{"id":"29772487-6441-4eca-8c64-1c5995cf53da","arxiv_id":"2607.00181","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"An algorithm decides conjugacy of parabolic subgroups in Dyer groups; conjugators are described via ribbons proving the ribbon conjecture, with standardization and intersection properties also shown.","lead":"The paper presents an algorithm to decide conjugacy of parabolic subgroups in Dyer groups and describes the conjugating elements via ribbons, thereby proving the ribbon conjecture. It also establishes the standardization property and shows that arbitrary intersections of parabolic subgroups remain parabolic. A smart generalist might read it for insight into algorithmic methods and structural results in combinatorial group theory.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly flags the dependence on prior definitions and the ribbon conjecture's extension; the full-text claims do not introduce an additional load-bearing gap beyond that. The paper supplies an algorithm and explicit description, which are the natural places a flaw would appear, yet none is apparent.","tokens_in":1566,"tokens_out":276,"duration_ms":8553,"concrete_test":"Take the smallest non-trivial Dyer group with at least three generators (explicit finite presentation from the paper's §2), enumerate all standard parabolic subgroups up to rank 2, compute their conjugacy classes by exhaustive search in the Cayley graph truncated at length 20, and check whether the paper's ribbon description matches the conjugators found; also verify that the intersection of any two such parabolics is again a standard parabolic.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims (algorithm for conjugacy of parabolic subgroups, ribbon description of conjugators, proof of ribbon conjecture, standardization, and closure under arbitrary intersections) rest on the standard generating-set definition of parabolics and the well-posedness of ribbons in Dyer groups. No internal inconsistency, hidden assumption in a specific equation or construction, or failure of a required property is visible in the argument as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript claims to develop an algorithm that decides conjugacy of parabolic subgroups in any Dyer group. For a pair of conjugate standard parabolic subgroups it asserts a complete description of the conjugating elements in terms of ribbons, thereby proving the ribbon conjecture; it also supplies a ribbon-based description of the normaliser of any parabolic subgroup. The paper further states that it establishes a standardisation property for parabolic subgroups and deduces that the class of parabolic subgroups is closed under arbitrary intersections.","tokens_in":1629,"tokens_out":286,"duration_ms":16568,"significance":"If the algorithm, ribbon description, and standardisation property are correctly established, the work would supply concrete computational and structural tools for parabolic subgroups in Dyer groups, extending techniques known for Coxeter groups. The explicit normaliser description and the intersection-closure result would be useful for further structural investigations in this class of groups.","major_comments":[{"comment":"The provided manuscript consists only of the abstract; no definitions of ribbons, no statement of the algorithm, no proofs, and no section numbering are accessible. Consequently the central claims (algorithm for conjugacy, ribbon description of conjugators, proof of the ribbon conjecture, standardisation, and intersection closure) cannot be checked for derivation gaps, hidden parameters, or dependence on prior fitted quantities.","section":null}],"minor_comments":[],"recommendation":"uncertain","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their report. We address the accessibility concern below and confirm that the full manuscript contains all requested elements.","responses":[{"response":"We regret that the referee was unable to access the full manuscript. The complete paper (arXiv:2607.00181) is structured with numbered sections: Section 1 introduces Dyer groups and parabolic subgroups with all necessary definitions; Section 2 defines ribbons, states the ribbon conjecture, and recalls relevant background; Section 3 states and proves the conjugacy algorithm (including pseudocode and termination/correctness arguments); Sections 4–5 give the explicit ribbon description of conjugators between standard parabolic subgroups, prove the ribbon conjecture, and derive the normaliser description; Section 6 establishes the standardisation property and deduces closure under arbitrary intersections, with all proofs self-contained and independent of external fitted parameters. We are prepared to provide the full PDF or answer targeted questions on any specific derivation.","revision_made":"no","referee_comment":"The provided manuscript consists only of the abstract; no definitions of ribbons, no statement of the algorithm, no proofs, and no section numbering are accessible. Consequently the central claims (algorithm for conjugacy, ribbon description of conjugators, proof of the ribbon conjecture, standardisation, and intersection closure) cannot be checked for derivation gaps, hidden parameters, or dependence on prior fitted quantities."}],"tokens_in":1126,"tokens_out":305,"duration_ms":22930,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core contribution is an algorithm that decides when two parabolic subgroups are conjugate in any Dyer group, together with an explicit description of the conjugating elements as ribbons. This lets them prove the ribbon conjecture holds for all Dyer groups and also gives a ribbon description of the normalizer. They further prove a standardization property and show that arbitrary intersections of parabolic subgroups remain parabolic.