R0051-26015 · q-20261015-advance-20261115-r06-minimum-vertex-cover-0090
bounded finite simple graph的全部vertex subsets與all-edge cover audit。操作=mincover-empty-multiple-boundary；資料={"mode":"multiple_minima","vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v0"]]}。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":"multiple_minima","vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v0"]]}。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], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v0]], null]
操作後
derived = [[v0, v1, v2, v3, v4], [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], 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], [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], 32]
操作後
derived = [[11, 13, 15, 21, 22, 23, 26, 27, 29, 30, 31], [[0, 0], [1, 2], [2, 1], [3, 3], [4, 0], [5, 4], [6, 1], [7, 4], [8, 0], [9, 2], [10, 1], [12, 0], [14, 1], [16, 0], [17, 2], [18, 3], [19, 3], [20, 0], [24, 0], [25, 2], [28, 0]], [[v0, v1, v3], [v0, v2, v3], [v0, v1, v2, v3], [v0, v2, v4], [v1, v2, v4], [v0, v1, v2, v4], [v1, v3, v4], [v0, v1, v3, v4], [v0, v2, v3, v4], [v1, v2, v3, v4], [v0, v1, v2, v3, v4]], [0, 0, 0, 5, 5, 1], 3, [[v0, v1, v3], [v0, v2, v3], [v0, v2, v4], [v1, v2, v4], [v1, v3, v4]]]
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 = [multiple_minima, [v0, v1, v2, v3, v4], [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]]]
操作後
derived = [3, [[v0, v1, v3], [v0, v2, v3], [v0, v2, v4], [v1, v2, v4], [v1, v3, v4]], 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。(multiple_minima, [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], 3, [[v0, v1, v3], [v0, v2, v3], [v0, v2, v4], [v1, v2, v4], [v1, v3, v4]], 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: f83d5d3963dde1eb708f8c440d15cde0960c9f0857087371f4d81412896f8901candidate record SHA-256: 3f828c813dab9f7312871830fc429d144d2107c8c416468ff289af9bc49cdb48semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5cLean: NOT_RUN · human review: NOT_PERFORMED