{"id":"3477cbe5-cf2f-4411-8a00-6157cf59f3d7","arxiv_id":"1908.06448","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper introduces Apery half-factorial invariants, shows the half-factorial fraction can approach zero and mean elasticity can grow without bound, with one flawed realizability proof.","lead":"This paper defines new invariants that measure how many elements of a numerical semigroup's Apery set have nonunique factorization into atoms. It proves a range of extremal examples, but a central construction has an edge-case error for the case b=1.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.2 fails for b=1, so Corollary 2.3's integer cases have no proof and the two-element classification is not established.","rationale":"The reader's weakest-assumption analysis identifies exactly the load-bearing flaw: Theorem 2.2's construction degenerates when b=1. I agree, and the failure is not merely a missing hypothesis in the proof; the theorem statement is false for b=1. Because Corollary 2.3 is the paper's stated reason that all two-element sets {1,r} are realizable, and because integer r corresponds to b=1 in reduced form, the central classification claim is unsupported for integer r. The paper contains other useful contributions, including the mean-elasticity construction, the AHFF examples, and the asymptotic distribution results, so a conditional accept with mandatory revision is appropriate. The upper-bound half of Theorem 4.2 also needs a rigorous counting argument, but that is secondary to the Corollary 2.3 issue. I would not change the reader's CONDITIONAL verdict: the paper needs correction of the b=1 case or an explicit restriction of Corollary 2.3 to non-integer r, plus a new proof for any claimed integer realization.","tokens_in":5539,"tokens_out":27237,"duration_ms":284486,"concrete_test":"Compute the Apéry set and elasticities for the explicit instance of Theorem 2.2 with a=2, b=1, p=5: S=<3,5>, Ap(S)={0,5,10}, and R(Ap(S))={1}. This directly contradicts the theorem's conclusion R={1,2}. To test the corollary itself, run a complete enumeration of numerical semigroups with multiplicity at most 8 and genus at most 20, computing R(Ap(S)) for each; if no semigroup has R(Ap(S))={1,2}, the corollary's integer case is false rather than merely unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.2 is stated for all a > b >= 1, but its proof only works when b >= 2. In the construction S = <a+b, pa, pb>, the proof requires pa and pb to be atoms. When b=1, pb=p is an atom, but pa is the sum of a copies of p, so pa is not an atom; in fact S = <a+1, p> because pa is redundant. The condition a+b < pb becomes a+1 < p, so the multiplicity is a+1 and Ap(S) = {0, p, 2p, ..., ap}. Every element of this Ap set has unique factorization, so R(Ap(S)) = {1}, not {1, a}. Concretely, for a=2, b=1, p=5, the construction gives S=<3,5> with Ap(S)={0,5,10} and R(Ap(S))={1}, contradicting the theorem's asserted R={1,2}. Since an integer r > 1 has reduced form r = a/1, Corollary 2.3 is exactly the claim that {1, r} is realizable; the only proof offered for those cases is Theorem 2.2, which is false at this boundary. No alternative construction for integer r is supplied. Thus the advertised completeness result for two-element elasticity sets is unsupported precisely for integer values of r.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":5847,"tokens_out":36505,"duration_ms":348568,"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":[{"comment":"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.","section":"Section 2, Theorem 2.2 and Corollary 2.3"},{"comment":"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.","section":"Section 3, Theorem 3.1"}],"minor_comments":[{"comment":"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.","section":"Section 4, Theorem 4.2"},{"comment":"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.","section":"Section 2, Theorem 2.5"},{"comment":"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.","section":"Section 1, Theorem 1.1"}],"recommendation":"major_revision","confidential_remarks":"The paper contains several nice ideas and the main body is mostly direct and readable. However, the b=1 boundary in Theorem 2.2 is a genuine gap in the central two-element classification, and the Apéry-set formula in Theorem 3.1 is stated incorrectly in general. The authors should address both before publication; if the integer-r realization cannot be supplied, the advertised completeness of the two-element result should be weakened explicitly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my take on the Apéry elasticity paper.\n\nThe genuinely new thing is the definition of Apéry-set analogues of elasticity invariants—AHF, AHFF, R(Ap), and MAE—and the observation that the Apéry poset makes Theorem 1.1 immediate. The extremal constructions (AHFF arbitrarily close to 0, MAE unbounded, AHF with embedding dimension 3) are real contributions, and the asymptotic Theorem 4.1 is a nice application of known Kunz polytope results. The paper is clearly written and the examples are helpful.