{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0001","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\":3,\"edges\":[[0,1],[0,2],[1,2]],\"vertex_walk\":[0,1,2,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 0], [0, 2, 1], 是, 是, 是, 是, 是, trail_allows_repeated_vertices_not_edges)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 0], [0, 2, 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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-trail-audit","seed":202611054001,"parameters":{"n":3,"edges":[[0,1],[0,2],[1,2]],"vertex_walk":[0,1,2,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,2],[1,2]],\"n\":3,\"vertex_walk\":[0,1,2,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0002","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\":4,\"edges\":[[0,1],[0,3],[1,2],[2,3]],\"vertex_walk\":[0,1,2,3,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 0], [0, 2, 3, 1], 是, 是, 是, 是, 是, trail_allows_repeated_vertices_not_edges)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 0], [0, 2, 3, 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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-trail-audit","seed":202611054002,"parameters":{"n":4,"edges":[[0,1],[0,3],[1,2],[2,3]],"vertex_walk":[0,1,2,3,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,3],[1,2],[2,3]],\"n\":4,\"vertex_walk\":[0,1,2,3,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0003","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\":5,\"edges\":[[0,1],[0,4],[1,2],[2,3],[3,4]],\"vertex_walk\":[0,1,2,3,4,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 0], [0, 2, 3, 4, 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, 0], [0, 2, 3, 4, 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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-trail-audit","seed":202611054003,"parameters":{"n":5,"edges":[[0,1],[0,4],[1,2],[2,3],[3,4]],"vertex_walk":[0,1,2,3,4,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,4],[1,2],[2,3],[3,4]],\"n\":5,\"vertex_walk\":[0,1,2,3,4,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0004","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\":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。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 0], [0, 2, 3, 4, 5, 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, 0], [0, 2, 3, 4, 5, 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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-trail-audit","seed":202611054004,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],"vertex_walk":[0,1,2,3,4,5,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"vertex_walk\":[0,1,2,3,4,5,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0005","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\":7,\"edges\":[[0,1],[0,6],[1,2],[2,3],[3,4],[4,5],[5,6]],\"vertex_walk\":[0,1,2,3,4,5,6,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 0], [0, 2, 3, 4, 5, 6, 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, 0], [0, 2, 3, 4, 5, 6, 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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-trail-audit","seed":202611054005,"parameters":{"n":7,"edges":[[0,1],[0,6],[1,2],[2,3],[3,4],[4,5],[5,6]],"vertex_walk":[0,1,2,3,4,5,6,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,6],[1,2],[2,3],[3,4],[4,5],[5,6]],\"n\":7,\"vertex_walk\":[0,1,2,3,4,5,6,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0006","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\":8,\"edges\":[[0,1],[0,7],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],\"vertex_walk\":[0,1,2,3,4,5,6,7,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 0], [0, 2, 3, 4, 5, 6, 7, 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, 0], [0, 2, 3, 4, 5, 6, 7, 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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-trail-audit","seed":202611054006,"parameters":{"n":8,"edges":[[0,1],[0,7],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"vertex_walk":[0,1,2,3,4,5,6,7,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,7],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],\"n\":8,\"vertex_walk\":[0,1,2,3,4,5,6,7,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0007","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\":9,\"edges\":[[0,1],[0,8],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,0]}。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, 0], [0, 2, 3, 4, 5, 6, 7, 8, 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, 0], [0, 2, 3, 4, 5, 6, 7, 8, 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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-trail-audit","seed":202611054007,"parameters":{"n":9,"edges":[[0,1],[0,8],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8]],"vertex_walk":[0,1,2,3,4,5,6,7,8,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,8],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8]],\"n\":9,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0008","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\":10,\"edges\":[[0,1],[0,9],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,0]}。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, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 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, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-trail-audit","seed":202611054008,"parameters":{"n":10,"edges":[[0,1],[0,9],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,9],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9]],\"n\":10,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0009","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\":11,\"edges\":[[0,1],[0,10],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,0]}。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, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 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, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 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":202611054009,"parameters":{"n":11,"edges":[[0,1],[0,10],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,10],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":11,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0010","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\":12,\"edges\":[[0,1],[0,11],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,0]}。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, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 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, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 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":202611054010,"parameters":{"n":12,"edges":[[0,1],[0,11],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,11],[1,2],[10,11],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":12,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,0]}"}},"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"}}
{"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0012","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\":14,\"edges\":[[0,1],[0,13],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,0]}。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, 13, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 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, 13, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 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":202611054012,"parameters":{"n":14,"edges":[[0,1],[0,13],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,13],[1,2],[10,11],[11,12],[12,13],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":14,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0013","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\":15,\"edges\":[[0,1],[0,14],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,0]}。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, 13, 14, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 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, 13, 14, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 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":202611054013,"parameters":{"n":15,"edges":[[0,1],[0,14],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,14],[1,2],[10,11],[11,12],[12,13],[13,14],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":15,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0014","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\":16,\"edges\":[[0,1],[0,15],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0]}。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, 13, 14, 15, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 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, 13, 14, 15, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 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":202611054014,"parameters":{"n":16,"edges":[[0,1],[0,15],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,15],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":16,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0015","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\":17,\"edges\":[[0,1],[0,16],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0]}。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, 13, 14, 15, 16, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 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, 13, 14, 15, 16, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 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":202611054015,"parameters":{"n":17,"edges":[[0,1],[0,16],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,16],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":17,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0016","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\":18,\"edges\":[[0,1],[0,17],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0]}。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, 13, 14, 15, 16, 17, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 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, 13, 14, 15, 16, 17, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 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":202611054016,"parameters":{"n":18,"edges":[[0,1],[0,17],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,17],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":18,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0017","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\":19,\"edges\":[[0,1],[0,18],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0]}。