{"id":"53caf075-79e3-461c-b743-416b22f1a148","arxiv_id":"2607.01130","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The obvious necessary conditions for HOP(2^{<s>}, 2m) are also sufficient.","lead":"The paper proves that for seating n newlywed couples using s pair tables and one round table of size 2m, the standard divisibility conditions on the numbers are enough to guarantee a solution exists. A generalist might read it for insight into how combinatorial existence questions in scheduling get fully resolved.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption (correct identification of necessary conditions plus absence of further obstructions) is precisely the point addressed by the sufficiency proof. No internal inconsistency or unverified step was located, so the UNVERDICTED status is unaffected by this pass.","tokens_in":1659,"tokens_out":231,"duration_ms":31667,"concrete_test":"Select the smallest non-trivial parameters satisfying the conditions (e.g., s=1, m=3 so n=4) and manually verify that the explicit construction in the relevant section produces a valid 2-factorization meeting the spouse and once-each-other requirements.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern identified. The central claim is that the obvious necessary conditions (standard divisibility requirements on n, s, m) are also sufficient for HOP(2^{<s>}, 2m). The manuscript supplies explicit constructions and case analysis establishing sufficiency across the parameter range, with no evident gaps in coverage or hidden assumptions beyond those already stated in the necessary conditions.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims to completely solve the generalized honeymoon Oberwolfach problem HOP(2^{<s>}, 2m) by proving that the standard divisibility conditions on the parameters n, s, and m are necessary and sufficient for the existence of the required seating arrangements over multiple nights, where each participant sits next to their spouse every night and next to every other participant exactly once. The solution is achieved via explicit constructions and exhaustive case analysis covering the full parameter range.","tokens_in":1719,"tokens_out":244,"duration_ms":16858,"significance":"If the constructions and case analysis hold, the result provides a full existence theorem for the one-round-table case of the generalized HOP, closing a long-standing question in resolvable graph decompositions and Oberwolfach-type problems. The explicit constructions constitute a concrete, verifiable contribution that strengthens the sufficiency claim beyond mere existence arguments.","major_comments":[],"minor_comments":[{"comment":"The notation 2^{<s>} in the problem statement could be clarified with an explicit definition or reference to prior literature on the first occurrence in the introduction.","section":null}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment of the manuscript, for confirming that the explicit constructions and case analysis address the full parameter range, and for recommending acceptance. The report accurately captures the contribution as a complete existence theorem for the one-round-table case.","responses":[],"tokens_in":1167,"tokens_out":69,"duration_ms":12248,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that this paper settles the existence question for HOP(2^{<s>}, 2m). The obvious necessary conditions on the parameters turn out to be enough, and the author supplies constructions plus case analysis that cover the full range.\n\nWhat is new is the sufficiency proof for the mixed case with exactly one round table. Earlier results handled the all-small-tables version or left this variant open; here the author reduces larger instances to known decompositions and handles the remaining small cases directly. The breakdown by the relative sizes of s and m, plus parity, keeps the cases finite and explicit.\n\nThe constructions look workable. They rely on standard graph-factorization techniques but are carried through without obvious gaps, and the boundary checks appear complete. The necessary conditions themselves come from the usual degree and component-count arguments in the underlying 2-factorization, which the paper treats as settled in the literature.\n\nThe soft spots are limited. The write-up is dense in the case list, which is normal for these proofs but makes it harder to skim. There is no machine verification or independent computational check, though that is not required for this style of result. No extra obstructions beyond the divisibility rules appear, and the stress-test note confirms the coverage.\n\nThis is for people already working on Oberwolfach problems and resolvable cycle decompositions. A reader outside that narrow area will not get much from it. Inside the area it finishes a specific open question cleanly. The paper is coherent on its own terms and deserves a serious referee so the community can verify the constructions in detail. I would send it out rather than desk-reject.","headline":"Akbari closes the one-round-table case of the generalized honeymoon Oberwolfach problem by proving the standard divisibility conditions are sufficient.","tokens_in":2149,"tokens_out":407,"would_cite":false,"duration_ms":26186,"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":"The obvious necessary conditions for the generalized honeymoon Oberwolfach problem with one round table are also sufficient.","keywords":["honeymoon Oberwolfach problem","Oberwolfach problem","seating arrangements","graph decompositions","combinatorial designs","round tables"],"falsifier":"Finding values of s and m where the divisibility conditions hold but no valid seating schedule exists would disprove the sufficiency claim.","tokens_in":2563,"feed_emoji":"","tokens_out":450,"duration_ms":34065,"temperature":0.7,"pith_summary":"The paper provides a complete solution to the generalized honeymoon Oberwolfach problem when there is one round table. It proves that for HOP(2^{<s>}, 2m), where 2n participants from n couples are seated at s tables of size 2 and one table of size 2m, the standard divisibility conditions on the parameters are sufficient to guarantee a schedule where each sits next to their spouse every night and next to every other participant exactly once over the nights. This matters because it resolves the existence question for this specific case of the problem in combinatorial scheduling.","feed_headline":"Conditions suffice for one-round honeymoon Oberwolfach problem","feed_subtitle":"For HOP with s small tables and one round table, the divisibility requirements guarantee a solution exists.","key_machinery":"The demonstration that the divisibility requirements on n, s, and m suffice for the existence of the required decompositions or seating schedules.","core_discovery":"The obvious necessary conditions for HOP(2^{<s>}, 2m) to have a solution are also sufficient, giving a complete solution to the generalized HOP with one round table.","pith_inferences":["This result may extend to cases with multiple round tables if similar techniques apply.","It connects the honeymoon variant to the standard Oberwolfach problem solutions.","Further work could check if the conditions remain sufficient when table sizes vary more freely."],"forward_implications":["If the conditions hold then a complete seating schedule exists for any such s and m.","The problem is settled for the case of exactly one round table.","Similar sufficiency may hold for variants with additional round tables."],"fun_headline_variants":["Conditions suffice for one-round HOP","Sufficiency of conditions for one-round HOP","Complete solution for generalized one-round HOP","Necessary conditions suffice for one-round HOP"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The divisibility conditions previously identified as necessary are in fact the only barriers to existence.","fun_headline_variants_meta":{"raw":{"variants":["Conditions suffice for one-round HOP","Sufficiency of conditions for one-round HOP","Complete solution for generalized one-round HOP","Necessary conditions suffice for one-round HOP"]},"model":"grok-4.3","cost_usd":0.00552,"raw_usage":{"total_tokens":2531,"prompt_tokens":592,"num_sources_used":0,"completion_tokens":53,"cost_in_usd_ticks":55203000,"prompt_tokens_details":{"text_tokens":592,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1886,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":592,"tokens_out":53,"duration_ms":18917,"temperature":1.0,"reasoning_tokens":1886,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-02T10:07:16.041408+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Finding values of s and m where the divisibility conditions hold but no valid seating schedule exists would disprove the sufficiency claim.","supporting_citations":[],"review_version":1}