{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261105-r04-eulerian-trail-0011", "pilot_number": "R0036-18961", "number": 18961, "family_id": "qf-mod36-euler-trail-audit", "title": { "zh-Hant": "第 18961 題：Euler trail與circuit", "en": "Question 18961: Revisiting Vertices Is Allowed; Repeating an Edge Is Not" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261105-r04-eulerian-trail-0011", "batch_id": "batch-20261014-advance-20261105-r04-eulerian-trail-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-eulerian-trail" ], "title_zh": "Euler trail與circuit", "stem_zh": "完整有限簡單無向圖、edge support與明示walk。操作=euler-trail-audit；資料={\"n\":13,\"edges\":[[0,1],[0,12],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1], 是, 是, 是, 是, 是, trail_allows_repeated_vertices_not_edges)", "explanation_zh": "Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1], 是, 是, 是, 是, 是, trail_allows_repeated_vertices_not_edges)；依完整明示資料獨立重算；trail可重複頂點但不可重複邊；0或2個奇度判準還要求非孤立頂點同一分量；孤立點不破壞非零邊support；零邊非空圖採length-zero circuit且不推Hamilton性質。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "Euler trail與circuit" ] }, "variant": { "family_id": "qf-mod36-euler-trail-audit", "seed": 202611054011, "parameters": { "n": 13, "edges": [ [ 0, 1 ], [ 0, 12 ], [ 1, 2 ], [ 2, 3 ], [ 3, 4 ], [ 4, 5 ], [ 5, 6 ], [ 6, 7 ], [ 7, 8 ], [ 8, 9 ], [ 9, 10 ], [ 10, 11 ], [ 11, 12 ] ], "vertex_walk": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0 ], "domain": "Complete_finite_simple_undirected_graph", "operation": "euler-trail-audit", "semantic_template": "qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,12],[1,2],[10,11],[11,12],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":13,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,0]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261105-r04-eulerian-trail-0011", "batch_id": "batch-20261014-advance-20261105-r04-eulerian-trail-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "65a6378c5bb876f30002628536aba4ed833767951399798f432aa98b1aed7861", "family_id": "qf-mod36-euler-trail-audit", "seed": 202611054011, "semantic_template": "qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,12],[1,2],[10,11],[11,12],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":13,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,0]}" }, "fields": { "title": "Revisiting Vertices Is Allowed; Repeating an Edge Is Not", "stem": "Use the complete finite simple undirected graph on Fin(13)={0,…,12}, with canonical sorted edges=[[0, 1], [0, 12], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).", "answer": { "kind": "expression", "value": "([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)", "explanation": "Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)." }, "skills": [ "Revisiting Vertices Is Allowed; Repeating an Edge Is Not" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "完整有限簡單無向圖、edge support與明示walk。操作=euler-trail-audit；資料={\"n\":13,\"edges\":[[0,1],[0,12],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the complete finite simple undirected graph on Fin(13)={0,…,12}, with canonical sorted edges=[[0, 1], [0, 12], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=euler-trail-audit;original snapshot and mathematics retained", "after": "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]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "complete ORIGINAL carrier Fin(13);canonical ORIGINAL simple undirected edges=[[0, 1], [0, 12], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12]]", "after": "edge indices are EXACT original list positions0..12;retain ALL vertices;no12-vertex or10-edge bound", "relation": "implies", "rule_id": "euler_original_simple_graph_indices_full_carrier", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Keep edge indices at their original numeric positions,not string-sorted vertex pairs.This family permits up to60 vertices rather than importing earlier bounds;empty-edge graphs appear only where the operation allows them." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "original walk step=0;consecutive vertices=[0, 1];undirected candidate edge=[0, 1]", "after": "matching original edge index=0;recorded edge prefix=[0];length-and-adjacency status=是", "relation": "implies", "rule_id": "euler_every_walk_step_missing_edge_not_recorded", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "original walk step=1;consecutive vertices=[1, 2];undirected candidate edge=[1, 2]", "after": "matching original edge index=2;recorded edge prefix=[0, 2];length-and-adjacency status=是", "relation": "implies", "rule_id": "euler_every_walk_step_missing_edge_not_recorded", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "original walk step=2;consecutive vertices=[2, 3];undirected candidate edge=[2, 3]", "after": "matching original edge index=3;recorded edge prefix=[0, 2, 3];length-and-adjacency status=是", "relation": "implies", "rule_id": "euler_every_walk_step_missing_edge_not_recorded", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "original walk step=3;consecutive vertices=[3, 4];undirected candidate edge=[3, 4]", "after": "matching original edge index=4;recorded edge prefix=[0, 2, 3, 4];length-and-adjacency status=是", "relation": "implies", "rule_id": "euler_every_walk_step_missing_edge_not_recorded", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "original walk step=4;consecutive vertices=[4, 5];undirected candidate edge=[4, 5]", "after": "matching original edge index=5;recorded edge prefix=[0, 2, 3, 4, 5];length-and-adjacency status=是", "relation": "implies", "rule_id": "euler_every_walk_step_missing_edge_not_recorded", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "original walk step=5;consecutive vertices=[5, 6];undirected candidate edge=[5, 6]", "after": "matching original edge index=6;recorded edge prefix=[0, 2, 3, 4, 