{"id":"4566df9e-2602-409b-a5e3-f7386c6dd0c6","arxiv_id":"2607.11093","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every t≥2 an extremal set A of size t produces a far-flung Gorenstein semigroup S(n(A),A) of type t attaining the multiplicity bound n(t); for t≥5 the same family also satisfies res(S)>l(S).","lead":"The paper constructs, for every type t≥2, a far-flung Gorenstein numerical semigroup attaining the Rohrbach upper bound on multiplicity. The same method yields counterexamples, for every t≥5, to the conjecture that residue never exceeds the complementary length of the canonical ideal.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The Reader correctly isolates Lemma 3(2) as the only non-trivial arithmetic step needed for maximal reduced type. That step is proved elementarily inside the manuscript (no appeal to tables or external theorems), and once it holds the rest of Theorem 2 follows by direct application of the earlier characterizations (Lemma 1, Proposition 2, Corollary 4). The finite checks for t=5…8 in Theorem 3 are likewise fully explicit and reproducible. No circularity, free parameters or unstated hypotheses remain. The Reader’s ACCEPT / HIGH / low-risk assessment is therefore unchanged.","tokens_in":11587,"tokens_out":527,"duration_ms":5662,"concrete_test":"Pick any known extremal set of size t=9 (e.g., from OEIS A123509 or Kohonen–Corander tables), form S=S(n(A),A), compute PF(S) and rPF(S) with NumericalSgps, and verify |PF(S)|=9 together with m(S)=n(9). Agreement confirms the general argument; any discrepancy would isolate a concrete failure of Lemma 3 or Corollary 4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central existence claim (Theorem 2) rests on an elementary, self-contained chain: extremal A of size t (0,1∈A by Lemma 3(1)), S:=S(n(A),A) satisfies m=F+1-m wait no F=2m-1, rPF={2m-1-a:a∈A}, and the two inequalities max A+1≤n(A) and max A+n(A)<2m become automatic once m=n(A) and Lemma 3(2) hold. Lemma 3(2) itself is proved by a short contradiction (if n(A)≤max A then the truncated set B would give a larger n-value, contradicting extremality). All subsequent steps (Corollary 4(3)\to Theorem 2) are pure set arithmetic inside the numerical-semigroup definitions; no external black-box results, hidden bounds, or computer-dependent steps for general t appear. The same construction also supplies the counter-examples of Section 4 without additional hypotheses. Consequently the load-bearing inequality flagged by the Reader is already secured inside the paper and does not threaten the claim.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper answers Question 5.5 of Herzog–Kumashiro–Stamate by constructing, for every t≥2, a far-flung Gorenstein numerical semigroup S of type t that attains the multiplicity bound m(S)=n(t) coming from the Rohrbach problem. The construction is the family S(m,A)=Δ(m)\\{(2m-1)-a:a∈A} with m=n(A) and A any extremal set of cardinality t. After establishing a characterization of the far-flung Gorenstein property via rPF(S) (Lemma 1) and the basic arithmetic of S(m,A) (Proposition 2), the authors prove that the two numerical conditions max A+1≤n(A) and max A+n(A)<2m force rPF(S)=PF(S) (Theorem 1 and Corollary 4); these conditions hold automatically for extremal A by Lemma 3, yielding Theorem 2. The same family is then used to produce, for every t≥5, a far-flung Gorenstein semigroup with res(S)>l(S), thereby answering the residual case of the Herzog–Hibi–Stamate question and supplying an explicit counter-example for type 4.","tokens_in":11902,"tokens_out":825,"duration_ms":6925,"significance":"The result settles a clean existence question that links the algebraic notion of far-flung Gorenstein rings to a classical problem in additive number theory. The construction is elementary, completely explicit, and works uniformly for all t≥2; it simultaneously yields an infinite family of counter-examples to the inequality res≤l. Because the proofs rely only on the definitions of n(A), rPF and the conductor, the argument is self-contained and immediately usable by other researchers working on numerical semigroups or one-dimensional Cohen–Macaulay rings.","major_comments":[],"minor_comments":[{"comment":"In the statement of Lemma 3(2) the inequality is written max A+2≤n(A); the proof actually shows the slightly stronger max A+1≤n(A). A one-line remark clarifying the relation would avoid a momentary pause for the reader.","section":null},{"comment":"The four explicit sets A5–A8 in the proof of Theorem 3 are asserted to satisfy 3(t-1)<n(At) and max At+2≤n(At). While these inequalities are elementary to check by hand, a short parenthetical verification (or a reference to the known values of n(t) for t≤8) would make the argument fully self-contained without external lookup.","section":null},{"comment":"Example 5 claims that the listed generators give res(S)=8>7=l(S) and s(S)=t(S)=4. Since the verification is described as “we can check,” it would be helpful to record the sets K(S)\\S and S\\tr(K(S)) explicitly (they are already written in the text) and to note that the computation was performed with NumericalSgps, as is done for other examples.","section":null},{"comment":"A few typographical inconsistencies appear: “F ar-flung” in the running head, “SUGA W ARA” with an extra space, and the occasional missing space after a period. These are purely cosmetic.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is short, self-contained and settles two open questions with a single elementary construction. It is a clear accept for a specialized algebra journal; no further refereeing rounds appear necessary beyond ordinary copy-editing."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper settles two concrete open questions in the far-flung Gorenstein literature with a single explicit construction. For every t≥2 and every extremal set A of size t, the semigroup S(n(A),A) is far-flung Gorenstein of type t and multiplicity exactly n(t). The same family also yields res(S)>l(S) for every t≥5, answering the residue-length comparison that had only a single type-5 counterexample before.\n\nWhat is new is the uniform method. The author defines S(m,A)=Δ(m) minus the points (2m-1)-a for a in A, proves the basic arithmetic (multiplicity m, Frobenius 2m-1, rPF exactly the translated A), then shows that the two inequalities max A+1≤n(A) and max A+n(A)<2m force both the far-flung property and maximal reduced type. When m is set to n(A) those inequalities become automatic by a short contradiction argument (Lemma 3). Everything is elementary set arithmetic; no black-box theorems or free parameters appear. The small-t sets A5–A8 are written out explicitly so the computer checks are reproducible by hand.\n\nThe soft spots are minor. The paper relies on the classical existence of extremal sets of every size (known up to 25 and conjecturally beyond), but that is external data, not a gap in the argument. The counterexample for type 4 is not far-flung, so it answers a slightly different question; that is stated clearly. Citation pattern is tight and appropriate.\n\nThis is for people already working on numerical semigroups, trace ideals, or the Rohrbach problem. Anyone who has looked at the HKS papers will get immediate value. The proofs are short enough for a reading group and solid enough that a serious editor should send it to referees without hesitation. I would cite the construction myself the next time I need an infinite family of far-flung examples with controlled type and multiplicity.","headline":"Clean, elementary existence proof that every Rohrbach number n(t) is attained by a far-flung Gorenstein numerical semigroup of type t, plus uniform counterexamples for the residue-versus-length inequality when t≥5.","tokens_in":12481,"tokens_out":554,"would_cite":true,"duration_ms":5132,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13H10","20M14","20M25"],"pacs":[],"model":"grok-4.5","headline":"For every type t≥2, a far-flung Gorenstein numerical semigroup attains the multiplicity upper bound n(t) coming from the Rohrbach problem.","keywords":["numerical semigroup","far-flung Gorenstein","reduced type","maximal reduced type","Rohrbach problem","multiplicity","trace ideal","canonical module"],"falsifier":"Exhibit a single extremal set $A$ of size $t \\ge 2$ for which $\\max A + 2 > n(A)$; then $S(n(A),A)$ fails to have maximal reduced type and the multiplicity-attaining claim for that $t$ collapses.","tokens_in":12500,"feed_emoji":"∑","tokens_out":667,"duration_ms":5002,"temperature":0.7,"texified_at":"2026-08-05T21:18:27.567967+00:00","pith_summary":"Numerical semigroups that are far-flung Gorenstein are known to have multiplicity bounded above by the Rohrbach number $n(t)$ of their type $t$. The paper answers whether this bound is sharp for every $t$ by giving an explicit construction: take any extremal set $A$ of size $t$ and form the semigroup obtained by deleting from the half-line of multiplicity $n(A)$ the points that correspond to the elements of $A$. The resulting semigroup is far-flung Gorenstein of type exactly $t$ and multiplicity exactly $n(t)$. The same family also produces, for every $t \\ge 5$, examples in which the residue of the trace ideal strictly exceeds the number of non-semigroup elements of the canonical ideal, answering a second open question in the negative.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":4444,"prompt_tokens":527,"completion_tokens":3917,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":3437}},"feed_headline":"Far-flung Gorenstein semigroups hit the Rohrbach bound for every type","feed_subtitle":"An explicit construction from extremal sets settles the sharpness question and yields counterexamples for t≥5","key_machinery":"The family $S(m,A) = \\Delta(m) \\setminus \\{(2m-1)-a : a \\in A\\}$, where $A$ is an extremal set of size $t$ containing $0$ and $m=n(A)$. Proposition 2 and Corollary 4 show that this semigroup is far-flung Gorenstein of maximal reduced type precisely when $m=n(A)$.","core_discovery":"For every integer $t \\ge 2$ and every extremal finite set $A$ of cardinality $t$, the numerical semigroup $S=S(n(A),A)$ is far-flung Gorenstein, has type $t$, and attains the multiplicity bound $m(S)=n(t)$. In particular, the Herzog–Kumashiro–Stamate upper bound is sharp for every type.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Far-flung Gorenstein semigroups attain HKS multiplicity bound for every type","Explicit construction yields type-t far-flung Gorenstein semigroups meeting the HKS bound","HKS upper bound on multiplicity is sharp for far-flung Gorenstein semigroups of all types","Extremal sets produce far-flung Gorenstein semigroups attaining m(S)=n(t) for all t","For every t the HKS bound is realized by a far-flung Gorenstein numerical semigroup"],"cache_read_input_tokens":128,"weakest_assumption_plain":"Every extremal set of size at least 2 has its largest element at least two less than its Rohrbach number; without that gap the constructed semigroup need not have maximal reduced type.","fun_headline_variants_meta":{"raw":{"variants":["Far-flung Gorenstein semigroups attain HKS multiplicity bound for every type","Explicit construction yields type-t far-flung Gorenstein semigroups meeting the HKS bound","HKS upper bound on multiplicity is sharp for far-flung Gorenstein semigroups of all types","Extremal sets produce far-flung Gorenstein semigroups attaining m(S)=n(t) for all t","For every t the HKS bound is realized by a far-flung Gorenstein numerical semigroup"]},"model":"grok-4.5","effort":"low","cost_usd":0.005684,"raw_usage":{"total_tokens":1422,"prompt_tokens":664,"num_sources_used":0,"completion_tokens":131,"cost_in_usd_ticks":56840000,"prompt_tokens_details":{"text_tokens":664,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":627,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":664,"tokens_out":131,"duration_ms":6152,"temperature":1.0,"reasoning_tokens":627,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T07:06:02.922280+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a single extremal set $A$ of size $t \\ge 2$ for which $\\max A + 2 > n(A)$; then $S(n(A),A)$ fails to have maximal reduced type and the multiplicity-attaining claim for that $t$ collapses.","supporting_citations":[],"review_version":1}