{"id":"ddecc103-c1ba-4df9-900d-d717dc5f26e1","arxiv_id":"2601.19629","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"In shifted numerical semigroups, being nearly Gorenstein or almost symmetric eventually repeats with period r_k, via a corrected pseudo-Frobenius bijection.","lead":"The authors study numerical semigroups obtained by shifting a fixed generator list by n, and prove that the nearly Gorenstein and almost symmetric properties are eventually periodic in the shift. The proof rests on a corrected bijection between pseudo-Frobenius elements that fixes an error in the prior literature and yields explicit Frobenius-number formulas.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reversed inequality in Step 2, Case A of Theorem 2.1 leaves the corrected PF bijection unproved as written.","rationale":"The reader's weakest assumption concerned Remark 1.3(1), the extension of Barron–O'Neill–Pelayo to non-minimal generator lists. That is a reasonable premise, but I found a more immediate issue inside the proof of Theorem 2.1 itself: a reversed inequality in Step 2, Case A. This is not an external assumption but an internal inconsistency in the proof of the corrected pseudo-Frobenius bijection, on which the periodicity theorems rest. The likely explanation is a typographical slip, and the theorem statement appears to be supported by examples and GAP computations, so I am not recommending rejection. However, as printed, the proof does not establish the bijection, and the subsequent formulas in Corollaries 2.6/2.8 and Theorem 3.4 inherit that gap. A conditional acceptance requiring correction of this inequality (or an explicit justification of the printed direction) is appropriate. No other issue in the main argument rose to this level.","tokens_in":21253,"tokens_out":28712,"duration_ms":266659,"concrete_test":"Re-derive Case A of Step 2 directly: with y = i+dr_k+r_j+(m(i+r_j)+d)(n+r_k) ∈ Ap(M_{n+r_k}, n+r_k) and z = y + (m(i)-m(i+r_k? no, m(i+r_j)))(n+r_k), use y-(n+r_k) ∉ M_{n+r_k} to show z ∈ M_{n+r_k} iff m(i) ≥ m(i+r_j). Compare this with the printed m(i) ≤ m(i+r_j). If they differ, then Theorem 2.1's proof is invalid as written. Optionally, run the same check in GAP/NumericalSgps for a redundant-generator family such as M_n = <n, n+3, n+4, n+6> with n large, enumerating PF(M_n) and PF(M_{n+r_k}) to confirm the corrected inequality direction.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Theorem 2.1, Step 2, Case A, the paper asserts that i+dr_k ∈ P_{n+r_k} is equivalent to m(i) ≤ m(i+r_j). This is the reverse of what (7) plus Apéry maximality give. Let n' = n+r_k and set y = i+dr_k+r_j + (m(i+r_j)+d)n'. By Theorem 1.2, y ∈ Ap(M_{n'}, n'). The membership in (7) is z = i+dr_k+r_j + (m(i)+d)n' = y + (m(i)-m(i+r_j))n'. Since y-n' ∉ M_{n'}, we have z ∈ M_{n'} iff m(i) ≥ m(i+r_j), i.e. m(i+r_j) ≤ m(i) — not m(i) ≤ m(i+r_j). As printed, (6) gives m(i+r_j) ≤ m(i), while the text says the image condition is m(i) ≤ m(i+r_j); the two are not equivalent, so the claimed bijection is not established. Because Theorem 2.1 drives Corollaries 2.6/2.8 and Theorem 3.4, this is load-bearing. The examples and the step's conclusion suggest it is a typographical slip, but the printed proof is internally inconsistent and needs correction.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies shifted families of numerical semigroups M_n = ⟨n, n+r_1, ..., n+r_k⟩. Its principal contributions are: (i) a corrected bijection φ_n: PF(M_n) → PF(M_{n+r_k}) (Theorem 2.1), replacing an incorrect map in O'Neill–Pelayo; (ii) explicit formulas for pseudo-Frobenius and Frobenius numbers under iteration φ_n^λ (Corollaries 2.6, 2.8, 2.11), with order preservation for n ≥ r_k^4 (Proposition 2.10); (iii) a propagation theorem for the nearly Gorenstein property: for n > N, if M_n is nearly Gorenstein then M_{n+λ r_k} is nearly Gorenstein, and an NG-vector transforms as (φ_n^λ(f_0), ..., φ_n^λ(f_k)) (Theorem 3.4); (iv) the corresponding statement for almost symmetric semigroups (Corollary 4.1), together with a result excluding even type for n ≥ r_k^4 (Proposition 4.3); and (v) a counterexample showing the residue is not eventually periodic (Example 5.2), plus results on canonical reductions and reduced type (§6). Explicit lower bounds are provided and many examples are checked with GAP.","tokens_in":21615,"tokens_out":15416,"duration_ms":151951,"significance":"If the issues identified below are fixed, the paper settles the asymptotic periodicity of nearly Gorenstein and almost symmetric properties for shifted numerical semigroup families in full generality. The corrected pseudo-Frobenius bijection is a