{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r02-bipartite-graph-0001","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,0,0,0,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 0, 0, 0, 0], [[0, 1], [1, 2], [2, 3], [3, 4]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([0, 0, 0, 0, 0, 0], [[0, 1], [1, 2], [2, 3], [3, 4]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312001,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,0,0,0,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0002","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,0,0,0,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 0, 0, 0, 0], [[1, 2], [2, 3], [3, 4]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([1, 0, 0, 0, 0, 0], [[1, 2], [2, 3], [3, 4]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312002,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,0,0,0,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0003","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,0,0,0,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 0, 0, 0], [[2, 3], [3, 4]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([0, 1, 0, 0, 0, 0], [[2, 3], [3, 4]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312003,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,0,0,0,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0004","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,0,0,0,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 0, 0, 0, 0], [[0, 1], [2, 3], [3, 4]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([1, 1, 0, 0, 0, 0], [[0, 1], [2, 3], [3, 4]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312004,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,0,0,0,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0005","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,1,0,0,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 1, 0, 0, 0], [[0, 1], [3, 4]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([0, 0, 1, 0, 0, 0], [[0, 1], [3, 4]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312005,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,1,0,0,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0006","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,1,0,0,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1, 0, 0, 0], [[3, 4]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([1, 0, 1, 0, 0, 0], [[3, 4]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312006,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,1,0,0,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0007","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,1,0,0,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 1, 0, 0, 0], [[1, 2], [3, 4]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([0, 1, 1, 0, 0, 0], [[1, 2], [3, 4]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312007,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,1,0,0,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0008","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,1,0,0,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1, 0, 0, 0], [[0, 1], [1, 2], [3, 4]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([1, 1, 1, 0, 0, 0], [[0, 1], [1, 2], [3, 4]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312008,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,1,0,0,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0009","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,0,1,0,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 0, 1, 0, 0], [[0, 1], [1, 2]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([0, 0, 0, 1, 0, 0], [[0, 1], [1, 2]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312009,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,0,1,0,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0010","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,0,1,0,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 0, 1, 0, 0], [[1, 2]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([1, 0, 0, 1, 0, 0], [[1, 2]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312010,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,0,1,0,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0011","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,0,1,0,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 0], [], 是, bipartition_certificate)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 0], [], 是, bipartition_certificate)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312011,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,0,1,0,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0012","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,0,1,0,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 0, 1, 0, 0], [[0, 1]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([1, 1, 0, 1, 0, 0], [[0, 1]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312012,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,0,1,0,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0013","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,1,1,0,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 1, 1, 0, 0], [[0, 1], [2, 3]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([0, 0, 1, 1, 0, 0], [[0, 1], [2, 3]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312013,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,1,1,0,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0014","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,1,1,0,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1, 1, 0, 0], [[2, 3]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([1, 0, 1, 1, 0, 0], [[2, 3]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312014,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,1,1,0,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0015","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,1,1,0,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 1, 1, 0, 0], [[1, 2], [2, 3]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([0, 1, 1, 1, 0, 0], [[1, 2], [2, 3]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312015,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,1,1,0,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0016","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,1,1,0,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1, 1, 0, 0], [[0, 1], [1, 2], [2, 3]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([1, 1, 1, 1, 0, 0], [[0, 1], [1, 2], [2, 3]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312016,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,1,1,0,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0017","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,0,0,1,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 0, 0, 1, 0], [[0, 1], [1, 2], [2, 3]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([0, 0, 0, 0, 1, 0], [[0, 1], [1, 2], [2, 3]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312017,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,0,0,1,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0018","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,0,0,1,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 0, 0, 1, 0], [[1, 2], [2, 3]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([1, 0, 0, 0, 1, 0], [[1, 2], [2, 3]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312018,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,0,0,1,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0019","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,0,0,1,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 0, 1, 0], [[2, 3]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([0, 1, 0, 0, 1, 0], [[2, 3]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312019,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,0,0,1,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0020","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,0,0,1,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 0, 0, 1, 0], [[0, 1], [2, 3]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([1, 1, 0, 0, 1, 0], [[0, 1], [2, 3]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312020,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,0,0,1,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0021","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,1,0,1,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 1, 0, 1, 0], [[0, 1]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([0, 0, 1, 0, 1, 0], [[0, 1]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312021,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,1,0,1,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0022","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,1,0,1,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1, 0, 1, 0], [], 是, bipartition_certificate)","explanation_zh":"Wave4收束核對=([1, 0, 1, 0, 1, 0], [], 是, bipartition_certificate)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312022,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,1,0,1,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0023","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,1,0,1,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 1, 0, 1, 0], [[1, 2]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([0, 1, 1, 0, 1, 0], [[1, 2]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312023,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,1,0,1,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0024","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,1,0,1,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1, 0, 1, 0], [[0, 1], [1, 2]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([1, 1, 1, 0, 1, 0], [[0, 1], [1, 2]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312024,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,1,0,1,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0025","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-partition-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,0,1,1,0]}。