R0035-18413 · q-20261014-advance-20261104-r06-spanning-tree-0013
完整finite connected simple parent graph與parent edge subset。操作=spantree-audit；資料={"n":5,"edges":[[0,1],[1,2],[1,3],[1,4],[2,3],[3,4]],"selected":[0,1,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。
G001 — 完整finite connected simple parent graph與parent edge subset。操作=spantree-audit；資料={"n":5,"edges":[[0,1],[1,2],[1,3],[1,4],[2,3],[3,4]],"selected":[0,1,4,5]}。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 operation=spantree-audit;original snapshot and mathematics retained
操作後
explicit ordered source output fields=[selected_indices, parent_connected, selected_connected, selected_acyclic, spanning_tree]
條件：無
原題的歷史介面標記保留在快照；本頁直接列原始欄位順序，不需JavaScript或外部表。欄位描述不自動成為數學證據，No標記須依本題的聲稱範圍解讀。
規則：source_ordered_interface_explicit_without_javascript · 依賴：G001
操作前
complete original vertex carrier Fin(5);canonical original edges=[[0, 1], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]];raw selected edge indices=[0, 1, 4, 5]
操作後
sorted unique ORIGINAL selection=[0, 1, 4, 5];selected edges=[[0, 1], [1, 2], [2, 3], [3, 4]];ALL parent vertices retained;no10-edge cap or weights
條件：無
本題頂點上界12但沒有maximum-matching的10邊上限。只選原邊、保留全部頂點及孤立點，選邊排序不改原索引。沒有權重或最小化目標。
規則：spantree_original_graph_whole_carrier_not_matching_domain · 依賴：G001, S001
操作前
parent graph:actual edge index=0;edge=[0, 1];reached from=0;new vertex=1
操作後
original-edge path from0=[0, 1];reached vertices=[0, 1]
條件：無
每個新點都有由真實邊串成的0起點路徑，不用頂點數或邊數猜connected。
規則：spantree_every_discovered_vertex_path_witness · 依賴：G001, S002
操作前
parent graph:actual edge index=1;edge=[1, 2];reached from=1;new vertex=2
操作後
original-edge path from0=[0, 1, 2];reached vertices=[0, 1, 2]
條件：無
每個新點都有由真實邊串成的0起點路徑，不用頂點數或邊數猜connected。
規則：spantree_every_discovered_vertex_path_witness · 依賴：G001, S003
操作前
parent graph:actual edge index=2;edge=[1, 3];reached from=1;new vertex=3
操作後
original-edge path from0=[0, 1, 3];reached vertices=[0, 1, 2, 3]
條件：無
每個新點都有由真實邊串成的0起點路徑，不用頂點數或邊數猜connected。
規則：spantree_every_discovered_vertex_path_witness · 依賴：G001, S004
操作前
parent graph:actual edge index=3;edge=[1, 4];reached from=1;new vertex=4
操作後
original-edge path from0=[0, 1, 4];reached vertices=[0, 1, 2, 3, 4]
條件：無
每個新點都有由真實邊串成的0起點路徑，不用頂點數或邊數猜connected。
規則：spantree_every_discovered_vertex_path_witness · 依賴：G001, S005
操作前
parent graph:complete carrier Fin(5);ALL graph edges=[[0, 1], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]];reached=[0, 1, 2, 3, 4];unreached=[]
操作後
every graph edge stays within reached or its complement;connected=是
條件：無
queue耗盡後，沒有邊跨越已達與未達集合；有未達點便無從0的路徑，無未達點則路徑見證覆蓋全集。單點空邊圖由0自己覆蓋而connected。
規則：spantree_reachability_and_closed_cut_completeness · 依賴：G001, S006
操作前
selected spanning subgraph:actual edge index=0;edge=[0, 1];reached from=0;new vertex=1
操作後
original-edge path from0=[0, 1];reached vertices=[0, 1]
條件：無
每個新點都有由真實邊串成的0起點路徑，不用頂點數或邊數猜connected。
規則：spantree_every_discovered_vertex_path_witness · 依賴：G001, S007
操作前
selected spanning subgraph:actual edge index=1;edge=[1, 2];reached from=1;new vertex=2
操作後
original-edge path from0=[0, 1, 2];reached vertices=[0, 1, 2]
條件：無