genuine contribution, since the earlier construction in the literature is wrong. The explicit transformation of NG-vectors is stronger than a qualitative periodicity statement and is likely to be useful in further work. The residue counterexample (Example 5.2) answers a natural question negatively. The paper is also commendably transparent: it gives explicit bounds, works out several examples, and its numerical claims are independently verified with GAP. The universal bound N < r_k^4 in Remark 3.3 is especially useful because the technical threshold N is defined through the PF set.","major_comments":[{"comment":"The printed inequality in Case A is reversed. The text first correctly notes that i+r_j+m(i)n ∈ M_n iff m(i+r_j) ≤ m(i). It then states that i+dr_k ∈ P_{n+r_k} is equivalent to m(i) ≤ m(i+r_j). From (7), the relevant Apéry element is w = i+dr_k+r_j+(m(i+r_j)+d)(n+r_k), so i+dr_k+r_j+(m(i)+d)(n+r_k) = w + (m(i)-m(i+r_j))(n+r_k) lies in M_{n+r_k} iff m(i) ≥ m(i+r_j), i.e. m(i+r_j) ≤ m(i). The printed inequality is therefore the reverse of the condition actually obtained. Since this equivalence is the main step proving the bijection, the proof as written is internally inconsistent. The examples and the surrounding argument strongly suggest a typographical slip, but the statement must be corrected before the theorem can be accepted.","section":"§2, Theorem 2.1, Step 2, Case A (equations (6)–(7))"},{"comment":"Corollaries 4.1 and 6.1 are stated for n > N, but their proofs invoke Proposition 2.11, which is proved only for n ≥ r_k^4. Remark 3.3 gives N < r_k^4, so the hypothesis n > N does not imply n ≥ r_k^4. Proposition 2.11 is needed to identify φ_n^λ(F(M_n)) with the Frobenius number of M_{n+λr_k}; without order preservation of PF numbers the proofs of the two corollaries do not cover their stated range. Example 6.2 illustrates the gap: it applies Corollary 6.1 with n = 26 and r_k = 4, while r_k^4 = 256. Please either extend Proposition 2.10 and Proposition 2.11 to the range n > N, or state Corollaries 4.1 and 6.1 under the stronger hypothesis n ≥ r_k^4.","section":"§3 Definition 3.1 with §4 Corollary 4.1 and §6.1 Corollary 6.1"},{"comment":"The identity m(i+r_k) = m(i)+1 for i > r_{k-1}r_k is quoted from [2, Theorem 4.3], which is stated for minimal generator lists, and the authors assert without proof that the proof works for their possibly redundant list r_1, ..., r_k. This is not a purely formal extension because m(i) in this paper is defined as the minimum length of factorizations with respect to the given list r_1,...,r_k, not with respect to the minimal generators of S; the paper itself notes the distinction in the example S = ⟨3,4,6⟩. This identity is used in essential steps, including Step 3 of Theorem 2.1, Corollaries 2.6, 2.8, 2.9, Remark 3.2, and Theorem 3.4. Please include a proof or a precise lemma stating the extension and its hypotheses.","section":"§1, Remark 1.3(1)"}],"minor_comments":[{"comment":"The title appears as “NEARL Y GORENSTEIN...” in the full text; it should read “NEARLY GORENSTEIN...”.","section":"Title page"},{"comment":"The threshold N is defined using P'_n and P''_n, which in turn depend on PF(M_n). This is legitimate, but it would help to state explicitly that N is not a closed-form constant in the r_i alone and that Remark 3.3 provides the uniform alternative n ≥ r_k^4.","section":"Definition 3.1"},{"comment":"The equivalence “i ∈ P_n iff i+(m(i)-1)n ∈ PF(M_n)” is used repeatedly. A short justification via the maximality of Apéry elements would improve readability.","section":"§2, Theorem 2.1, Step 1"},{"comment":"In the shifted family data, “r1 = 1, r2 = 3, r4, d = 1” appears to contain a typo: it should be r3 = 4.","section":"§4, Example 4.2"}],"recommendation":"major_revision","confidential_remarks":"The central results are valuable and appear to be correct, but the manuscript currently has two load-bearing issues that need work: a reversed inequality in the proof of Theorem 2.1 and a mismatch between the hypotheses of Proposition 2.11 and the thresholds used in Corollaries 4.1 and 6.1. A third issue, the unproved extension in Remark 1.3(1), also affects several proofs. None of these appears to be fatal; they are repairable with a focused revision. The corrected bijection and propagation theorem are likely to be cited, so I would be comfortable recommending acceptance after the authors address these points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis is a genuinely useful paper. It proves eventual periodicity of nearly Gorenstein and almost symmetric properties for shifted numerical semigroups at arbitrary k, corrects a wrong pseudo-Frobenius bijection from O'Neill–Pelayo, and gives explicit Frobenius/pseudo-Frobenius formulas. The k=2 case was known; the arbitrary-k result is new, and Example 5.2, showing that the residue is not eventually periodic, is a nice counterexample with a clean quasilinearity question attached.\n\nThe paper does a lot of things well. The proof of Theorem 3.4 is detailed and the four-case propagation argument is careful. The examples are reproducible with GAP, and the bounds on n are explicit. The correction of the O'Neill–Pelayo map is important on its own; using the Apéry-set formulation to define the bijection is the right move.\n\nThat said, there is a concrete flaw in the proof of Theorem 2.1. In Step 2, first bullet, the paper claims that membership of i+dr_k in P_{n+r_k} is equivalent to m(i) ≤ m(i+r_j). But the same reasoning that gave the opposite inequality for i∈P_n applies here: from (7) and the Apéry element i+dr_k+r_j + (m(i+r_j)+d)(n+r_k), adding (m(i)-m(i+r_j))(n+r_k) shows the condition is m(i+r_j) ≤ m(i). As printed, the two equivalences are inconsistent, so the proof does not establish the bijection. This looks like a typo—the corrected inequality matches the first bullet and the GAP examples—but it is load-bearing because Theorem 2.1 drives Corollaries 2.6/2.8 and Theorem 3.4.\n\nSecond, Remark 1.3(1) extends Barron–O'Neill–Pelayo [2, Theorem 4.3] from minimal to non-minimal generator lists with a one-line 'their proof works' assertion. That is likely true, but it is not shown, and the m(i+dr_k)=m(i)+d identity is used throughout.\n\nNeither issue sinks the paper. The main theorems are plausible and well-supported by examples. But the proof of Theorem 2.1 needs to be fixed, and the cited extension needs either a proof or a careful reference.\n\nFor you: worth reading if you work on numerical semigroups or semigroup rings. Send it to a serious referee, with a request to fix the inequality and the Remark 1.3 extension. I would not desk-reject.","headline":"Useful paper on shifted numerical semigroups, but Theorem 2.1's proof has a reversed inequality that must be fixed before publication.","tokens_in":22066,"tokens_out":7538,"would_cite":true,"duration_ms":68545,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20M25","20M14","13H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For sufficiently large n, if a shifted numerical semigroup M_n is nearly Gorenstein or almost symmetric, then every later member M_{n+λ r_k} inherits the property.","keywords":["numerical semigroup","shifted family","pseudo-Frobenius numbers","nearly Gorenstein","almost symmetric","Apery set","trace ideal","reduced type"],"falsifier":"Take a shifted family with redundant generators—for instance S=⟨3,4,6⟩—and search for i>r_{k−1}r_k=24 with m(i+6)≠m(i)+1; or compute PF(M_n) and PF(M_{n+r_k}) for some n>N and compare them with the paper's φ_n formula. A single mismatch would falsify the corrected bijection and with it the propagation theorem.","tokens_in":21197,"feed_emoji":"🔁","tokens_out":8749,"duration_ms":80456,"temperature":0.7,"pith_summary":"For integers 0<r_1<...<r_k, consider the shifted family M_n=⟨n,n+r_1,...,n+r_k⟩. The paper proves that for all sufficiently large n, if M_n is nearly Gorenstein, then M_{n+λ r_k} is nearly Gorenstein for every λ≥0, and the same eventual periodicity holds for almost symmetric semigroups. The engine is a corrected bijection between the pseudo-Frobenius numbers of M_n and M_{n+r_k}, replacing a flawed construction in the earlier literature, together with explicit formulas that translate the Frobenius and pseudo-Frobenius data of M_n into those of M_{n+λ r_k}. This matters because nearly Gorenstein and almost symmetric properties were known to be eventually periodic only in special low-generator cases, and here they become a general asymptotic phenomenon with an explicit threshold N that is always smaller than r_k^4.","feed_headline":"Large n nearly Gorenstein forces all later shifts nearly Gorenstein","feed_subtitle":"For every shifted family, once M_n is nearly Gorenstein, so are M_{n+r_k}, M_{n+2r_k}, … — with explicit bounds.","key_machinery":"The load-bearing mechanism is the corrected bijection φ_n: PF(M_n)→PF(M_{n+r_k}), induced by a map ψ_n on the Apéry-set representatives P_n={i∈Ap(S,dn) | i≡f mod n for some f∈PF(M_n)}. The map splits P_n into a low part P'_n={i<dn−r_k} where the representative is unchanged, and a high part P''_n where i moves to i+dr_k, with d=gcd(r_1,...,r_k). The integer m(i)—the minimum length of a factorization of i using the full list r_1,...,r_k, even when that list is redundant—records how the shift changes each