R0051-26008 · q-20261015-advance-20261115-r06-minimum-vertex-cover-0083
bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={"mode":"empty_graph","vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[]}。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-empty-multiple-boundary；資料={"mode":"empty_graph","vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[]}。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-empty-multiple-boundary]
操作後
derived = [[mode, edges, minimum_size, minimum_covers, mode_certificate, boundary]]
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, v4, v5, v6, v7], [], null]
操作後
derived = [[v0, v1, v2, v3, v4, v5, v6, v7], [], null]
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, v4, v5, v6, v7], [], 256]
操作後
derived = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255], [], [[], [v0], [v1], [v0, v1], [v2], [v0, v2], [v1, v2], [v0, v1, v2], [v3], [v0, v3], [v1, v3], [v0, v1, v3], [v2, v3], [v0, v2, v3], [v1, v2, v3], [v0, v1, v2, v3], [v4], [v0, v4], [v1, v4], [v0, v1, v4], [v2, v4], [v0, v2, v4], [v1, v2, v4], [v0, v1, v2, v4], [v3, v4], [v0, v3, v4], [v1, v3, v4], [v0, v1, v3, v4], [v2, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4], [v5], [v0, v5], [v1, v5], [v0, v1, v5], [v2, v5], [v0, v2, v5], [v1, v2, v5], [v0, v1, v2, v5], [v3, v5], [v0, v3, v5], [v1, v3, v5], [v0, v1, v3, v5], [v2, v3, v5], [v0, v2, v3, v5], [v1, v2, v3, v5], [v0, v1, v2, v3, v5], [v4, v5], [v0, v4, v5], [v1, v4, v5], [v0, v1, v4, v5], [v2, v4, v5], [v0, v2, v4, v5], [v1, v2, v4, v5], [v0, v1, v2, v4, v5], [v3, v4, v5], [v0, v3, v4, v5], [v1, v3, v4, v5], [v0, v1, v3, v4, v5], [v2, v3, v4, v5], [v0, v2, v3, v4, v5], [v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], [v6], [v0, v6], [v1, v6], [v0, v1, v6], [v2, v6], [v0, v2, v6], [v1, v2, v6], [v0, v1, v2, v6], [v3, v6], [v0, v3, v6], [v1, v3, v6], [v0, v1, v3, v6], [v2, v3, v6], [v0, v2, v3, v6], [v1, v2, v3, v6], [v0, v1, v2, v3, v6], [v4, v6], [v0, v4, v6], [v1, v4, v6], [v0, v1, v4, v6], [v2, v4, v6], [v0, v2, v4, v6], [v1, v2, v4, v6], [v0, v1, v2, v4, v6], [v3, v4, v6], [v0, v3, v4, v6], [v1, v3, v4, v6], [v0, v1, v3, v4, v6], [v2, v3, v4, v6], [v0, v2, v3, v4, v6], [v1, v2, v3, v4, v6], [v0, v1, v2, v3, v4, v6], [v5, v6], [v0, v5, v6], [v1, v5, v6], [v0, v1, v5, v6], [v2, v5, v6], [v0, v2, v5, v6], [v1, v2, v5, v6], [v0, v1, v2, v5, v6], [v3, v5, v6], [v0, v3, v5, v6], [v1, v3, v5, v6], [v0, v1, v3, v5, v6], [v2, v3, v5, v6], [v0, v2, v3, v5, v6], [v1, v2, v3, v5, v6], [v0, v1, v2, v3, v5, v6], [v4, v5, v6], [v0, v4, v5, v6], [v1, v4, v5, v6], [v0, v1, v4, v5, v6], [v2, v4, v5, v6], [v0, v2, v4, v5, v6], [v1, v2, v4, v5, v6], [v0, v1, v2, v4, v5, v6], [v3, v4, v5, v6], [v0, v3, v4, v5, v6], [v1, v3, v4, v5, v6], [v0, v1, v3, v4, v5, v6], [v2, v3, v4, v5, v6], [v0, v2, v3, v4, v5, v6], [v1, v2, v3, v4, v5, v6], [v0, v1, v2, v3, v4, v5, v6], [v7], [v0, v7], [v1, v7], [v0, v1, v7], [v2, v7], [v0, v2, v7], [v1, v2, v7], [v0, v1, v2, v7], [v3, v7], [v0, v3, v7], [v1, v3, v7], [v0, v1, v3, v7], [v2, v3, v7], [v0, v2, v3, v7], [v1, v2, v3, v7], [v0, v1, v2, v3, v7], [v4, v7], [v0, v4, v7], [v1, v4, v7], [v0, v1, v4, v7