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, 13, 14, 15, 16, 17, 18, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 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, 13, 14, 15, 16, 17, 18, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 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":202611054017,"parameters":{"n":19,"edges":[[0,1],[0,18],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,18],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":19,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0018","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\":20,\"edges\":[[0,1],[0,19],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0]}。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, 13, 14, 15, 16, 17, 18, 19, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 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, 13, 14, 15, 16, 17, 18, 19, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-trail-audit","seed":202611054018,"parameters":{"n":20,"edges":[[0,1],[0,19],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,19],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":20,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0019","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\":21,\"edges\":[[0,1],[0,20],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0]}。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, 13, 14, 15, 16, 17, 18, 19, 20, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 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, 13, 14, 15, 16, 17, 18, 19, 20, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-trail-audit","seed":202611054019,"parameters":{"n":21,"edges":[[0,1],[0,20],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,20],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":21,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0020","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\":22,\"edges\":[[0,1],[0,21],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0]}。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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-trail-audit","seed":202611054020,"parameters":{"n":22,"edges":[[0,1],[0,21],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,21],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":22,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0021","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\":23,\"edges\":[[0,1],[0,22],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0]}。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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-trail-audit","seed":202611054021,"parameters":{"n":23,"edges":[[0,1],[0,22],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,22],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":23,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0022","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\":24,\"edges\":[[0,1],[0,23],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0]}。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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-trail-audit","seed":202611054022,"parameters":{"n":24,"edges":[[0,1],[0,23],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,23],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":24,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0023","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\":25,\"edges\":[[0,1],[0,24],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0]}。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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-trail-audit","seed":202611054023,"parameters":{"n":25,"edges":[[0,1],[0,24],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,24],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[23,24],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":25,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0024","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\":26,\"edges\":[[0,1],[0,25],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24],[24,25]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0]}。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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-trail-audit","seed":202611054024,"parameters":{"n":26,"edges":[[0,1],[0,25],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24],[24,25]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,25],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[23,24],[24,25],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":26,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0025","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\":27,\"edges\":[[0,1],[0,26],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24],[24,25],[25,26]],\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0]}。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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-trail-audit","seed":202611054025,"parameters":{"n":27,"edges":[[0,1],[0,26],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24],[24,25],[25,26]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-trail-audit","semantic_template":"qf-mod36-euler-trail-audit/{\"edges\":[[0,1],[0,26],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[23,24],[24,25],[25,26],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":27,\"vertex_walk\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,0]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0026","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-odd-degree-classification；資料={\"n\":2,\"edges\":[[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 1], [0, 1], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 1], [0, 1], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054026,"parameters":{"n":2,"edges":[[0,1]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1]],\"n\":2}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0027","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-odd-degree-classification；資料={\"n\":3,\"edges\":[[0,1],[1,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 1], [0, 2], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 1], [0, 2], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054027,"parameters":{"n":3,"edges":[[0,1],[1,2]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2]],\"n\":3}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0028","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-odd-degree-classification；資料={\"n\":4,\"edges\":[[0,1],[1,2],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 1], [0, 3], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 1], [0, 3], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054028,"parameters":{"n":4,"edges":[[0,1],[1,2],[2,3]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[2,3]],\"n\":4}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0029","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-odd-degree-classification；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 1], [0, 4], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 1], [0, 4], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054029,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3],[3,4]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":5}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0030","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-odd-degree-classification；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 1], [0, 5], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 1], [0, 5], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054030,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0031","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-odd-degree-classification；資料={\"n\":7,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 1], [0, 6], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 1], [0, 6], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054031,"parameters":{"n":7,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6]],\"n\":7}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0032","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-odd-degree-classification；資料={\"n\":8,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 1], [0, 7], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 1], [0, 7], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054032,"parameters":{"n":8,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],\"n\":8}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0033","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-odd-degree-classification；資料={\"n\":9,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 1], [0, 8], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 1], [0, 8], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054033,"parameters":{"n":9,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8]],\"n\":9}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0034","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-odd-degree-classification；資料={\"n\":10,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 9], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 9], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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-odd-degree-classification","seed":202611054034,"parameters":{"n":10,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9]],\"n\":10}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0035","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-odd-degree-classification；資料={\"n\":11,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 10], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 10], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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-odd-degree-classification","seed":202611054035,"parameters":{"n":11,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":11}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0036","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-odd-degree-classification；資料={\"n\":12,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 11], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 11], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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-odd-degree-classification","seed":202611054036,"parameters":{"n":12,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":12}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0037","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-odd-degree-classification；資料={\"n\":13,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 12], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 12], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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-odd-degree-classification","seed":202611054037,"parameters":{"n":13,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":13}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0038","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-odd-degree-classification；資料={\"n\":14,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 13], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 13], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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-odd-degree-classification","seed":202611054038,"parameters":{"n":14,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":14}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0039","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-odd-degree-classification；資料={\"n\":15,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 14], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 14], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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-odd-degree-classification","seed":202611054039,"parameters":{"n":15,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":15}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0040","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-odd-degree-classification；資料={\"n\":16,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 15], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 15], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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-odd-degree-classification","seed":202611054040,"parameters":{"n":16,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":16}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0041","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-odd-degree-classification；資料={\"n\":17,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 16], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 16], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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-odd-degree-classification","seed":202611054041,"parameters":{"n":17,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":17}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0042","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-odd-degree-classification；資料={\"n\":18,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 17], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 17], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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-odd-degree-classification","seed":202611054042,"parameters":{"n":18,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":18}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0043","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-odd-degree-classification；資料={\"n\":19,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 18], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 18], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054043,"parameters":{"n":19,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":19}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0044","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-odd-degree-classification；資料={\"n\":20,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 19], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 19], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054044,"parameters":{"n":20,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":20}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0045","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-odd-degree-classification；資料={\"n\":21,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 20], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 20], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054045,"parameters":{"n":21,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":21}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0046","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-odd-degree-classification；資料={\"n\":22,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 21], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 21], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054046,"parameters":{"n":22,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":22}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0047","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-odd-degree-classification；資料={\"n\":23,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 22], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 22], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054047,"parameters":{"n":23,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":23}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0048","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-odd-degree-classification；資料={\"n\":24,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 23], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 23], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054048,"parameters":{"n":24,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":24}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0049","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-odd-degree-classification；資料={\"n\":25,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 24], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 24], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054049,"parameters":{"n":25,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[23,24],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":25}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0050","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-odd-degree-classification；資料={\"n\":26,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24],[24,25]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 25], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)","explanation_zh":"Wave5第五層核對=([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 25], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-odd-degree-classification","seed":202611054050,"parameters":{"n":26,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24],[24,25]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-odd-degree-classification","semantic_template":"qf-mod36-euler-odd-degree-classification/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[23,24],[24,25],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":26}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0051","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-support-connectivity；資料={\"n\":3,\"edges\":[[0,1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], 是, 否, [0, 1], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1], 是, 否, [0, 1], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-support-connectivity","seed":202611054051,"parameters":{"n":3,"edges":[[0,1]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1]],\"n\":3}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0052","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-support-connectivity；資料={\"n\":4,\"edges\":[[0,1],[1,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 否, [0, 2], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2], 是, 否, [0, 2], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-support-connectivity","seed":202611054052,"parameters":{"n":4,"edges":[[0,1],[1,2]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2]],\"n\":4}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0053","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-support-connectivity；資料={\"n\":5,\"edges\":[[0,1],[1,2],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3], 是, 否, [0, 3], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3], 是, 否, [0, 3], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-support-connectivity","seed":202611054053,"parameters":{"n":5,"edges":[[0,1],[1,2],[2,3]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[2,3]],\"n\":5}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0054","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-support-connectivity；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4], 是, 否, [0, 4], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4], 是, 否, [0, 4], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-support-connectivity","seed":202611054054,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0055","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-support-connectivity；資料={\"n\":7,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5], 是, 否, [0, 5], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5], 是, 否, [0, 5], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-support-connectivity","seed":202611054055,"parameters":{"n":7,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":7}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0056","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-support-connectivity；資料={\"n\":8,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6], 是, 否, [0, 6], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6], 是, 否, [0, 6], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-support-connectivity","seed":202611054056,"parameters":{"n":8,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6]],\"n\":8}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0057","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-support-connectivity；資料={\"n\":9,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], 是, 否, [0, 7], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7], 是, 否, [0, 7], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-support-connectivity","seed":202611054057,"parameters":{"n":9,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],\"n\":9}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0058","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-support-connectivity；資料={\"n\":10,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8]]}。