y_n^r)$ fails to tend to $1$, the main theorem is false. A sharper check targets the uniformity claim: compute $\gamma_\epsilon(n) = 1 - \alpha_{n,\epsilon n}/\binom{n}{\epsilon n - 1}$ over a fine grid of $\epsilon \in [0.1, 0.9]$ and verify that its maximum tends to $0$.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that atomicity in the restricted numerical power monoid $\mathcal{P}_{\text{fin},0}(\mathbb{N}_0)$ is generic: for each fixed $\epsilon \in (0,1)$, the number $\alpha_{n,\epsilon n}$ of atoms of size $\epsilon n$ with maximum at most $n$ satisfies $\alpha_{n,\epsilon n} = \binom{n}{\epsilon n - 1}(1 - \gamma_\epsilon(n))$ with $\gamma_\epsilon(n) = O(n^{-1})$. Since $\binom{n}{\epsilon n - 1}$ counts all subsets of $\{1,\dots,n\}$ of size $\epsilon n - 1$ together with $0$, this says that asymptotically almost every candidate set of the right size is an atom. Summing this identity over all $k$ shows the sequence $(\alpha_{n,k})_k$ is unimodal in the sense that $\alpha_{n,k} < \alpha_{n,k+1}$ for $k < n/2$ and $\alpha_{n,k} > \alpha_{n,k+1}$ for $k > n/2$, with at most $o(n)$ exceptions. The paper then converts this into a moment statement: writing each $\alpha_{n,k}$ as $\binom{n}{k-1} - \beta_{n,k}$, it shows the weighted error terms vanish, yielding $\mathbb{E}(X_n^r)/\mathbb{E}(Y_n^r) \to 1$ for every fixed $r$, where $X_n$ chooses an atom of $\mathcal{P}_{\text{fin},0}(\mathbb{N}_0)$ with maximum at most $n$ uniformly and returns its size, and $Y_n \sim \text{Bin}(n,1/2)$. The paper also proves the structural result that restricted numerical power monoids have no primal elements and the only unrestricted numerical power monoid with a primal element is $\mathcal{P}_{\text{fin}}(\mathbb{N}_0)$, whose unique primal element is $\{1\}$.

Load-bearing premise

The moment theorem depends on the error term $\gamma_\epsilon(n)$ in the block-counting estimate being not only $O(n^{-1})$ for each fixed $\epsilon$ but also uniformly so over $\epsilon \in [0.1, 0.9]$; the proof chooses constants separately for each $\epsilon$ and then takes a maximum over the continuum without supplying a uniformity argument.

Editorial extensions

If this is right

  • For every fixed positive integer $r$, $\mathbb{E}(X_n^r)/\mathbb{E}(Y_n^r)$ tends to $1$, where $Y_n \sim \text{Bin}(n,1/2)$, so the size distribution of a uniformly random atom with maximum at most $n$ is asymptotically binomial in all moments.
  • The sequence $(\alpha_{n,k})_{1\le k \le n+1}$ is increasing up to $k \approx n/2$ and decreasing afterward, for all but $o(n)$ values of $k$; in particular it is unimodal on $n(1-o(1))$ indices.
  • The exact formulas $\alpha_{n,3} = \binom{n}{2} - \lfloor n/2 \rfloor$ and $\alpha_{n,4} = \binom{n}{3} - \tfrac{1}{2}\binom{n}{2} + \tfrac{1}{2}\lfloor n/2 \rfloor$ hold for all $n$, so the first two nontrivial block sizes are known precisely.
  • Restricted numerical power monoids $\mathcal{P}_{\text{fin},0}(N)$ contain no primal elements, and the only numerical power monoid with any primal element is $\mathcal{P}_{\text{fin}}(\mathbb{N}_0)$, whose unique primal element is $\{1\}$.

Reading between the lines

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

  • The moment convergence for every fixed $r$ suggests, though the paper does not prove, full convergence in distribution of $X_n$ to $\text{Bin}(n,1/2)$, for example in total variation distance or in the Kolmogorov metric.
  • The block-counting method may be portable to restricted power monoids of other numerical monoids $N$, where the density result $q_n \to 1$ is already known; a natural test is whether the limiting size distribution of atoms with maximum at most $n$ is again binomial, possibly with parameter depending on $n$ and the Frobenius number of $N$.
  • The simultaneous absence of absolute irreducibles and primal elements in restricted numerical power monoids suggests these monoids are very far from being pre-Schreier; one might ask whether they contain primary elements, a weaker notion that the paper does not address.
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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

4 major / 3 minor

Summary. The paper studies arithmetic properties of power monoids of numerical monoids. In Section 3 it proves that restricted numerical power monoids contain no primal elements, and that among unrestricted numerical power monoids only P_fin(N_0) contains a primal element, namely {1}. In Sections 4 and 5 it analyzes the atomic density of P_fin,0(N_0). The authors define the block sizes alpha_{n,k} of atoms of size k with maximum at most n, prove an upper bound in Theorem 4.3, an asymptotic lower bound for 0.111n < k < 0.5n in Theorem 4.6, and an asymptotic formula alpha_{n,epsilon n} = (1-o(1)) binom(n,epsilon n-1) for each fixed epsilon in (0,1) in Theorem 5.2. From this they derive almost-unimodality of the sequences (alpha_{n,k})_k and the main moment theorem: for the random variable X_n giving the size of a uniformly random atom with maximum at most n, E(X_n^r)/E(Y_n^r) -> 1, where Y_n ~ Bin(n,1/2).

