R0051-26085 · q-20261015-advance-20261115-r06-minimum-vertex-cover-0035
bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v0"]],"candidate":["v0","v1","v2","v3"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。
G001 — bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-nonminimum-witness；資料={"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v0"]],"candidate":["v0","v1","v2","v3"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。
操作前
source roles = [mincover-nonminimum-witness]
操作後
derived = [[candidate, cover, candidate_size, minimum_size, smaller_cover, minimum, certificate]]
條件：無
本頁直接列完整來源欄序，不需JavaScript；contract的6/4/6/6與IVT的8/7/8由原mode選，不補欄也不由答案形狀猜。
規則：source_ordered_fields_without_JS · 依賴：G001
操作前
source roles = [[v0, v1, v2, v3], [[v0, v1], [v1, v2], [v2, v3], [v3, v0]], [v0, v1, v2, v3]]
操作後
derived = [[v0, v1, v2, v3], [[v0, v1], [v0, v3], [v1, v2], [v2, v3]], [v0, v1, v2, v3]]
條件：無
非空carrier至多12；端點UTF16排序後edge族English排序，loop/反向duplicate拒絕。未提供candidate時保持null，不擅補空candidate。
規則：Mincover_complete_simple_graph_UTF16_English_and_optional_candidate · 依賴：G001, S001
操作前
source roles = [[v0, v1, v2, v3], [[v0, v1], [v0, v3], [v1, v2], [v2, v3]], 16]
操作後
derived = [[5, 7, 10, 11, 13, 14, 15], [[0, 0], [1, 2], [2, 1], [3, 3], [4, 0], [6, 1], [8, 0], [9, 2], [12, 0]], [[v0, v2], [v0, v1, v2], [v1, v3], [v0, v1, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3]], [0, 0, 2, 4, 1], 2, [[v0, v2], [v1, v3]]]
條件：無
逐全部mask核所有edges；每個失敗mask保首uncovered edge index反例，通過者全列source順序。較小size計數0給下界，實際最小解給上界；全部ties依numeric mask保留，不改English/大小序。
規則：Mincover_ALL_masks_acceptance_refutations_ALLcovers_ALLminimizers · 依賴：G001, S002
操作前
source roles = [[v0, v1, v2, v3], [[v0, v1], [v0, v3], [v1, v2], [v2, v3]]]
操作後
derived = [[[[v0, v1], 是], [[v0, v3], 是], [[v1, v2], 是], [[v2, v3], 是]], 4, 2, [v0, v2]]
條件：無
先核candidate真cover；較小解取完整source清單的首minimum，不用任意小集合代替。
規則：Mincover_actual_candidate_complete_cover_and_smaller_witness · 依賴：G001, S003
S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。([v0, v1, v2, v3], 是, 4, 2, [v0, v2], 否, explicit_smaller_cover_refutes_minimum_status)
依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。
覆核日期：2026-10-26
原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: a3905c063f1f759940edd460cc85b469a07518d23558eea82eefb018007b0332candidate record SHA-256: b45b67c246247c359f34e416fc6413e10bd7c802599ef172bcba95f3cfd14052semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5cLean: NOT_RUN · human review: NOT_PERFORMED