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], 是, 否, [0, 8], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8], 是, 否, [0, 8], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-support-connectivity","seed":202611054058,"parameters":{"n":10,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8]],\"n\":10}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0059","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-support-connectivity；資料={\"n\":11,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9]]}。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], 是, 否, [0, 9], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 是, 否, [0, 9], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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-support-connectivity","seed":202611054059,"parameters":{"n":11,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9]],\"n\":11}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0060","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-support-connectivity；資料={\"n\":12,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]]}。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], 是, 否, [0, 10], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 是, 否, [0, 10], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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-support-connectivity","seed":202611054060,"parameters":{"n":12,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":12}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0061","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-support-connectivity；資料={\"n\":13,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11]]}。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], 是, 否, [0, 11], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是, 否, [0, 11], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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-support-connectivity","seed":202611054061,"parameters":{"n":13,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":13}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0062","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-support-connectivity；資料={\"n\":14,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12]]}。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], 是, 否, [0, 12], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 是, 否, [0, 12], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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-support-connectivity","seed":202611054062,"parameters":{"n":14,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":14}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0063","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-support-connectivity；資料={\"n\":15,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13]]}。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, 13], 是, 否, [0, 13], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 是, 否, [0, 13], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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-support-connectivity","seed":202611054063,"parameters":{"n":15,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":15}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0064","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-support-connectivity；資料={\"n\":16,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14]]}。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, 13, 14], 是, 否, [0, 14], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], 是, 否, [0, 14], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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-support-connectivity","seed":202611054064,"parameters":{"n":16,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":16}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0065","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-support-connectivity；資料={\"n\":17,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15]]}。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, 13, 14, 15], 是, 否, [0, 15], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], 是, 否, [0, 15], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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-support-connectivity","seed":202611054065,"parameters":{"n":17,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":17}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0066","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-support-connectivity；資料={\"n\":18,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16]]}。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, 13, 14, 15, 16], 是, 否, [0, 16], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 是, 否, [0, 16], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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-support-connectivity","seed":202611054066,"parameters":{"n":18,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":18}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0067","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-support-connectivity；資料={\"n\":19,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17]]}。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, 13, 14, 15, 16, 17], 是, 否, [0, 17], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], 是, 否, [0, 17], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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-support-connectivity","seed":202611054067,"parameters":{"n":19,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":19}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0068","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-support-connectivity；資料={\"n\":20,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18]]}。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, 13, 14, 15, 16, 17, 18], 是, 否, [0, 18], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], 是, 否, [0, 18], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-support-connectivity","seed":202611054068,"parameters":{"n":20,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":20}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0069","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-support-connectivity；資料={\"n\":21,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19]]}。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, 13, 14, 15, 16, 17, 18, 19], 是, 否, [0, 19], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], 是, 否, [0, 19], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-support-connectivity","seed":202611054069,"parameters":{"n":21,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":21}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0070","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-support-connectivity；資料={\"n\":22,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20]]}。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, 13, 14, 15, 16, 17, 18, 19, 20], 是, 否, [0, 20], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], 是, 否, [0, 20], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-support-connectivity","seed":202611054070,"parameters":{"n":22,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":22}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0071","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-support-connectivity；資料={\"n\":23,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21]]}。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, 13, 14, 15, 16, 17, 18, 19, 20, 21], 是, 否, [0, 21], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], 是, 否, [0, 21], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-support-connectivity","seed":202611054071,"parameters":{"n":23,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":23}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0072","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-support-connectivity；資料={\"n\":24,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22]]}。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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], 是, 否, [0, 22], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], 是, 否, [0, 22], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-support-connectivity","seed":202611054072,"parameters":{"n":24,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":24}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0073","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-support-connectivity；資料={\"n\":25,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23]]}。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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], 是, 否, [0, 23], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], 是, 否, [0, 23], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-support-connectivity","seed":202611054073,"parameters":{"n":25,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":25}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0074","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-support-connectivity；資料={\"n\":26,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24]]}。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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], 是, 否, [0, 24], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], 是, 否, [0, 24], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-support-connectivity","seed":202611054074,"parameters":{"n":26,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[23,24],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":26}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0075","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-support-connectivity；資料={\"n\":27,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24],[24,25]]}。