{"id":"4915c631-5768-4f87-bffb-b7791a7aff09","arxiv_id":"2507.10909","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Locally compact and non-totally-disconnected strongly topologically orderable gyrogroups are shown to be metrizable and to contain a suitable set, a discrete generator set that is closed together with the identity.","lead":"Locally compact and not totally disconnected strongly topologically orderable gyrogroups always contain a suitable set: a discrete collection that generates a dense subgyrogroup. The result extends known topological-group theorems to non-associative gyrogroups and gives progress on the gyrogroup version of a long-open question.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Suitable-set existence for the two headline classes rests on the unpublished [18, Theorem 1]; if that citation is unsound, Corollary 4.5 and Theorem 4.10 collapse even though the metrizability reductions are plausible.","rationale":"The most load-bearing point is structural: the paper's advertised 'affirmative' existence result is reached only by a chain that reduces the problem to metrizability and then applies a theorem that is not proved or publicly available. Every internal construction in Sections 3 and 4 is in service of that reduction; the final suitable-set existence itself is never constructed or verified here. If [18, Theorem 1] is true and applies, the paper is correct (modulo minor repairs); if not, the headline claim is unsupported. I therefore agree with the reader's identification of [18] as the weakest assumption. I also reviewed the other flagged items: the borrowed order construction in Theorem 3.5 is not actually used in the main suitability theorems, and the countability argument in Lemma 4.9 contains a false inference ('the set {n_α} is countable, hence eventually constant'); the latter is readily patched by noting that a strictly monotone ω1-sequence in N cannot exist, so it does not create an independent rejection. The conditional verdict is the right one; no verdict change is needed.","tokens_in":10404,"tokens_out":18067,"duration_ms":214956,"concrete_test":"Obtain the full proof of [18, Theorem 1] from the authors or the final accepted version, and verify it applies verbatim to every metrizable strongly topological gyrogroup, in particular checking that the construction does not require associativity, commutativity, separability, or first-countability beyond metrizability. If the proof is available and sound, the last step of Corollary 4.5, Theorem 4.6 and Theorem 4.10 is justified; if it is not, these results should be stated as conditional on [18] or proved in the paper.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Corollary 4.5, Theorem 4.6 and Theorem 4.10 all end by invoking [18, Theorem 1], 'each metrizable strongly topological gyrogroup has a suitable set.' This is exactly the existence assertion advertised in the abstract, and the paper supplies no proof, no statement beyond the quoted clause, and no public version of [18]. The new metrizability results (Theorem 4.4, Lemma 4.9, Theorem 4.10) are the paper's contribution, but the final suitable-set step is as secure as an accepted-but-unavailable citation. If [18, Theorem 1] carries any unstated hypothesis, or if its proof is flawed, the central claim of the paper does not follow from the present arguments. This is a missing-support dependency rather than an internal inconsistency.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies suitable sets in strongly topologically orderable gyrogroups. It proves a dichotomy: such a gyrogroup is either metrizable or has a totally ordered local base of clopen L-subgyrogroups (Theorem 3.4), gives characterizations of topological orderability for non-metrizable strongly topological gyrogroups (Theorem 3.5), proves hereditary paracompactness (Theorem 3.6), and establishes metrizability of not totally disconnected (Theorem 4.4) and locally compact (Theorem 4.10) strongly topologically orderable gyrogroups. It then concludes these classes have suitable sets using [18, Theorem 1], and proves a hereditary result for dense subgyrogroups (Theorem 5.4).","tokens_in":10575,"tokens_out":17852,"duration_ms":181236,"significance":"If the gaps identified below are repaired, the paper would establish a natural gyrogroup analogue of the classical result that each not totally disconnected