{"id":"edd1dded-8b32-4954-91a7-d4683fea68e6","arxiv_id":"2607.10306","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For regular anti-phase templates P(G,A), |Stab|=n iff every adjacent quotient generates G; a coset-based lower bound is sharp for |G|≤5 and recovers f(4)=10.","lead":"The paper builds highly symmetric stable-marriage instances from finite groups and proves that the number of stable matchings equals the group order exactly when every adjacent quotient generates the group. It also supplies a sharp counting lower bound that recovers the classical maximum of 10 for n=4 via the Klein four-group.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claim (exact generator criterion + coset-index lower bound) is proved by short, elementary arguments that are fully checkable from the text. Lemma 5.5 is indeed load-bearing, yet its proof contains no soft spot: the only ingredients are the definition of the index function, the anti-phase ranking of women, and the ordinary blocking-pair condition. The converse construction then simply takes every proper nonempty union of left cosets and produces a distinct non-canonical stable matching; disjointness across different maximal indices is immediate from the two-value support of each such matching. Computational verification for all groups of order ≤6 (and the classical f(4)=10 attained by the Klein template) supplies independent corroboration. Residual open problems (exact formula for larger groups, asymptotic constructions) are clearly flagged and do not affect the proved statements. Consequently the reader's ACCEPT / HIGH-confidence verdict stands unchanged.","tokens_in":14182,"tokens_out":613,"duration_ms":7129,"concrete_test":"Independently re-derive the implication g∈D_b ⇒ g q_b ∈D_b from the blocking-pair condition of Lemma 5.3 alone (without invoking maximality of b until the final step); if the same closure is obtained, the coset lemma is confirmed. As a secondary check, re-run the script's enumeration for the single non-sharp example G=Z_6, A=(0,2,4,1,3,5) and verify that |Stab|=24 while the two-level family contributes exactly the bound value 20.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption correctly isolates Lemma 5.5 as the single rigidity statement on which the converse (Theorem 5.7) and counting bound (Theorem 5.8) rest. The lemma's short proof is elementary and self-contained: for any stable index function d with maximal value b≥1, the woman w_g a_{b-1} is preferred by m_g, so stability forces her partner m_x to satisfy d_x=b and therefore x=g q_b, closing D_b under right multiplication by q_b. The subsequent construction of the two-level matchings μ^(b,K) for every proper nonempty union of left cosets of ⟨q_b⟩ is fully explicit, and the families for distinct b are disjoint because their value sets {b-1,b} are distinct. No hidden assumption, circularity, or gap appears; the argument holds for arbitrary finite groups without commutativity. The accompanying pure-Python script further corroborates the coset lemma, the equivalence, and the bound for all groups of order ≤6 (and sampled order-8 groups).","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies highly symmetric stable-marriage instances obtained from regular actions of a finite group G of order n together with an ordering A of its elements. It defines the regular anti-phase template P(G,A), shows that the anti-phase choice of women’s lists is the unique one among automorphism-maximal profiles that yields the constant rank-sum identity r_M + r_W = n+1, and proves that the n canonical matchings exhaust the stable set if and only if every adjacent quotient q_b generates G. The necessity direction is obtained by an explicit two-level construction of non-canonical stable matchings indexed by proper nonempty unions of left cosets of ⟨q_b⟩. A counting lower bound |Stab| ≥ n + ∑_b (2[G:⟨q_b⟩]-2) follows at once; the bound is sharp for all groups of order ≤5 and both groups of order 4, recovering the classical maximum f(4)=10 for the Klein four-group. The paper also corrects an earlier impression that cyclicity alone forces a chain lattice: the phenomenon is governed by primality of |G| (or, for composite order, by the particular ordering A). All computational claims are verified by an accompanying pure-Python script.","tokens_in":14425,"tokens_out":747,"duration_ms":11174,"significance":"The work supplies a clean group-theoretic characterization of a natural family of Latin, automorphism-maximal profiles and gives the first exact criterion (and its converse) for when such a profile has exactly n stable matchings. The coset lemma and the explicit two-level construction are elementary yet non-obvious; together they convert a previously ad-hoc generator criterion into a sharp if-and-only-if statement that holds for arbitrary finite groups. The counting bound is sharp in all small cases and furnishes a structural explanation of f(4)=10. The accompanying machine-checkable script and the careful correction of the cyclicity-versus-primality picture further strengthen the contribution. The results sit at a useful intersection of combinatorial matching theory, Latin squares, and finite-group actions, and they open several concrete open problems on exact counts and lattice structure.","major_comments":[],"minor_comments":[{"comment":"In the statement of Theorem 4.7 the re-indexing that converts b_s = a^{-1}_{n-1-s} into the displayed form is correct but slightly terse; a one-line intermediate equation would help a reader who is not already fluent with the normal form.","section":null},{"comment":"Example 5.12 and Observation 7.2 both list concrete orderings of Z_4; it would be convenient to have a single table that also records the resulting rotation posets (or at least their heights) so that the dependence of lattice shape on A is visible at a glance.","section":null},{"comment":"The product construction of Section 6 is standard and correctly stated, yet it is never used later; either a brief forward reference to a decomposition question or a one-sentence remark that it is recorded only for completeness would avoid the impression of an orphaned section.","section":null},{"comment":"A few typographical inconsistencies remain (e.g., “anti-phase” versus “antiphase”, occasional missing spaces around mathematical operators). They do not affect readability but should be cleaned in the final version.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is self-contained, the proofs are elementary and complete, and the computational verification is unusually thorough for a pure-mathematics paper. I see no reason to request further refereeing; the only remaining issues are purely presentational."