請依 [parts,conflicting_edges,proper,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 0, 1, 1, 0], [[0, 1], [1, 2], [3, 4]], 否, this_partition_fails_not_nonbipartite_proof)","explanation_zh":"Wave4收束核對=([0, 0, 0, 1, 1, 0], [[0, 1], [1, 2], [3, 4]], 否, this_partition_fails_not_nonbipartite_proof)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-partition-audit","seed":202610312025,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,0,1,1,0],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-partition-audit","semantic_template":"qf-mod31-bipartite-partition-audit/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,1,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0026","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,0,0,0,0],\"edge_index\":0}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1], 0, 0, 否, 否)","explanation_zh":"Wave4收束核對=(0, [0, 1], 0, 0, 否, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312026,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,0,0,0,0],"edge_index":0,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":0,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0027","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,0,0,0,0],\"edge_index\":1}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(1, [1, 2], 0, 0, 否, 否)","explanation_zh":"Wave4收束核對=(1, [1, 2], 0, 0, 否, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312027,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,0,0,0,0],"edge_index":1,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":1,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0028","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,0,0,0,0],\"edge_index\":2}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(2, [2, 3], 0, 0, 否, 否)","explanation_zh":"Wave4收束核對=(2, [2, 3], 0, 0, 否, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312028,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,0,0,0,0],"edge_index":2,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":2,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0029","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,0,0,0,0],\"edge_index\":3}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(3, [3, 4], 0, 0, 否, 否)","explanation_zh":"Wave4收束核對=(3, [3, 4], 0, 0, 否, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312029,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,0,0,0,0],"edge_index":3,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":3,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0030","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,1,0,0,0],\"edge_index\":0}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1], 0, 0, 否, 否)","explanation_zh":"Wave4收束核對=(0, [0, 1], 0, 0, 否, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312030,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,1,0,0,0],"edge_index":0,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":0,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0031","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,1,0,0,0],\"edge_index\":1}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(1, [1, 2], 0, 1, 是, 否)","explanation_zh":"Wave4收束核對=(1, [1, 2], 0, 1, 是, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312031,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,1,0,0,0],"edge_index":1,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":1,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0032","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,1,0,0,0],\"edge_index\":2}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(2, [2, 3], 1, 0, 是, 否)","explanation_zh":"Wave4收束核對=(2, [2, 3], 1, 0, 是, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312032,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,1,0,0,0],"edge_index":2,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":2,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0033","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,1,0,0,0],\"edge_index\":3}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(3, [3, 4], 0, 0, 否, 否)","explanation_zh":"Wave4收束核對=(3, [3, 4], 0, 0, 否, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312033,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,1,0,0,0],"edge_index":3,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":3,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0034","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,0,1,0,0],\"edge_index\":0}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1], 0, 0, 否, 否)","explanation_zh":"Wave4收束核對=(0, [0, 1], 0, 0, 否, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312034,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,0,1,0,0],"edge_index":0,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":0,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0035","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,0,1,0,0],\"edge_index\":1}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(1, [1, 2], 0, 0, 否, 否)","explanation_zh":"Wave4收束核對=(1, [1, 2], 0, 0, 否, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312035,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,0,1,0,0],"edge_index":1,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":1,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0036","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,0,1,0,0],\"edge_index\":2}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(2, [2, 3], 0, 1, 是, 是)","explanation_zh":"Wave4收束核對=(2, [2, 3], 0, 1, 是, 是)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312036,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,0,1,0,0],"edge_index":2,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":2,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0037","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,0,1,0,0],\"edge_index\":3}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(3, [3, 4], 1, 0, 是, 否)","explanation_zh":"Wave4收束核對=(3, [3, 4], 1, 0, 是, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312037,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,0,1,0,0],"edge_index":3,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":3,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0038","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,1,1,0,0],\"edge_index\":0}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1], 0, 0, 否, 否)","explanation_zh":"Wave4收束核對=(0, [0, 1], 0, 0, 否, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312038,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,1,1,0,0],"edge_index":0,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":0,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0039","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,1,1,0,0],\"edge_index\":1}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(1, [1, 2], 0, 1, 是, 否)","explanation_zh":"Wave4收束核對=(1, [1, 2], 0, 1, 是, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312039,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,1,1,0,0],"edge_index":1,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":1,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0040","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,1,1,0,0],\"edge_index\":2}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(2, [2, 3], 1, 1, 否, 否)","explanation_zh":"Wave4收束核對=(2, [2, 3], 1, 1, 否, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312040,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,1,1,0,0],"edge_index":2,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":2,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0041","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,1,1,0,0],\"edge_index\":3}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(3, [3, 4], 1, 0, 是, 否)","explanation_zh":"Wave4收束核對=(3, [3, 4], 1, 0, 是, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312041,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,1,1,0,0],"edge_index":3,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":3,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0042","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,0,0,1,0],\"edge_index\":0}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1], 0, 0, 否, 否)","explanation_zh":"Wave4收束核對=(0, [0, 1], 0, 0, 否, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312042,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,0,0,1,0],"edge_index":0,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":0,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0043","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,0,0,1,0],\"edge_index\":1}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(1, [1, 2], 0, 0, 否, 否)","explanation_zh":"Wave4收束核對=(1, [1, 2], 0, 0, 否, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312043,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,0,0,1,0],"edge_index":1,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":1,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0044","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,0,0,1,0],\"edge_index\":2}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(2, [2, 3], 0, 0, 否, 否)","explanation_zh":"Wave4收束核對=(2, [2, 3], 0, 0, 否, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312044,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,0,0,1,0],"edge_index":2,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":2,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0045","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,0,0,1,0],\"edge_index\":3}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(3, [3, 4], 0, 1, 是, 否)","explanation_zh":"Wave4收束核對=(3, [3, 4], 0, 1, 是, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312045,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,0,0,1,0],"edge_index":3,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":3,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0046","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,1,0,1,0],\"edge_index\":0}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1], 0, 0, 否, 否)","explanation_zh":"Wave4收束核對=(0, [0, 1], 0, 0, 否, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312046,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,1,0,1,0],"edge_index":0,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":0,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0047","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,1,0,1,0],\"edge_index\":1}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(1, [1, 2], 0, 1, 是, 是)","explanation_zh":"Wave4收束核對=(1, [1, 2], 0, 1, 是, 是)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312047,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,1,0,1,0],"edge_index":1,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":1,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0048","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,1,0,1,0],\"edge_index\":2}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(2, [2, 3], 1, 0, 是, 否)","explanation_zh":"Wave4收束核對=(2, [2, 3], 1, 0, 是, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312048,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,1,0,1,0],"edge_index":2,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":2,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0049","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,1,0,1,0],\"edge_index\":3}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(3, [3, 4], 0, 1, 是, 否)","explanation_zh":"Wave4收束核對=(3, [3, 4], 0, 1, 是, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312049,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,1,0,1,0],"edge_index":3,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":3,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0050","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-edge-crossing；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,0,1,1,0],\"edge_index\":0}。請依 [edge_index,edge,left_part,right_part,crosses,whole_partition_proper] 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1], 0, 0, 否, 否)","explanation_zh":"Wave4收束核對=(0, [0, 1], 0, 0, 否, 否)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-edge-crossing","seed":202610312050,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,0,1,1,0],"edge_index":0,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-edge-crossing","semantic_template":"qf-mod31-bipartite-edge-crossing/{\"edge_index\":0,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,1,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0051","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312051,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0052","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[3,4]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312052,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[3,4]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[3,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0053","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[4,5]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312053,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[4,5]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0054","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[3,4],[4,5]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312054,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[3,4],[4,5]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0055","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[3,5]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312055,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[3,5]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[3,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0056","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[3,4],[3,5]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312056,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[3,4],[3,5]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[3,4],[3,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0057","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[4,5],[3,5]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312057,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[4,5],[3,5]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[3,5],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0058","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[3,4],[4,5],[3,5]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312058,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[3,4],[4,5],[3,5]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[3,4],[3,5],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0059","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[0,3]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312059,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[0,3]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0060","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[3,4],[0,3]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312060,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[3,4],[0,3]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0061","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[4,5],[0,3]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312061,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[4,5],[0,3]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0062","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[3,4],[4,5],[0,3]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312062,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[3,4],[4,5],[0,3]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0063","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[3,5],[0,3]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312063,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[3,5],[0,3]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0064","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[3,4],[3,5],[0,3]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312064,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[3,4],[3,5],[0,3]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4],[3,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0065","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[4,5],[3,5],[0,3]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312065,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[4,5],[3,5],[0,3]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,5],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0066","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[3,4],[4,5],[3,5],[0,3]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312066,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[3,4],[4,5],[3,5],[0,3]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4],[3,5],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0067","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[1,4]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312067,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[1,4]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[1,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0068","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[3,4],[1,4]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312068,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[3,4],[1,4]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[1,4],[3,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0069","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[4,5],[1,4]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312069,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[4,5],[1,4]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[1,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0070","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[3,4],[4,5],[1,4]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312070,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[3,4],[4,5],[1,4]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[1,4],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0071","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[3,5],[1,4]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312071,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[3,5],[1,4]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[1,4],[3,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0072","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[3,4],[3,5],[1,4]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312072,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[3,4],[3,5],[1,4]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[1,4],[3,4],[3,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0073","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[4,5],[3,5],[1,4]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312073,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[4,5],[3,5],[1,4]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[1,4],[3,5],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0074","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[3,4],[4,5],[3,5],[1,4]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312074,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[3,4],[4,5],[3,5],[1,4]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[1,2],[1,4],[3,4],[3,5],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0075","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-odd-cycle；資料={\"n\":6,\"edges\":[[0,1],[1,2],[0,2],[0,3],[1,4]],\"cycle\":[0,1,2]}。