每個新點都有由真實邊串成的0起點路徑，不用頂點數或邊數猜connected。
規則：spantree_every_discovered_vertex_path_witness · 依賴：G001, S008
操作前
selected spanning subgraph:actual edge index=2;edge=[2, 3];reached from=2;new vertex=3
操作後
original-edge path from0=[0, 1, 2, 3];reached vertices=[0, 1, 2, 3]
條件：無
每個新點都有由真實邊串成的0起點路徑，不用頂點數或邊數猜connected。
規則：spantree_every_discovered_vertex_path_witness · 依賴：G001, S009
操作前
selected spanning subgraph:actual edge index=3;edge=[3, 4];reached from=3;new vertex=4
操作後
original-edge path from0=[0, 1, 2, 3, 4];reached vertices=[0, 1, 2, 3, 4]
條件：無
每個新點都有由真實邊串成的0起點路徑，不用頂點數或邊數猜connected。
規則：spantree_every_discovered_vertex_path_witness · 依賴：G001, S010
操作前
selected spanning subgraph:complete carrier Fin(5);ALL graph edges=[[0, 1], [1, 2], [2, 3], [3, 4]];reached=[0, 1, 2, 3, 4];unreached=[]
操作後
every graph edge stays within reached or its complement;connected=是
條件：無
queue耗盡後，沒有邊跨越已達與未達集合；有未達點便無從0的路徑，無未達點則路徑見證覆蓋全集。單點空邊圖由0自己覆蓋而connected。
規則：spantree_reachability_and_closed_cut_completeness · 依賴：G001, S011
操作前
original selected edge index=0;edge=[0, 1];previous forest component roots=[0, 1]
操作後
closes cycle=否;actual cycle witness=[]
條件：無
兩端原已在同forest分量時，舊forest路徑加此真邊成cycle；否則連兩分量仍無cycle。逐全部選邊，不能只以n−1數量判。
規則：spantree_every_edge_cycle_or_forest_extension · 依賴：G001, S012
操作前
original selected edge index=1;edge=[1, 2];previous forest component roots=[0, 2]
操作後
closes cycle=否;actual cycle witness=[]
條件：無
兩端原已在同forest分量時，舊forest路徑加此真邊成cycle；否則連兩分量仍無cycle。逐全部選邊，不能只以n−1數量判。
規則：spantree_every_edge_cycle_or_forest_extension · 依賴：G001, S013
操作前
original selected edge index=4;edge=[2, 3];previous forest component roots=[0, 3]
操作後
closes cycle=否;actual cycle witness=[]
條件：無
兩端原已在同forest分量時，舊forest路徑加此真邊成cycle；否則連兩分量仍無cycle。逐全部選邊，不能只以n−1數量判。
規則：spantree_every_edge_cycle_or_forest_extension · 依賴：G001, S014
操作前
original selected edge index=5;edge=[3, 4];previous forest component roots=[0, 4]
操作後
closes cycle=否;actual cycle witness=[]
條件：無
兩端原已在同forest分量時，舊forest路徑加此真邊成cycle；否則連兩分量仍無cycle。逐全部選邊，不能只以n−1數量判。
規則：spantree_every_edge_cycle_or_forest_extension · 依賴：G001, S015
操作前
same complete vertex carrier Fin(5);parent connected=是;selected connected=是;all cycle witnesses=[]
操作後
selected acyclic=是;spanning tree validity=是
條件：無
parent若不連通便無spanning tree；parent連通時一次選取失敗不否定其他選取存在。這裡同時核全部頂點、selected connected與acyclic，孤立點沒有被刪掉。
規則：spantree_connectivity_acyclicity_full_vertices_conjunction · 依賴：G001, S016
S017 · 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。S016 · 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。S015 · 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。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。([0, 1, 4, 5], 是, 是, 是, 是)
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: e3e843f668025868f892cd23862df6407eb4fc2157b4a48370c784e18c277f3acandidate record SHA-256: 1f973f9becdab3dbb7912f80408872e61b156223a9f006fb96e423f3167477eesemantic review receipt SHA-256: 713db6309b22bf8b31f0253d4aaa7aca9011ab908587ae0e9d4a365edce1f9bcquestion-first witness SHA-256: 0338a3db344ab237113fc63be29153c3684da96e388109e9dcd913a0a693aed2Lean: NOT_RUN · human review: NOT_PERFORMED