R0051-26068 · q-20261015-advance-20261115-r06-minimum-vertex-cover-0068
bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-minimal-not-minimum；資料={"vertices":["c0","l0_0","l0_1","l0_2","c1","l1_0","l1_1"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c1","l1_0"],["c1","l1_1"]],"candidate":["l0_0","l0_1","l0_2","l1_0","l1_1"]}。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-minimal-not-minimum；資料={"vertices":["c0","l0_0","l0_1","l0_2","c1","l1_0","l1_1"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c1","l1_0"],["c1","l1_1"]],"candidate":["l0_0","l0_1","l0_2","l1_0","l1_1"]}。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-minimal-not-minimum]
操作後
derived = [[candidate, deletion_rows, inclusion_minimal, 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 = [[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1]], [l0_0, l0_1, l0_2, l1_0, l1_1]]
操作後
derived = [[c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1]], [l0_0, l0_1, l0_2, l1_0, l1_1]]
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 = [[c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1]], 128]
操作後
derived = [[3, 7, 11, 15, 19, 23, 27, 30, 31, 35, 39, 43, 47, 51, 55, 59, 62, 63, 67, 71, 75, 79, 83, 87, 91, 94, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 124, 125, 126, 127], [[0, 0], [1, 3], [2, 0], [4, 1], [5, 3], [6, 1], [8, 0], [9, 3], [10, 0], [12, 2], [13, 3], [14, 2], [16, 0], [17, 3], [18, 0], [20, 1], [21, 3], [22, 1], [24, 0], [25, 3], [26, 0], [28, 3], [29, 3], [32, 0], [33, 4], [34, 0], [36, 1], [37, 4], [38, 1], [40, 0], [41, 4], [42, 0], [44, 2], [45, 4], [46, 2], [48, 0], [49, 4], [50, 0], [52, 1], [53, 4], [54, 1], [56, 0], [57, 4], [58, 0], [60, 4], [61, 4], [64, 0], [65, 3], [66, 0], [68, 1], [69, 3], [70, 1], [72, 0], [73, 3], [74, 0], [76, 2], [77, 3], [78, 2], [80, 0], [81, 3], [82, 0], [84, 1], [85, 3], [86, 1], [88, 0], [89, 3], [90, 0], [92, 3], [93, 3], [96, 0], [98, 0], [100, 1], [102, 1], [104, 0], [106, 0], [108, 2], [110, 2], [112, 0], [114, 0], [116, 1], [118, 1], [120, 0], [122, 0]], [[c0, c1], [c0, c1, l0_0], [c0, c1, l0_1], [c0, c1, l0_0, l0_1], [c0, c1, l0_2], [c0, c1, l0_0, l0_2], [c0, c1, l0_1, l0_2], [c1, l0_0, l0_1, l0_2], [c0, c1, l0_0, l0_1, l0_2], [c0, c1, l1_0], [c0, c1, l0_0, l1_0], [c0, c1, l0_1, l1_0], [c0, c1, l0_0, l0_1, l1_0], [c0, c1, l0_2, l1_0], [c0, c1, l0_0, l0_2, l1_0], [c0, c1, l0_1, l0_2, l1_0], [c1, l0_0, l0_1, l0_2, l1_0], [c0, c1, l0_0, l0_1, l0_2, l1_0], [c0, c1, l1_1], [c0, c1, l0_0, l1_1], [c0, c1, l0_1, l1_1], [c0, c1, l0_0, l0_1, l1_1], [c0, c1, l0_2, l1_1], [c0, c1, l0_0, l0_2, l1_1], [c0, c1, l0_1, l0_2, l1_1], [c1, l0_0, l0_1, l0_2, l1_1], [c0, c1, l0_0, l0_1, l0_2, l1_1], [c0, l1_0, l1_1], [c0, c1, l1_0, l1_1], [c0, l0_0, l1_0, l1_1], [c0, c1, l0_0, l1_0, l1_1], [c0, l0_1, l1_0, l1_1], [c0, c1, l0_1, l1_0, l1_1], [c0, l0_0, l0_1, l1_0, l1_1], [c0, c1, l0_0, l0_1, l1_0, l1_1], [c0, l0_2, l1_0, l1_1], [c0, c1, l0_2, l1_0, l1_1], [c0, l0_0, l0_2, l1_0, l1_1], [c0, c1, l0_0, l0_2, l1_0, l1_1], [c0, l0_1, l0_2, l1_0, l1_1], [c0, c1, l0_1, l0_2, l1_0, l1_1], [l0_0, l0_1, l0_2, l1_0, l1_1], [c0, l0_0, l0_1, l0_2, l1_0, l1_1], [c1, l0_0, l0_1, l0_2, l1_0, l1_1], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1]], [0, 0, 1, 6, 14, 16, 7, 1], 2, [[c0, c1]]]
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 = [[l0_0, l0_1, l0_2, l1_0, l1_1], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1]]]
操作後
derived = [[[[c0, l0_0], 是], [[c0, l0_1], 是], [[c0, l0_2], 是], [[c1, l1_0], 是], [[c1, l1_1], 是]], 5, 2, [c0, c1]]
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
操作前
source roles = [[l0_0, l0_1, l0_2, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l1_0, l1_1], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0], 否]]]
操作後
derived = [是, 2, [c0, c1], 否]
條件：無
逐刪都失cover，所以inclusion-minimal；但完整枚舉已有嚴格更小cover，所以不是minimum。cover對superset單調使逐刪可驗全部proper subset責任，不等同最大／最優大小。
規則：Mincover_ALL_single_deletions_minimal_BUT_smaller_cover_exists · 依賴：G001, S004
S005 · 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。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。([l0_0, l0_1, l0_2, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l1_0, l1_1], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0], 否]], 是, 2, [c0, c1], 否, inclusion_minimal_is_not_minimum)
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: 5fd4c1c7288235012c8cf77cf2f67ac8abdd19cde7b8554d0268717ad5c09f8bcandidate record SHA-256: bc40416aa1fb8d066157f93fca3e067f1a9e416f5fd99cece77b97ee2c664fcesemantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5cLean: NOT_RUN · human review: NOT_PERFORMED