topologically orderable group is metrizable and hence has a suitable set, and would extend suitable-set existence to locally compact strongly topologically orderable gyrogroups. The structural dichotomy in Theorem 3.4 and the hereditary paracompactness result in Theorem 3.6 are potentially useful contributions. However, the advertised existence theorems depend at the last step on an unpublished accepted paper by two of the present authors, and the proof of the key orderability characterization in Theorem 3.5 is delegated to the group case without verification for non-associative gyrogroups. The metrizability reductions are plausible, but the paper is not self-contained as it stands.","major_comments":[{"comment":"The construction of the compatible linear order is delegated to [13, Theorem 6] with the phrase 'a complete similar method', but [13] treats topological groups. The non-associative gyrogroup operation affects both the transitivity of the lexicographic order and the verification that the order topology agrees with the given topology; no argument is given that the method transfers. Since Theorem 3.5 is used in Theorem 3.6, Theorem 3.7, and Lemma 5.3, this is a load-bearing gap.","section":"Theorem 3.5, implication (3)⇒(1)"},{"comment":"The proof states '(2) ⇒ (3) by Theorem 3.4 and Lemma 3.3 respectively', but Theorem 3.4 assumes G is topologically orderable, which is exactly what (2) is supposed to imply in the equivalence being proved. Thus the cited theorem cannot be used in this direction without circularity. A direct proof that a totally ordered local base plus strong gyrogroup invariance yields a totally ordered local base of clopen L-subgyrogroups is missing.","section":"Theorem 3.5, implication (2)⇒(3)"},{"comment":"Each of these results concludes the existence of a suitable set from metrizability by invoking [18, Theorem 1], an accepted paper by two of the present authors that has no public version and whose statement is only partially quoted. The advertised existence results in the abstract therefore depend on a result that is neither proved nor available for verification. The authors should include the full statement and proof of [18, Theorem 1] or provide a publicly accessible preprint.","section":"Corollary 4.5, Theorem 4.6, Theorem 4.10"},{"comment":"The construction of the totally ordered invariant local base V is not rigorous. The conditions 'Uασ ⊆ Wαβ ⊆ Uαβ' mix ordinals and sets (with σ=β+1), and the argument does not show that the selected family is a base, is totally ordered by inclusion, or inherits the invariance property. Since Lemma 3.3 underpins Theorem 3.4, this proof needs to be rewritten in detail.","section":"Lemma 3.3"},{"comment":"In the totally disconnected case the proof infers from Lemma 4.9 that G is metrizable. Lemma 4.9 only provides a clopen L-subgyrogroup H homeomorphic to the Cantor set. The missing steps are that the left cosets of H form an open cover of G by metrizable subspaces, so G is locally metrizable, and by Theorem 3.6 it is paracompact; metrizability then follows from the Smirnov metrization theorem. As written, the inference is unsupported.","section":"Theorem 4.10"},{"comment":"The statement assumes D⊆L, yet the proof contains the case 'If x∉L' for x∈D, which is impossible under the assumption, and condition (iii) 'D ⊆ F in G' is unclear. The intended statement appears to be about transferring a discrete set D in G to a discrete set F inside a dense subgyrogroup L, but the lemma as stated is internally inconsistent. This affects the validity of Theorem 5.4.","section":"Lemma 5.3"}],"minor_comments":[{"comment":"There are numerous typos: 'oederable' in Question 1.1; 'exisits' in Section 2; 'subgyrgroup' in Lemmas 4.8 and 4.9; 'Vedenissov' for Vedenissoff; 'Narural Science' in reference [18]; 'stronglly' in the Section 5 heading.","section":"Throughout"},{"comment":"The introduction claims that a strongly topologically orderable gyrogroup has a suitable set if and only if each dense subgyrogroup has one, but Theorem 5.4 proves only the forward direction; the converse is not shown and is not trivial, so either prove it or adjust the claim.","section":"Section 5 introduction"},{"comment":"The line 'D ⊂ H = G' should be 'D ⊆ H' since H is a dense subgyrogroup and need not equal G.","section":"Theorem 5.4 proof"},{"comment":"The terms 'cofinality from above' and 'cofinality from below' are used without definition; add a definition or a precise reference to [13]. Also, 'lim dτ = 1' should be phrased as 'dτ → 1' with quantifiers over neighborhoods.","section":"Lemma 3.1"},{"comment":"The phrase 'zero-dimension' should be 'zero-dimensional'.","section":"Lemma 4.8"}],"recommendation":"major_revision","confidential_remarks":"The central existence results in Corollary 4.5, Theorem 4.6, and Theorem 4.10 rely on [18, Theorem 1], an accepted paper by two of the present authors that is not publicly available. Since the abstract's headline claims are exactly these existence statements, I recommend that the editor obtain a copy of [18] during the revision process or require the authors to include a proof or preprint. The proof of Theorem 3.5 (3)⇒(1) is also a serious omission that should be addressed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this. The paper does real work: it proves that every not totally disconnected and every locally compact strongly topologically orderable gyrogroup is metrizable (Theorems 4.4 and 4.10). Those are genuine new results, and the gyrogroup non-associativity shows up in the proofs, e.g., Lemmas 3.1 and 4.8. The hereditary paracompactness theorem (3.6) and the dense-subgyrogroup result (Theorem 5.4) are also worth having.\n\nThe soft spot is not the metrizability; it's the last step. Corollary 4.5, Theorem 4.6, and Theorem 4.10 all conclude \"has a suitable set\" by invoking [18, Theorem 1]—an accepted paper by two of the authors, with no public version—that says every metrizable strongly topological gyrogroup has a suitable set. If that theorem has unstated hypotheses or a flaw, the advertised conclusions don't follow from this paper. This is a citation dependency, not internal circularity, and self-citation of a real result is fine. But because the abstract advertises suitable-set existence, the authors should at least state [18, Theorem 1] precisely and ideally include a proof or make the preprint available. A referee should be asked to check this.\n\nA second, smaller gap: Theorem 3.5 (3) implies (1) says \"define a linear order by a complete similar method by [13, Theorem 6]\" without actually verifying that the Nyikos-Reichel order construction survives gyrogroup non-associativity. This may be fixable, but as written it's a handwave. It doesn't affect the main suitable-set results, since those assume topological orderability rather than constructing it, but it does affect the characterization in Theorem 3.5 and Theorem 3.6.\n\nThere are typos (e.g., \"gyrgroup\" in Lemma 4.9) and the English is rough in places, but nothing confusing. The citation pattern is reasonable; the heavy use of [17] for orderable spaces is legitimate since a gyrogroup is first of all a topological space.\n\nWho's this for? Specialists in topological gyrogroups and point-set topology working on suitable sets. It deserves a serious referee. My recommendation: send it to review, with instructions to verify the [18, Theorem 1] dependency and the (3) implies (1) step. If those hold up, it's a publishable contribution.","headline":"Solid metrizability reductions for strongly topologically orderable gyrogroups, but the advertised suitable-set existence leans on an unpublished self-cited theorem.","tokens_in":11118,"tokens_out":3726,"would_cite":false,"duration_ms":41601,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22A15","54F05","54H11","54H99"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every locally compact or not totally disconnected strongly topologically orderable gyrogroup is metrizable and contains a suitable set; dense subgyrogroups inherit suitable sets.","keywords":["strongly topologically orderable gyrogroup","suitable set","L-subgyrogroup","metrizable","hereditarily paracompact","locally compact","totally disconnected","topological gyrogroup"],"falsifier":"Exhibit a metrizable strongly topological gyrogroup that has no suitable set; because the paper's corollaries convert the relevant orderable gyrogroups into metrizable ones and then stop, any such example would falsify the suitable-set conclusions and [18, Theorem 1]. A natural place to search is among countable, non-discrete metrizable gyrogroups, where the candidate discrete generating subsets can be listed directly.","tokens_in":10176,"feed_emoji":"🌀","tokens_out":14081,"duration_ms":143668,"temperature":0.7,"pith_summary":"Suitable