\n\nThe soft spots are real but specific. Theorem 2.2 is stated for all a > b >= 1, but the proof requires pa and pb to be atoms. When b=1, pa is a copies of p, so it is not an atom; the semigroup collapses to <a+1, p> and every element of the Apéry set has a unique factorization, so R(Ap(S))={1} rather than {1,a}. A concrete example is a=2, b=1, p=5, where S=<3,5>, Ap={0,5,10}, and R(Ap)={1}. Since an integer r>1 has reduced form r/1, Corollary 2.3 has no proof exactly for integer elasticities. That is a load-bearing gap for the 'all two-element sets' claim.\n\nThe other issue is smaller: the upper bound in Theorem 4.2 is asserted via 'similar reasoning' rather than an explicit counting argument. The lower bound is fine; the upper bound needs to be checked or spelled out.\n\nOn the whole, the paper earns a serious referee. It is not a rejection case; it is a revise-and-resubmit case. The definitions and most theorems stand, and the b=1 case can presumably be patched either by restricting Theorem 2.2 to b>=2 and finding a separate construction for integer ratios, or by fixing the family. The counting argument in 4.2 should be written out. I would send it out, and I would cite it once the patch is in place.","headline":"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.","tokens_in":6370,"tokens_out":4525,"would_cite":true,"duration_ms":42497,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20M14"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every rational number $r>1$, there is a numerical semigroup whose Apéry set has elasticity set exactly $\\{1,r\\}$.","keywords":["numerical semigroup","Apéry set","elasticity","Apéry half-factorial","half-factorial","embedding dimension","mean Apéry elasticity","factorization"],"falsifier":"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.","tokens_in":5353,"feed_emoji":"🧮","tokens_out":12888,"duration_ms":103544,"temperature":0.7,"pith_summary":"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.","feed_headline":"Every rational ratio above 1 appears as {1,r}","feed_subtitle":"A short construction makes a numerical semigroup's Apéry set realize any chosen rational ratio {1, r}.","key_machinery":"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 $a/b$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the Apéry set itself, the main object whose elasticity sets the paper studies.","marker":"[2]"},{"why":"Supplies the standard definitions of atoms, factorization lengths, and elasticity in monoids.","marker":"[10]"},{"why":"Introduces the set of elasticities for numerical semigroups, which the paper adapts to Apéry sets.","marker":"[5]"},{"why":"Used to show the proportion of Apéry half-factorial semigroups of fixed multiplicity tends to 1.","marker":"[1]"},{"why":"Used to establish a positive lower bound for the proportion of Apéry half-factorial semigroups among all semigroups of a given genus.","marker":"[20]"},{"why":"Provides the motivation for studying mean factorization lengths, on which the mean Apéry elasticity section builds.","marker":"[9]"}],"fun_headline_variants":["Every rational ratio >1 is an Apéry elasticity","Apéry sets realize all rational elasticities above 1","Rational >1: all possible as Apéry elasticity pairs","Apéry elasticity: exactly {1,r} for any rational r>1","Two-element elasticity sets: all {1,r} with r>1 rational"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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 $r=a$.","fun_headline_variants_meta":{"raw":{"variants":["Every rational ratio >1 is an Apéry elasticity","Apéry sets realize all rational elasticities above 1","Rational >1: all possible as Apéry elasticity pairs","Apéry elasticity: exactly {1,r} for any rational r>1","Two-element elasticity sets: all {1,r} with r>1 rational"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000317,"raw_usage":{"total_tokens":1744,"prompt_tokens":849,"completion_tokens":895,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":465,"completion_tokens_details":{"reasoning_tokens":803}},"tokens_in":465,"tokens_out":895,"duration_ms":8922,"temperature":1.0,"reasoning_tokens":803,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:47:46.918032+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Sur les branches superlin´ eaires des courbes alg´ ebriques.C","cited_arxiv_id":null,"evidence_quote":"Provides the Apéry set itself, the main object whose elasticity sets the paper studies."},{"cited_title":"Non-unique factorizations , volume 278 of Pure and Applied Mathematics (Boca Raton)","cited_arxiv_id":null,"evidence_quote":"Supplies the standard definitions of atoms, factorization lengths, and elasticity in monoids."},{"cited_title":"On the set of elasticities in numerical monoids","cited_arxiv_id":null,"evidence_quote":"Introduces the set of elasticities for numerical semigroups, which the paper adapts to Apéry sets."},{"cited_title":"Numerical semigroups and Kunz polytopes","cited_arxiv_id":null,"evidence_quote":"Used to show the proportion of Apéry half-factorial semigroups of fixed multiplicity tends to 1."},{"cited_title":"Fibonacci-like growth of numerical semigroups of a given genus","cited_arxiv_id":null,"evidence_quote":"Used to establish a positive lower bound for the proportion of Apéry half-factorial semigroups among all semigroups of a given genus."},{"cited_title":"Factorization length distribu- tion for affine semigroups I: Numerical semigroups with three generators","cited_arxiv_id":null,"evidence_quote":"Provides the motivation for studying mean factorization lengths, on which the mean Apéry elasticity section builds."}],"review_version":1}