], [v2, v4, v7], [v0, v2, v4, v7], [v1, v2, v4, v7], [v0, v1, v2, v4, v7], [v3, v4, v7], [v0, v3, v4, v7], [v1, v3, v4, v7], [v0, v1, v3, v4, v7], [v2, v3, v4, v7], [v0, v2, v3, v4, v7], [v1, v2, v3, v4, v7], [v0, v1, v2, v3, v4, v7], [v5, v7], [v0, v5, v7], [v1, v5, v7], [v0, v1, v5, v7], [v2, v5, v7], [v0, v2, v5, v7], [v1, v2, v5, v7], [v0, v1, v2, v5, v7], [v3, v5, v7], [v0, v3, v5, v7], [v1, v3, v5, v7], [v0, v1, v3, v5, v7], [v2, v3, v5, v7], [v0, v2, v3, v5, v7], [v1, v2, v3, v5, v7], [v0, v1, v2, v3, v5, v7], [v4, v5, v7], [v0, v4, v5, v7], [v1, v4, v5, v7], [v0, v1, v4, v5, v7], [v2, v4, v5, v7], [v0, v2, v4, v5, v7], [v1, v2, v4, v5, v7], [v0, v1, v2, v4, v5, v7], [v3, v4, v5, v7], [v0, v3, v4, v5, v7], [v1, v3, v4, v5, v7], [v0, v1, v3, v4, v5, v7], [v2, v3, v4, v5, v7], [v0, v2, v3, v4, v5, v7], [v1, v2, v3, v4, v5, v7], [v0, v1, v2, v3, v4, v5, v7], [v6, v7], [v0, v6, v7], [v1, v6, v7], [v0, v1, v6, v7], [v2, v6, v7], [v0, v2, v6, v7], [v1, v2, v6, v7], [v0, v1, v2, v6, v7], [v3, v6, v7], [v0, v3, v6, v7], [v1, v3, v6, v7], [v0, v1, v3, v6, v7], [v2, v3, v6, v7], [v0, v2, v3, v6, v7], [v1, v2, v3, v6, v7], [v0, v1, v2, v3, v6, v7], [v4, v6, v7], [v0, v4, v6, v7], [v1, v4, v6, v7], [v0, v1, v4, v6, v7], [v2, v4, v6, v7], [v0, v2, v4, v6, v7], [v1, v2, v4, v6, v7], [v0, v1, v2, v4, v6, v7], [v3, v4, v6, v7], [v0, v3, v4, v6, v7], [v1, v3, v4, v6, v7], [v0, v1, v3, v4, v6, v7], [v2, v3, v4, v6, v7], [v0, v2, v3, v4, v6, v7], [v1, v2, v3, v4, v6, v7], [v0, v1, v2, v3, v4, v6, v7], [v5, v6, v7], [v0, v5, v6, v7], [v1, v5, v6, v7], [v0, v1, v5, v6, v7], [v2, v5, v6, v7], [v0, v2, v5, v6, v7], [v1, v2, v5, v6, v7], [v0, v1, v2, v5, v6, v7], [v3, v5, v6, v7], [v0, v3, v5, v6, v7], [v1, v3, v5, v6, v7], [v0, v1, v3, v5, v6, v7], [v2, v3, v5, v6, v7], [v0, v2, v3, v5, v6, v7], [v1, v2, v3, v5, v6, v7], [v0, v1, v2, v3, v5, v6, v7], [v4, v5, v6, v7], [v0, v4, v5, v6, v7], [v1, v4, v5, v6, v7], [v0, v1, v4, v5, v6, v7], [v2, v4, v5, v6, v7], [v0, v2, v4, v5, v6, v7], [v1, v2, v4, v5, v6, v7], [v0, v1, v2, v4, v5, v6, v7], [v3, v4, v5, v6, v7], [v0, v3, v4, v5, v6, v7], [v1, v3, v4, v5, v6, v7], [v0, v1, v3, v4, v5, v6, v7], [v2, v3, v4, v5, v6, v7], [v0, v2, v3, v4, v5, v6, v7], [v1, v2, v3, v4, v5, v6, v7], [v0, v1, v2, v3, v4, v5, v6, v7]], [1, 8, 28, 56, 70, 56, 28, 8, 1], 0, [[]]]
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 = [empty_graph, [v0, v1, v2, v3, v4, v5, v6, v7], []]
操作後
derived = [0, [[]], 是]
條件：無
empty_graph指無邊而非空vertices；唯一minimum是空集。multiple_minima列全部至少兩個解，不把一個witness當唯一解，也不引König／weighted／directed版本。
規則：Mincover_original_empty_OR_multiple_boundary_not_uniqueness_guess · 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。(empty_graph, [], 0, [[]], 是, empty_graph_has_unique_empty_minimum_cover_while_other_graphs_may_have_multiple_minima_without_Konig_or_weighted_claims)
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: 0d35260f2dec8c4de03912bd61478bd760ea447bbdbb7a71b33235f78990a1f7candidate record SHA-256: fc4cd44331dcb64d7b1b1b062b3b9489e183aaa08da94323bb5c51139028adb1semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5cLean: NOT_RUN · human review: NOT_PERFORMED