R0036-18954 · q-20261014-advance-20261105-r04-eulerian-trail-0004
完整有限簡單無向圖、edge support與明示walk。操作=euler-trail-audit；資料={"n":6,"edges":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],"vertex_walk":[0,1,2,3,4,5,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。
G001 — 完整有限簡單無向圖、edge support與明示walk。操作=euler-trail-audit；資料={"n":6,"edges":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],"vertex_walk":[0,1,2,3,4,5,0]}。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=euler-trail-audit;original snapshot and mathematics retained
操作後
explicit ordered source output fields=[vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary]
條件：無
原題歷史介面標記仍在快照；這裡直接給原始欄位順序，主文不需JavaScript或外部表。描述欄位不自動證成任何數學聲稱。
規則：source_ordered_interface_explicit_without_javascript · 依賴：G001
操作前
complete ORIGINAL carrier Fin(6);canonical ORIGINAL simple undirected edges=[[0, 1], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]]
操作後
edge indices are EXACT original list positions0..5;retain ALL vertices;no12-vertex or10-edge bound
條件：無
逐原邊位置固定索引，不用頂點對的字串排序代替來源數字順序。此題族允許到60頂點，不沿用前題族上界；零邊圖僅在允許的操作出現。
規則：euler_original_simple_graph_indices_full_carrier · 依賴：G001, S001
操作前
original walk step=0;consecutive vertices=[0, 1];undirected candidate edge=[0, 1]
操作後
matching original edge index=0;recorded edge prefix=[0];length-and-adjacency status=是
條件：無
每個步長都查原圖；不存在的邊只使adjacency失敗，不補一個假索引到used。used只記實際找到的邊，故它的覆蓋檢查不能取代walk完整性。
規則：euler_every_walk_step_missing_edge_not_recorded · 依賴：G001, S002
操作前
original walk step=1;consecutive vertices=[1, 2];undirected candidate edge=[1, 2]
操作後
matching original edge index=2;recorded edge prefix=[0, 2];length-and-adjacency status=是
條件：無
每個步長都查原圖；不存在的邊只使adjacency失敗，不補一個假索引到used。used只記實際找到的邊，故它的覆蓋檢查不能取代walk完整性。
規則：euler_every_walk_step_missing_edge_not_recorded · 依賴：G001, S003
操作前
original walk step=2;consecutive vertices=[2, 3];undirected candidate edge=[2, 3]
操作後
matching original edge index=3;recorded edge prefix=[0, 2, 3];length-and-adjacency status=是
條件：無
每個步長都查原圖；不存在的邊只使adjacency失敗，不補一個假索引到used。used只記實際找到的邊，故它的覆蓋檢查不能取代walk完整性。
規則：euler_every_walk_step_missing_edge_not_recorded · 依賴：G001, S004
操作前
original walk step=3;consecutive vertices=[3, 4];undirected candidate edge=[3, 4]
操作後
matching original edge index=4;recorded edge prefix=[0, 2, 3, 4];length-and-adjacency status=是
條件：無
每個步長都查原圖；不存在的邊只使adjacency失敗，不補一個假索引到used。used只記實際找到的邊，故它的覆蓋檢查不能取代walk完整性。
規則：euler_every_walk_step_missing_edge_not_recorded · 依賴：G001, S005
操作前
original walk step=4;consecutive vertices=[4, 5];undirected candidate edge=[4, 5]
操作後
matching original edge index=5;recorded edge prefix=[0, 2, 3, 4, 5];length-and-adjacency status=是
條件：無
每個步長都查原圖；不存在的邊只使adjacency失敗，不補一個假索引到used。used只記實際找到的邊，故它的覆蓋檢查不能取代walk完整性。
規則：euler_every_walk_step_missing_edge_not_recorded · 依賴：G001, S006
操作前
original walk step=5;consecutive vertices=[5, 0];undirected candidate edge=[0, 5]
操作後
matching original edge index=1;recorded edge prefix=[0, 2, 3, 4, 5, 1];length-and-adjacency status=是
條件：無
每個步長都查原圖；不存在的邊只使adjacency失敗，不補一個假索引到used。used只記實際找到的邊，故它的覆蓋檢查不能取代walk完整性。
規則：euler_every_walk_step_missing_edge_not_recorded · 依賴：G001, S007
操作前
original complete walk=[0, 1, 2, 3, 4, 5, 0];required vertex-list length=7;actual length=7;found ORIGINAL edge indices=[0, 2, 3, 4, 5, 1]
操作後
length-and-all-adjacency=是;recorded indices unique=是;recorded complete single-use coverage=是;vertex revisited=是;VALID Euler trail=是
條件：無
Euler trail可重複頂點，真正禁止重複邊。最後valid是長度／全部adjacency與完整單次用邊的合取；例如沒有找到的額外步仍可能留下used剛好覆蓋全圖，但walk仍無效。一次壞walk不否認此圖另有Euler walk。
規則：euler_trail_full_conjunction_repeated_vertices_permitted · 依賴：G001, S008
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, 2, 3, 4, 5, 0], [0, 2, 3, 4, 5, 1], 是, 是, 是, 是, 是, trail_allows_repeated_vertices_not_edges)
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: ee2d75c555d705cb9dab4c841d39b88dba92f07916d33bf0c36773de3580a43fcandidate record SHA-256: cd2ab3be80821e9637f4c645fbae77f3906f1e9728e2b99786c13659542ba846semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2Lean: NOT_RUN · human review: NOT_PERFORMED