sets are discrete subsets of a topological gyrogroup that generate a dense subgyrogroup; they are analogues of building dense subgroups from discrete generating sets. The paper proves that strongly topologically orderable gyrogroups, meaning gyrogroups with a compatible total order and a gyro-operation whose gyrations preserve a neighborhood base, always admit suitable sets in the locally compact case and in the not-totally-disconnected case. The strategy is to show these gyrogroups are metrizable, then invoke a cited theorem that every metrizable strongly topological gyrogroup has a suitable set. This extends a classical result that every not-totally-disconnected topologically orderable group has a suitable set.","feed_headline":"Locally compact orderable gyrogroups: metrizable, with suitable sets","feed_subtitle":"These discrete generators of dense subgyrogroups exist in the locally compact case, extending a classical group result.","key_machinery":"The load-bearing device is the L-subgyrogroup: a subgyrogroup whose gyration automorphisms keep it invariant, so that its left cosets partition the gyrogroup. In a strongly topologically orderable gyrogroup, the total order is converted into a totally ordered local base at the identity consisting of clopen L-subgyrogroups, and this base does the heavy lifting: it yields hereditary paracompactness, forces first-countability in the not-totally-disconnected case, and in the locally compact totally disconnected case produces a compact clopen L-subgyrogroup that is either discrete or homeomorphic to the Cantor set, giving metrizability.","core_discovery":"On its own terms, the paper establishes a structural dichotomy: every strongly topologically orderable gyrogroup is either metrizable or carries a totally ordered local base at the identity made of clopen L-subgyrogroups invariant under all gyrations, and in either case it is hereditarily paracompact. From this structure the paper proves that a not-totally-disconnected strongly topologically orderable gyrogroup contains an open first-countable L-subgyrogroup and is metrizable, and that every locally compact strongly topologically orderable gyrogroup is metrizable, with the totally disconnected compact case splitting as either discrete or homeomorphic to the Cantor set. The paper then concludes, using [18, Theorem 1], that both classes admit suitable sets; it also shows that dense subgyrogroups inherit closed suitable sets.","pith_inferences":["Going beyond the paper: because every topological group is a gyrogroup with trivial gyrations, the same proof supplies suitable sets for all locally compact and all not-totally-disconnected topologically orderable groups, matching the classical result the paper generalizes.","Going beyond the paper: if the cited theorem that every metrizable strongly topological gyrogroup has a suitable set is fully proved, the paper's route would show that every metrizable strongly topologically orderable gyrogroup has a suitable set, leaving orderability itself as the only hypothesis that matters.","Going beyond the paper: the totally ordered clopen L-subgyrogroup base constructed here is a concrete structure that might yield suitable sets directly in non-metrizable cases, without passing through metrizability and the external final theorem."],"forward_implications":["Every not-totally-disconnected strongly topologically orderable gyrogroup contains a suitable set, generalizing the known group case.","Every locally compact strongly topologically orderable gyrogroup is metrizable and therefore contains a suitable set.","If the identity element of a strongly topologically orderable gyrogroup is a $G_\\delta$-set, then the gyrogroup is metrizable and has a suitable set.","If a strongly topologically orderable gyrogroup has a closed suitable set, then every dense subgyrogroup of it also has a closed suitable set.","Every strongly topologically orderable gyrogroup is hereditarily paracompact."],"supporting_citations":[{"why":"Supplies the theorem that every metrizable strongly topological gyrogroup has a suitable set, which is the final step of Corollary 4.5, Theorem 4.6, and Theorem 4.10.","marker":"[18]"},{"why":"Supplies the classical orderable-space facts used to show the identity component is open and that these spaces are first-countable when not totally disconnected.","marker":"[17]"},{"why":"Provides