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The paper gives a clean structural account of regular anti-phase templates P(G,A). What is new is the characterization that, among automorphism-maximal profiles, constant rank-sum is exactly the anti-phase choice (Thm 4.7), the exact converse of the generator criterion for arbitrary finite groups (Thm 5.7), and the coset-index lower bound (Thm 5.8) that recovers the classical maximum f(4)=10 for the Klein group by construction rather than pure enumeration. The coset lemma (5.5) is the single rigidity step; its short proof is elementary and self-contained, and the two-level matchings for proper coset unions are written out explicitly. The argument never needs commutativity.\n\nThe math is solid. Canonical stability, the matching property of the two-level constructions, and disjointness of the families for distinct b are all checked carefully. The accompanying pure-Python script verifies the coset lemma, the equivalence, the bound, and the small-order census (sharp for |G|<=5 and both groups of order 4). The correction that chain lattices are governed by primality of |G|, not cyclicity of G, is useful and well-illustrated by the Z4 examples.\n\nSoft spots are minor and already flagged by the author. The bound is not always tight (Z6 gives 20 vs 24), so the two-level family does not exhaust Stab in general; the exact formula and the structure of the rotation poset remain open. Asymptotic comparison with f(n) is left as a problem. None of this undercuts the proved statements.\n\nThis is for people who work on Latin profiles, rotation posets, or group actions in matching theory. It is self-contained, reproducible, and settles the open problems it claims to settle. I would send it to a serious referee without hesitation; the contribution is real and the write-up is careful.","headline":"Clean group-theoretic characterization of a natural family of Latin SMP instances, with an exact generator criterion, its converse, and a sharp coset-index lower bound that recovers f(4)=10.","tokens_in":15044,"tokens_out":493,"would_cite":true,"duration_ms":5116,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A05","05E18","91B68","06D05"],"pacs":[],"model":"grok-4.5","headline":"The number of stable matchings in a group-built preference template equals n exactly when every adjacent quotient generates the group.","keywords":["stable marriage problem","Latin preference profile","anti-phase template","regular group action","generator criterion","sex-equal matching","rotation poset"],"falsifier":"Exhibit any finite group G, any ordering A, and any stable matching of P(G,A) whose highest index level set is not a union of left cosets of ⟨qb⟩; or find a group of order ≤5 for which the counting lower bound is not attained.","tokens_in":15059,"feed_emoji":"⚖️","tokens_out":752,"duration_ms":5542,"temperature":0.7,"pith_summary":"This paper builds highly symmetric stable-marriage instances from any finite group of order n together with an ordering of its elements. The resulting regular anti-phase templates always possess n canonical stable matchings; the paper proves that these are the only stable matchings if and only if each successive “adjacent quotient” of the ordering generates the whole group. When some quotient generates only a proper subgroup, the paper constructs extra stable matchings by taking unions of left cosets, and thereby obtains a concrete lower bound on the size of the stable set. The bound is sharp for every group of order at most five and for both groups of order four; in particular it recovers the classical maximum of ten stable matchings for size-four instances via the Klein four-group. The same analysis shows that chain lattices appear for every ordering precisely when the group order is prime, not merely when the group is cyclic. The anti-phase construction itself is shown to be the unique choice, among all automorphism-maximal profiles, that forces every man–woman pair to have constant rank sum n+1.","feed_headline":"Stable matchings equal n exactly when quotients generate the group","feed_subtitle":"A coset lemma yields a sharp lower bound and recovers the classical maximum of 10 for n=4 via the Klein group.","key_machinery":"The coset lemma: for any stable matching the top level set of its index function is a union of left cosets of the cyclic subgroup generated by the adjacent quotient at the maximal index. This single rigidity statement supplies both the sufficiency and the necessity of the generator criterion and the counting bound.","core_discovery":"For every finite group G of order n and every ordering A of its elements, the regular anti-phase template P(G,A) has exactly n stable matchings if and only if every adjacent quotient qb generates G. When some qb fails to generate, an explicit family of non-canonical stable matchings can be built from the proper unions of left cosets of ⟨qb⟩, yielding the sharp lower bound |Stab|≥n+∑ b(2[G:⟨qb⟩]-2).","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Exact n stables iff adjacent quotients generate G","Anti-phase templates hit n stables precisely via generators","Quotient generation decides whether stable set equals group order","Failed generators yield extra stables from left coset unions","Sharp coset bound recovers 10 stables for Klein group of order 4"],"cache_read_input_tokens":128,"weakest_assumption_plain":"Every stable matching’s highest-rank set of men must be a union of left cosets of the subgroup generated by the corresponding adjacent quotient; if a counter-example matching existed, both the converse and the lower bound would fail.","fun_headline_variants_meta":{"raw":{"variants":["Exact n stables iff adjacent quotients generate G","Anti-phase templates hit n stables precisely via generators","Quotient generation decides whether stable set equals group order","Failed generators yield extra stables from left coset unions","Sharp coset bound recovers 10 stables for Klein group of order 4"]},"model":"grok-4.5","effort":"low","cost_usd":0.007946,"raw_usage":{"total_tokens":1925,"prompt_tokens":805,"num_sources_used":0,"completion_tokens":82,"cost_in_usd_ticks":79460000,"prompt_tokens_details":{"text_tokens":805,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1038,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":805,"tokens_out":82,"duration_ms":8625,"temperature":1.0,"reasoning_tokens":1038,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T12:44:40.616669+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit any finite group G, any ordering A, and any stable matching of P(G,A) whose highest index level set is not a union of left cosets of ⟨qb⟩; or find a group of order ≤5 for which the counting lower bound is not attained.","supporting_citations":[],"review_version":1}