pseudo-Frobenius element. The identity m(i+r_k)=m(i)+1 for i>r_{k−1}r_k, combined with this two-part dynamics, gives the explicit shift formulas and transfers an NG-vector from M_n to M_{n+r_k","core_discovery":"The central claim is that 'nearly Gorenstein' and 'almost symmetric' are eventually periodic along every shifted family: there is an explicit integer N (always below r_k^4) such that, for n>N, if M_n is nearly Gorenstein then M_{n+λ r_k} is nearly Gorenstein for every λ≥0, and similarly for almost symmetric semigroups. The theorem is constructive. If (f_0,...,f_k) is a nearly Gorenstein vector for M_n—meaning each f_i is pseudo-Frobenius and h_i+f_i−f∈M_n for every generator h_i and every f∈PF(M_n)—then the image of that vector under the paper's corrected bijection φ_n^λ is an NG-vector for M_{n+λ r_k}. The proof also corrects an erroneous bijection in the literature between PF(M_n) and PF(M","pith_inferences":["Because the proof transports an actual NG-vector rather than just the property, the same transport may apply to other properties defined by a vector certificate over PF(M_n), such as 'positioned' numerical semigroups (canonical reduction).","Any earlier result that relied on the flawed bijection between PF(M_n) and PF(M_{n+r_k}) should be re-checked with the corrected map; some of those statements may need new proofs or may become false.","The counterexample to periodicity of the residue suggests a stronger asymptotic law than the paper proves: one could computationally test, across many r_1,...,r_k, whether res(M_{n+λ r_k}) is always eventually linear in λ, not merely bounded by a polynomial.","Since N<r_k^4 always but examples show much smaller thresholds, a natural next step is to sharpen the uniform bound or to identify the exact onset of periodicity."],"forward_implications":["Eventual periodicity: for any fixed r_1,...,r_k, the tail of the family is completely determined once n exceeds N; if M_n has one of the two properties, every later M_{n+λ r_k} has it.","Explicit formulas: a pseudo-Frobenius number f=i+(m(i)−1)n−n of M_n becomes f+(m(i)−1)λ r_k when i<dn−r_k, and f+(m(i)+(λ+1)d−1)λ r_k+λdn when i≥dn−r_k.","The Frobenius number always comes from the high part P''_n for n≥r_k^4, giving a closed formula F(M_{n+λ r_k})=F(M_n)+(m(i)+(λ+1)d−1)λ r_k+λdn.","The reduced type of k[[M_n]] (number of pseudo-Frobenius numbers inside [F(M_n)−n,F(M_n)]) is eventually constant along the shifted family, with a counting formula in terms of P''_n and the lengths m(j).","The residue is not eventually periodic: for the family M_n=⟨n,n+2,n+3,n+7⟩, res(M_{63+7λ})=λ+9, growing linearly; the paper asks whether the residue is eventually linear (quasilinear) in general."],"fun_headline_variants":["Shifted semigroups: near Gorenstein persists for all later shifts","Once nearly Gorenstein, always nearly Gorenstein in shifted families","Eventual periodicity of near Gorenstein in shifted numerical semigroups","Shifted families: N-G implies all future shifts N-G","Correcting a bijection: near Gorenstein propagates along shifts"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is Remark 1.3(1), which extends the lemma m(i+r_k)=m(i)+1 for i>r_{k−1}r_k from the case of minimal generators to arbitrary (possibly redundant) generator lists by stating that the known proof 'works also in our context' without reproducing it; the shift formulas, the corrected bijection, and Theorem 3.4 all rest on this extension.","fun_headline_variants_meta":{"raw":{"variants":["Shifted semigroups: near Gorenstein persists for all later shifts","Once nearly Gorenstein, always nearly Gorenstein in shifted families","Eventual periodicity of near Gorenstein in shifted numerical semigroups","Shifted families: N-G implies all future shifts N-G","Correcting a bijection: near Gorenstein propagates along shifts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000157,"raw_usage":{"total_tokens":1040,"prompt_tokens":711,"completion_tokens":329,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":455,"completion_tokens_details":{"reasoning_tokens":243}},"tokens_in":455,"tokens_out":329,"duration_ms":4021,"temperature":1.0,"reasoning_tokens":243,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T07:34:40.411583+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a shifted family with redundant generators—for instance S=⟨3,4,6⟩—and search for i>r_{k−1}r_k=24 with m(i+6)≠m(i)+1; or compute PF(M_n) and PF(M_{n+r_k}) for some n>N and compare them with the paper's φ_n formula. A single mismatch would falsify the corrected bijection and with it the propagation theorem.","supporting_citations":[],"review_version":1}