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, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], 是, 否, [0, 25], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)","explanation_zh":"Wave5第五層核對=([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], 是, 否, [0, 25], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-support-connectivity","seed":202611054075,"parameters":{"n":27,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24],[24,25]],"domain":"Complete_finite_simple_undirected_graph","operation":"euler-support-connectivity","semantic_template":"qf-mod36-euler-support-connectivity/{\"edges\":[[0,1],[1,2],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[2,3],[20,21],[21,22],[22,23],[23,24],[24,25],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]],\"n\":27}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0076","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-zero-edge-convention；資料={\"n\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(1, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054076,"parameters":{"n":1,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":1}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0077","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-zero-edge-convention；資料={\"n\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(2, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054077,"parameters":{"n":2,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":2}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0078","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-zero-edge-convention；資料={\"n\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(3, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054078,"parameters":{"n":3,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":3}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0079","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-zero-edge-convention；資料={\"n\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(4, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054079,"parameters":{"n":4,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":4}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0080","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-zero-edge-convention；資料={\"n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(5, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054080,"parameters":{"n":5,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":5}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0081","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-zero-edge-convention；資料={\"n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(6, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054081,"parameters":{"n":6,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":6}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0082","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-zero-edge-convention；資料={\"n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(7, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054082,"parameters":{"n":7,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":7}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0083","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-zero-edge-convention；資料={\"n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(8, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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":1,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054083,"parameters":{"n":8,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":8}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0084","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-zero-edge-convention；資料={\"n\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(9, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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-zero-edge-convention","seed":202611054084,"parameters":{"n":9,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":9}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0085","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-zero-edge-convention；資料={\"n\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(10, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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-zero-edge-convention","seed":202611054085,"parameters":{"n":10,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":10}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0086","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-zero-edge-convention；資料={\"n\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(11, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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-zero-edge-convention","seed":202611054086,"parameters":{"n":11,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":11}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0087","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-zero-edge-convention；資料={\"n\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(12, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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-zero-edge-convention","seed":202611054087,"parameters":{"n":12,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":12}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0088","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-zero-edge-convention；資料={\"n\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(13, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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-zero-edge-convention","seed":202611054088,"parameters":{"n":13,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":13}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0089","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-zero-edge-convention；資料={\"n\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(14, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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-zero-edge-convention","seed":202611054089,"parameters":{"n":14,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":14}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0090","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-zero-edge-convention；資料={\"n\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(15, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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-zero-edge-convention","seed":202611054090,"parameters":{"n":15,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":15}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0091","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-zero-edge-convention；資料={\"n\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(16, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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-zero-edge-convention","seed":202611054091,"parameters":{"n":16,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":16}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0092","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-zero-edge-convention；資料={\"n\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(17, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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-zero-edge-convention","seed":202611054092,"parameters":{"n":17,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":17}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0093","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-zero-edge-convention；資料={\"n\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(18, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054093,"parameters":{"n":18,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":18}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0094","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-zero-edge-convention；資料={\"n\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(19, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054094,"parameters":{"n":19,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":19}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0095","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-zero-edge-convention；資料={\"n\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(20, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054095,"parameters":{"n":20,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":20}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0096","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-zero-edge-convention；資料={\"n\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(21, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054096,"parameters":{"n":21,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":21}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0097","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-zero-edge-convention；資料={\"n\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(22, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054097,"parameters":{"n":22,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":22}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0098","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-zero-edge-convention；資料={\"n\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(23, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054098,"parameters":{"n":23,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":23}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0099","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-zero-edge-convention；資料={\"n\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(24, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054099,"parameters":{"n":24,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":24}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r04-eulerian-trail-0100","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-zero-edge-convention；資料={\"n\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)","explanation_zh":"Wave5第五層核對=(25, 0, 是, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)；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":3,"skills":["Euler trail與circuit"]},"variant":{"family_id":"qf-mod36-euler-zero-edge-convention","seed":202611054100,"parameters":{"n":25,"domain":"Complete_finite_simple_undirected_graph","operation":"euler-zero-edge-convention","semantic_template":"qf-mod36-euler-zero-edge-convention/{\"n\":25}"}},"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"}}