R0049-25491 · q-20261015-advance-20261114-r07-vertex-cover-0041
有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v0"]],"candidate":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。
G001 — 有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v0"]],"candidate":[]}。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 = [vcover-uncovered-edge-witness]
操作後
derived = [[candidate, edge_coverage_rows, uncovered_edge, cover, boundary]]
條件：無
本頁列完整來源欄序，不需JavaScript；LP三種mode雖同為6欄，角色仍由原mode選，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, v8, v9], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v0]], []]
操作後
derived = [[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v9], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], []]
條件：無
carrier非空且至多12，edge端點依JS UTF16<排，再按來源English canonical排序；loop或反向重複edge拒絕，不把圖改成有向或edge cover。
規則：Vcover_complete_simple_undirected_graph_UTF16_then_English_order · 依賴：G001, S001
操作前
source roles = [0, [v0, v1], []]
操作後
derived = [[否, 否], 否]
條件：無
這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。
規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S002
操作前
source roles = [1, [v0, v9], []]
操作後
derived = [[否, 否], 否]
條件：無
這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。
規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S003
操作前
source roles = [2, [v1, v2], []]
操作後
derived = [[否, 否], 否]
條件：無
這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。
規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S004
操作前
source roles = [3, [v2, v3], []]
操作後
derived = [[否, 否], 否]
條件：無
這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。
規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S005
操作前
source roles = [4, [v3, v4], []]
操作後
derived = [[否, 否], 否]
條件：無
這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。
規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S006
操作前
source roles = [5, [v4, v5], []]
操作後
derived = [[否, 否], 否]
條件：無
這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。
規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S007
操作前
source roles = [6, [v5, v6], []]
操作後
derived = [[否, 否], 否]
條件：無
這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。
規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S008
操作前
source roles = [7, [v6, v7], []]
操作後
derived = [[否, 否], 否]
條件：無
這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。
規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S009
操作前
source roles = [8, [v7, v8], []]
操作後
derived = [[否, 否], 否]
條件：無
這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。
規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S010
操作前
source roles = [9, [v8, v9], []]
操作後
derived = [[否, 否], 否]
條件：無
這條edge至少一個endpoint被選才covered，逐完整edge表核對，不用頂點數猜測。
規則：Vcover_EVERY_edge_selected_endpoint_disjunction · 依賴：G001, S011
操作前
source roles = [[[[v0, v1], 否], [[v0, v9], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否], [[v8, v9], 否]]]
操作後
derived = [否, [v0, v1], []]
條件：無
全邊合取，首uncovered edge保來源canonical次序；沒有則null，不用False代null。isolated點沒有責任edge，故可選可不選。
規則：Vcover_COMPLETE_edge_conjunction_FIRST_uncovered_or_NULL · 依賴：G001, S012
操作前
source roles = [[v0, v1], []]
操作後
derived = [否]
條件：無
首反例edge兩端都不在C，已足夠否定全稱vertex cover；outside carrier或duplicate candidate不是此合法false witness。
規則：Vcover_actual_uncovered_edge_refutes_universal · 依賴：G001, S013
S014 · 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。S013 · 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。S012 · 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。S011 · 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。S010 · 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。S009 · 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。S008 · 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。S007 · 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。S006 · 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。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。([], [[[v0, v1], 否], [[v0, v9], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否], [[v8, v9], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)
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: 4c4d4beab6af0635c35d985252fbf35691ee762693868c4b129cb110c8c68589candidate record SHA-256: 6d9146c4e7ff4ce6581cdd1ea38fcc411cf90ed1912eb5b8168af1e6610992e5semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3Lean: NOT_RUN · human review: NOT_PERFORMED