the ordered-group machinery of cofinalities and integral orders that the proof adapts to gyrogroups.","marker":"[13]"},{"why":"Provides the theory of L-subgyrogroups, including left-coset partitions and gyration-invariance facts, on which the subgroup arguments rest.","marker":"[15]"},{"why":"Defines suitable sets for topological gyrogroups and supplies the strongly-topological-gyrogroup setting.","marker":"[11]"},{"why":"Gives the metrization theorem for first-countable topological gyrogroups used repeatedly.","marker":"[5]"},{"why":"Supplies quotient facts used to obtain clopen subgyrogroups from open ones in the local-base construction.","marker":"[10]"},{"why":"Supplies zero-dimensionality and orderability facts for metrizable totally disconnected spaces used in Theorem 3.7 and Theorem 4.7.","marker":"[7]"}],"fun_headline_variants":["Locally compact orderable gyrogroups always have suitable sets","Orderable gyrogroups: metrizable or clopen base, both yield suitable sets","Suitable sets exist for locally compact and non-totally-disconnected cases","New result: every locally compact orderable gyrogroup is metrizable and has suitable sets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The suitable-set conclusions rest on the cited theorem [18] that every metrizable strongly topological gyrogroup has a suitable set; the paper proves metrizability but not that theorem, so the main corollaries would fail if that external theorem fails.","fun_headline_variants_meta":{"raw":{"variants":["Locally compact orderable gyrogroups always have suitable sets","Orderable gyrogroups: metrizable or clopen base, both yield suitable sets","Suitable sets exist for locally compact and non-totally-disconnected cases","New result: every locally compact orderable gyrogroup is metrizable and has suitable sets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000227,"raw_usage":{"total_tokens":1410,"prompt_tokens":821,"completion_tokens":589,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":437,"completion_tokens_details":{"reasoning_tokens":502}},"tokens_in":437,"tokens_out":589,"duration_ms":6485,"temperature":1.0,"reasoning_tokens":502,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:23:01.288891+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a metrizable strongly topological gyrogroup that has no suitable set; because the paper's corollaries convert the relevant orderable gyrogroups into metrizable ones and then stop, any such example would falsify the suitable-set conclusions and [18, Theorem 1]. A natural place to search is among countable, non-discrete metrizable gyrogroups, where the candidate discrete generating subsets can be listed directly.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that every metrizable strongly topological gyrogroup has a suitable set, which is the final step of Corollary 4.5, Theorem 4.6, and Theorem 4.10."},{"cited_title":"Venkataraman, M","cited_arxiv_id":null,"evidence_quote":"Supplies the classical orderable-space facts used to show the identity component is open and that these spaces are first-countable when not totally disconnected."},{"cited_title":"Nyikos, H.C","cited_arxiv_id":null,"evidence_quote":"Provides the ordered-group machinery of cofinalities and integral orders that the proof adapts to gyrogroups."},{"cited_title":"Suksumran, K","cited_arxiv_id":null,"evidence_quote":"Provides the theory of L-subgyrogroups, including left-coset partitions and gyration-invariance facts, on which the subgroup arguments rest."},{"cited_title":"Suitable sets for strongly topological gyrogroups","cited_arxiv_id":"2005.13767","evidence_quote":"Defines suitable sets for topological gyrogroups and supplies the strongly-topological-gyrogroup setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the metrization theorem for first-countable topological gyrogroups used repeatedly."},{"cited_title":"Quotients with respect to strongly $L$-subgyrogroups","cited_arxiv_id":"2210.03648","evidence_quote":"Supplies quotient facts used to obtain clopen subgyrogroups from open ones in the local-base construction."},{"cited_title":"Herrlich, Ordnungsf¨ ahigkeit total-diskontinuierlicher R¨ aume","cited_arxiv_id":null,"evidence_quote":"Supplies zero-dimensionality and orderability facts for metrizable totally disconnected spaces used in Theorem 3.7 and Theorem 4.7."}],"review_version":1}