Significance. If the main results are correct, the moment theorem is an elegant and surprisingly clean answer to a natural atomic-density question: the size of a uniformly random bounded atom in P_fin,0(N_0) is asymptotically binomial with parameters n and 1/2. This substantially refines the density results of Shitov and of Bienvenu-Geroldinger, and the almost-unimodality corollary is a valuable byproduct. The primal-element theorem is a natural analogue of Bienvenu-Geroldinger's absolute-irreducibility theorem and appears plausible. The paper is clearly organized and uses appropriate tools (Stirling estimates, entropy bounds, and concentration inequalities). However, the central probabilistic estimate in Theorem 5.2 has a missing union bound, and Theorem 5.4 relies on an unproved uniformity assertion; these gaps prevent the main asymptotic claims from being accepted as proved.

major comments (4)
  1. [§5.1, Theorem 5.2, Case 2] The estimate P(A in S_2) = O(n^{-1}) is not justified. The proof fixes a small factor B of size at most r with max B <= omega n, represents it by gaps b_1,...,b_r, and applies Lemma 5.1 to conclude that, for that fixed B, there are binom(n,epsilon n-1) O(n^{-1}) admissible A. But S_2 is the union over all such B, and the number of choices of B is Omega(n^r). A union bound therefore gives binom(n,epsilon n-1) O(n^{r-1}), not O(n^{-1}). Since r is chosen later to satisfy gamma > 2 epsilon^r, the factor n^{r-1} is not o(1). Consequently the asymptotic formula for alpha_{n,epsilon n} is not proved, and the later applications in Corollary 5.3 and Theorem 5.4 inherit this gap.
  2. [§5.2, Theorem 5.4] The proof of Theorem 5.4 assumes that the convergence gamma_epsilon(n) -> 0 is uniform over epsilon in the interval [1/10,9/10]. The proof defines gamma(n) = max{gamma_epsilon(n) : epsilon in [1/10,9/10]} and asserts gamma(n) -> 0. Pointwise convergence for each fixed epsilon does not imply convergence of the supremum over a continuum, and the proof of Theorem 5.2 supplies constants that may depend on epsilon. A uniformity argument, or a strengthened version of Theorem 5.2 with error bounds valid uniformly on compact subintervals of (0,1), is needed. As written, the conclusion s_2(n) -> 0 is not established.
  3. [§5.2, proof of Theorem 5.4, first display] The algebraic identity used in the first display is incorrect. With j = k-1, the left side sum_{k=1}^{n+1} k^r binom(n,k-1) equals sum_{j=0}^n (j+1)^r binom(n,j) = 2^n E[(Y_n+1)^r], not 2 E[Y_n^r] (or the expression implied by the display). The proof should replace E[Y_n^r] by E[(Y_n+1)^r] and then use E[(Y_n+1)^r]/E[Y_n^r] -> 1 for fixed r. As it stands, the displayed equality is false and the subsequent expression for the limit needs correction.
  4. [§4.2, Theorem 4.3] The lower-bound construction for good sets is invalid when n is odd. In that case both S_0 and S_1 have cardinality 3, while the proof treats S_1,...,S_{floor(n/2)-1} as if each contributes exactly two elements. If S_1 is among the chosen indices, the constructed set has size k+1 rather than k, for both parities of k. Thus the claimed lower bound of binom(floor(n/2)-1, floor(k/2)-1) non-atoms of size k is not established. The theorem may still be true, but the supplied construction does not prove it.
minor comments (3)
  1. [§5.2, proof of Theorem 5.4] In the first display, the notation alpha_{n,r} should be alpha_{n,k}; the subscript is constant but should range over k.
  2. [§5.1, proof of Theorem 5.2, Case 2] In the subcase max B >= omega n, the inclusion C subset [0,(1-omega)n/r] appears to be a typo; the correct bound is C subset [0,(1-omega)n], and the subsequent estimate should be adjusted accordingly.
  3. [§5.2] The random variable is defined as X_n, but the probability mass function is written using P(X_{N,n} = k); the notation should be standardized.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main results are proved directly on top of independent external results (Bienvenu–Geroldinger, Shitov) and standard binomial estimates, with no equation reducing to its own input.