請依 [cycle,cycle_edges,length,certificate,conclusion] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)","explanation_zh":"Wave4收束核對=([0, 1, 2], [[0, 1], [1, 2], [0, 2]], 3, 是, odd_cycle_proves_nonbipartite)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-odd-cycle","seed":202610312075,"parameters":{"n":6,"edges":[[0,1],[1,2],[0,2],[0,3],[1,4]],"cycle":[0,1,2],"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-odd-cycle","semantic_template":"qf-mod31-bipartite-odd-cycle/{\"cycle\":[0,1,2],\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0076","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,0,0,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312076,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,0,0,0,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,0,0,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0077","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,0,0,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312077,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,0,0,0,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,0,0,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0078","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,0,0,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312078,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,0,0,0,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,0,0,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0079","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,0,0,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312079,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,0,0,0,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,0,0,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0080","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,1,0,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312080,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,1,0,0,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,0,0,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0081","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,1,0,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312081,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,1,0,0,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,0,0,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0082","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,1,0,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312082,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,1,0,0,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,0,0,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0083","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,1,0,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312083,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,1,0,0,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,0,0,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0084","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,0,1,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312084,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,0,1,0,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,1,0,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0085","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,0,1,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312085,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,0,1,0,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,1,0,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0086","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,0,1,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 是, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 是, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312086,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,0,1,0,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,1,0,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0087","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,0,1,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312087,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,0,1,0,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,1,0,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0088","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,1,1,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312088,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,1,1,0,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,1,0,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0089","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,1,1,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312089,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,1,1,0,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,1,0,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0090","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,1,1,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312090,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,1,1,0,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,1,0,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0091","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,1,1,0,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312091,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,1,1,0,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,1,0,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0092","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,0,0,1,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312092,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,0,0,1,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,0,1,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0093","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,0,0,1,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312093,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,0,0,1,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,0,0,1,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0094","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,0,0,1,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312094,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,0,0,1,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,0,0,1,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0095","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,0,0,1,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312095,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,0,0,1,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,0,0,1,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0096","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,1,0,1,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312096,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,1,0,1,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,1,0,1,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0097","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,0,1,0,1,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 是, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 是, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312097,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,0,1,0,1,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,0,1,0,1,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0098","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,1,1,0,1,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312098,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,1,1,0,1,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,1,1,0,1,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0099","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[1,1,1,0,1,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312099,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[1,1,1,0,1,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[1,1,1,0,1,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式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-20261031-r02-bipartite-graph-0100","batch_id":"batch-20261014-advance-20261031-r02-bipartite-graph-foundations","language":"zh-Hant","source_mko_ids":["mko-bipartite-graph"],"title_zh":"二分圖","stem_zh":"完整有限無向無loop簡單圖。操作=bipartite-isolated-boundary；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"parts\":[0,0,0,1,1,0],\"vertex\":5}。請依 [vertex,incident_edges,assigned_part,whole_partition_proper,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)","explanation_zh":"Wave4收束核對=(5, [], 0, 否, isolated_vertex_may_be_in_either_part_and_empty_part_allowed_for_edgeless_graph)；依完整明示資料重算；兩部互斥且完整cover所有頂點並逐邊跨部；失敗partition不證非二分；奇cycle才是本輪否證。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二分圖"]},"variant":{"family_id":"qf-mod31-bipartite-isolated-boundary","seed":202610312100,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4]],"parts":[0,0,0,1,1,0],"vertex":5,"domain":"Complete_finite_simple_graph_bipartition","operation":"bipartite-isolated-boundary","semantic_template":"qf-mod31-bipartite-isolated-boundary/{\"edges\":[[0,1],[1,2],[2,3],[3,4]],\"n\":6,\"parts\":[0,0,0,1,1,0],\"vertex\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}