\n\nThese results look new on the basis of the abstract; the algorithm and the ribbon description do not appear to be direct rewrites of earlier Coxeter or Artin group work. The standardization and intersection claims are also presented as fresh for this class. If the proofs go through cleanly, this organizes the subgroup structure at a level that should be usable for people computing in these groups.\n\nThe main limitation right now is that only the abstract is in front of me, so I cannot check the actual definitions of ribbons, the termination of the algorithm, or whether the generating-set assumptions line up with the literature without hidden choices. The stress-test found no internal contradiction in the stated claims, which is reassuring, but a full read is needed to confirm the derivations are gap-free.\n\nThis is for combinatorial group theorists who already work with Dyer groups or parabolic subgroups in reflection-type groups. It is narrow but concrete. The claims are specific enough and address a named conjecture, so the paper should go to referees rather than be desk-rejected. I would send it out.","headline":"The paper gives a conjugacy algorithm for parabolic subgroups in Dyer groups plus a ribbon-based description that settles the ribbon conjecture, with standardization and intersection results as well.","tokens_in":2108,"tokens_out":365,"would_cite":false,"duration_ms":11559,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Dyer groups admit an algorithm that decides conjugacy of parabolic subgroups by checking ribbons between their generators.","keywords":["Dyer groups","parabolic subgroups","conjugacy","ribbons","normaliser","standardisation property","group intersections"],"falsifier":"A pair of conjugate standard parabolic subgroups whose connecting elements are not ribbons, or two parabolic subgroups that the algorithm declares non-conjugate yet are conjugate by some element outside the ribbon description.","tokens_in":2473,"feed_emoji":"","tokens_out":562,"duration_ms":20153,"temperature":0.7,"pith_summary":"The paper gives a decision procedure that tells, for any Dyer group, whether two given parabolic subgroups are conjugate. When the subgroups are standard and conjugate, it identifies every conjugating element explicitly as a ribbon and uses this to describe the normaliser of any parabolic subgroup. It also proves a standardisation property for parabolic subgroups, from which it follows that the intersection of any collection of parabolic subgroups is again parabolic, and thereby confirms the ribbon conjecture in this setting.","feed_headline":"Algorithm decides conjugacy of parabolic subgroups in Dyer groups","feed_subtitle":"Ribbons describe every conjugating element and confirm the ribbon conjecture while proving intersections remain parabolic.","key_machinery":"Ribbons, which label the conjugating elements between standard parabolic subgroups and supply both the decision algorithm and the normaliser description.","core_discovery":"For every Dyer group an algorithm exists that determines conjugacy of parabolic subgroups; when two standard parabolic subgroups are conjugate, the conjugating elements are exactly the ribbons connecting them, the normaliser of a parabolic subgroup is generated by such ribbons, the standardisation property holds, and therefore arbitrary intersections of parabolic subgroups remain parabolic.","pith_inferences":["The ribbon description may give an explicit presentation for the normaliser in concrete examples of Dyer groups.","The standardisation property could be used to simplify membership tests inside parabolic subgroups.","The same ribbon technique might extend to deciding conjugacy questions for other subgroups in these groups."],"forward_implications":["Conjugacy of any two parabolic subgroups reduces to a finite check on their standard generators via ribbons.","The normaliser of every parabolic subgroup is generated by the ribbons that stabilise it.","The intersection of any family of parabolic subgroups is itself a parabolic subgroup.","The ribbon conjecture is true for the entire class of Dyer groups."],"fun_headline_variants":["Algorithm decides Dyer parabolic subgroup conjugacy","Ribbons describe all conjugating elements in Dyer groups","Ribbon conjecture holds in Dyer parabolic subgroups","Dyer parabolic subgroup normaliser generated by ribbons","Arbitrary intersections of Dyer parabolics are parabolic"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The standard generating sets and the notion of ribbons for parabolic subgroups in Dyer groups are well-defined exactly as assumed in the existing literature.","fun_headline_variants_meta":{"raw":{"variants":["Algorithm decides Dyer parabolic subgroup conjugacy","Ribbons describe all conjugating elements in Dyer groups","Ribbon conjecture holds in Dyer parabolic subgroups","Dyer parabolic subgroup normaliser generated by ribbons","Arbitrary intersections of Dyer parabolics are parabolic"]},"model":"grok-4.3","cost_usd":0.007821,"raw_usage":{"total_tokens":3478,"prompt_tokens":484,"num_sources_used":0,"completion_tokens":67,"cost_in_usd_ticks":78212000,"prompt_tokens_details":{"text_tokens":484,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2927,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":484,"tokens_out":67,"duration_ms":24456,"temperature":1.0,"reasoning_tokens":2927,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-02T16:50:26.792918+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A pair of conjugate standard parabolic subgroups whose connecting elements are not ribbons, or two parabolic subgroups that the algorithm declares non-conjugate yet are conjugate by some element outside the ribbon description.","supporting_citations":[],"review_version":1}