full rationale

The paper's derivation chain is self-contained and not circular. The primal-element results in Theorem 3.5 are proved directly by contradiction and structural arguments about divisors of discrete intervals and sets of the form {0,max P}, using only the cited definition of primality from Cohn and the prior theorem [3, Theorem 4.11] as motivation, not as a substitute for the proof. The atomic-density thread uses (4.3), α_n = 2^n(1-o(1)), as an external input from Bienvenu–Geroldinger [3] and Shitov [16]; this is independent support, and the paper does not derive it from its own moment theorem. The central quantity β_{n,k}, defined by α_{n,k} = binom(n,k-1) - β_{n,k}, is not a fitted parameter: Theorem 5.2 proves directly, via the partition into S1, S2, S3 and Lemma 5.1, that β_{n,εn}/binom(n,εn-1) = O(n^{-1}). Theorem 5.4 then rewrites E(X_n^r) algebraically, and the conclusion that E(X_n^r)/E(Y_n^r) → 1 is exactly the statement that the error terms s1(n), s2(n), s3(n) vanish; this is a derived asymptotic comparison, not a definitional identity. Constants such as 0.111 and 2ε^r arise from an entropy inequality and from the structural cases, not from fitting to the target moments. The reader's uniformity concern about γ_ε(n) over ε in [0.1,0.9] and the skeptic's counting concern about the number of possible small factors B in Theorem 5.2 are legitimate potential proof gaps, but they are correctness risks, not circularity: the theorem is neither assumed nor equivalent to its input. No self-citation is load-bearing: the cited previous work [8] is background on Puiseux monoids, and no uniqueness theorem imported from the same authors is used to force the conclusion. Therefore the paper merits a circularity score of 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central results rest on standard probabilistic inequalities and on two external bounds from Shitov [16] and Bienvenu-Geroldinger [3]; no new entities or fitted parameters are introduced.

assumptions (4)
  • domain assumption Shitov's bound [16, Lemma 2.2] on the number of pairs (A,B) with |A+B|=k and max A + max B <= n
    Used as the starting point for the non-atom count in Theorem 4.6; the lemma is cited but not restated or proved in the paper.
  • domain assumption The asymptotic alpha_n = 2^n (1 - o(1)) from [3, Theorem 6.1.1]
    Invoked in (4.3) and in the moment computation of Theorem 5.4 to normalize E(X_n^r).
  • standard math Hoeffding's inequality for binomial tail bounds
    Used in Theorem 5.4 to show the contributions of k <= epsilon n and k >= (1-epsilon)n vanish exponentially; the inequality is invoked without statement or citation.
  • standard math Stirling's approximation and Chebyshev's inequality
    Used throughout Sections 4 and 5 for binomial asymptotics and concentration of the random variable A_{b,z} in Lemma 5.1.

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Pith. "Pith review of On primality and atomicity of numerical power monoids." pith.science (2026). https://pith.science/paper/EAIBTDL4

@misc{pith2026241205857,
  author       = {Pith},
  title        = {Pith review of: On primality and atomicity of numerical power monoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EAIBTDL4}},
  note         = {Machine review of arXiv:2412.05857}
}
abstract

In the first part of this paper, we establish a variation of a recent result by Bienvenu and Geroldinger on the (almost) non-existence of absolute irreducibles in (restricted) power monoids of numerical monoids: we argue the (almost) non-existence of primal elements in the same class of power monoids. The second part of this paper, devoted to the study of the atomic density of $\mathcal{P}_{\text{fin}, 0}(\mathbb{N}_0)$, is motivated by work of Shitov, a recent paper by Bienvenu and Geroldinger, and some questions pointed out by Geroldinger and Tringali. In the same, we study atomic density through the lens of the natural partition $\{ \mathcal{A}_{n,k} : k \in \mathbb{N}_0\}$ of $\mathcal{A}_n$, the set of atoms of $\mathcal{P}_{\text{fin}, 0}(\mathbb{N}_0)$ with maximum at most $n$: \[ \mathcal{A}_{n,k} = \{A \in \mathcal{A} : \max A \le n \text{ and } |A| = k\} \] for all $n,k \in \mathbb{N}$, where $\mathcal{A}$ is the set of atoms of $\mathcal{P}_{\text{fin}, 0}(\mathbb{N}_0)$. We pay special attention to the sequence $(\alpha_{n,k})_{n,k \ge 1}$, where $\alpha_{n,k}$ denote the size of the block $\mathcal{A}_{n,k}$. First, we establish some bounds and provide some asymptotic results for $(\alpha_{n,k})_{n,k \ge 1}$. Then, we take some probabilistic approach to argue that, for each $n \in \mathbb{N}$, the sequence $(\alpha_{n,k})_{k \ge 1}$ is almost unimodal. Finally, for each $n \in \mathbb{N}$, we consider the random variable $X_n : \mathcal{A}_n \to \mathbb{N}_0$ defined by the assignments $X_n : A \mapsto |A|$, whose probability mass function is $\mathbb{P}(X_n=k) = \alpha_{n,k}/| \mathcal{A}_n|$. We conclude proving that, for each $m \in \mathbb{N}$, the sequence of moments $(\mathbb{E}(X_n^m))_{n \ge 1}$ behaves asymptotically as that of a sequence $(\mathbb{E}(Y_n^m))_{n \ge 1}$, where $Y_n$ is a binomially distributed random variable with parameters $n$ and $\frac12$.

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