5, 6];length-and-adjacency status=是", "relation": "implies", "rule_id": "euler_every_walk_step_missing_edge_not_recorded", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "original walk step=6;consecutive vertices=[6, 7];undirected candidate edge=[6, 7]", "after": "matching original edge index=7;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7];length-and-adjacency status=是", "relation": "implies", "rule_id": "euler_every_walk_step_missing_edge_not_recorded", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "original walk step=7;consecutive vertices=[7, 8];undirected candidate edge=[7, 8]", "after": "matching original edge index=8;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8];length-and-adjacency status=是", "relation": "implies", "rule_id": "euler_every_walk_step_missing_edge_not_recorded", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "original walk step=8;consecutive vertices=[8, 9];undirected candidate edge=[8, 9]", "after": "matching original edge index=9;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9];length-and-adjacency status=是", "relation": "implies", "rule_id": "euler_every_walk_step_missing_edge_not_recorded", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "original walk step=9;consecutive vertices=[9, 10];undirected candidate edge=[9, 10]", "after": "matching original edge index=10;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10];length-and-adjacency status=是", "relation": "implies", "rule_id": "euler_every_walk_step_missing_edge_not_recorded", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "original walk step=10;consecutive vertices=[10, 11];undirected candidate edge=[10, 11]", "after": "matching original edge index=11;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11];length-and-adjacency status=是", "relation": "implies", "rule_id": "euler_every_walk_step_missing_edge_not_recorded", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "original walk step=11;consecutive vertices=[11, 12];undirected candidate edge=[11, 12]", "after": "matching original edge index=12;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];length-and-adjacency status=是", "relation": "implies", "rule_id": "euler_every_walk_step_missing_edge_not_recorded", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "original walk step=12;consecutive vertices=[12, 0];undirected candidate edge=[0, 12]", "after": "matching original edge index=1;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1];length-and-adjacency status=是", "relation": "implies", "rule_id": "euler_every_walk_step_missing_edge_not_recorded", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "original complete walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0];required vertex-list length=14;actual length=14;found ORIGINAL edge indices=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1]", "after": "length-and-all-adjacency=是;recorded indices unique=是;recorded complete single-use coverage=是;vertex revisited=是;VALID Euler trail=是", "relation": "implies", "rule_id": "euler_trail_full_conjunction_repeated_vertices_permitted", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "An Euler trail may revisit vertices but not edges.Final validity conjoins length/all-adjacency with complete single-use edge coverage.Extra missing steps can leave a recorded list that covers every edge while invalidating the walk.One bad supplied walk does not disprove existence of another Euler walk." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0 ], [ 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1 ], true, true, true, true, true, "trail_allows_repeated_vertices_not_edges" ] }, "conclusion": { "canonical_value": "([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1], 是, 是, 是, 是, 是, trail_allows_repeated_vertices_not_edges)", "explanation": { "zh-Hant": "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。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "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。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261105-r04-eulerian-trail-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261105-r04-eulerian-trail-foundations/questions.jsonl", "batch_sha256": "290234a8be5114292ccdef5d758669877237520e20f4d7270260f7f29b4c745d", "original_line": 11, "original_line_sha256": "838e8dd96002bf7db46f40ae87c1f6f026f937398a3e2fc35dd4a6414ebd365d", "question_compact_sha256": "65a6378c5bb876f30002628536aba4ed833767951399798f432aa98b1aed7861", "english_row_compact_sha256": "983c3e42cf7b5a35f4b70cdc6442b49f6a994803416193aa737cd1608b12ac3e", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261105-r04-eulerian-trail-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261105-r04-eulerian-trail-foundations.en.jsonl" }, "mko_bindings": { "mko-eulerian-trail": { "sha256": "bd4c7accbd4091e00d7a37abe72e796dc663a93d6cdb230cfaa24a0ec125ad41", "english_sha256": "e03f13cd5f3980be232f919922b722c842b0d6d7a7dca8b3ba1e5ea56bf7f340", "original_path": "public/data/mko/eulerian-trail.json", "english_path": "public/data/i18n/editions/mko-eulerian-trail.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "49e5d7674cefa01298c2386543f777d2bdfc99446b4027156b80142f10031994", "original_candidate_sha256": "e922e5c2ce22bfb732a6cf505b87f7ef66924fe0eabdde3a88544dad32b51047", "original_candidate_line_sha256": "e922e5c2ce22bfb732a6cf505b87f7ef66924fe0eabdde3a88544dad32b51047" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-008-p37", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301", "question_first_witness_sha256": "acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 16, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "fd3ca770fda2aeafbf71923890df4d419cd1502768f2767c9c9b3f10d63834be", "arithmetic_steps": 0 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }