{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0001","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"x\":0,\"candidate\":[0,1]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, [0, 1], [[0], [0, 1]], 是)","explanation_zh":"第二批擴展核對=(4, 0, 是, [0, 1], [[0], [0, 1]], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192001,"parameters":{"n":4,"opens":[[],[0],[0,1],[0,1,2],[0,1,2,3]],"x":0,"candidate":[0,1],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,1],\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0002","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"x\":1,\"candidate\":[2]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [2], [], 否)","explanation_zh":"第二批擴展核對=(4, 1, 是, [2], [], 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192002,"parameters":{"n":4,"opens":[[],[1],[0,1],[0,1,2],[0,1,2,3]],"x":1,"candidate":[2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[2],\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0003","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"x\":2,\"candidate\":[0,2]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 是, [0, 2], [[2], [0, 2]], 是)","explanation_zh":"第二批擴展核對=(4, 2, 是, [0, 2], [[2], [0, 2]], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192003,"parameters":{"n":4,"opens":[[],[2],[0,2],[0,1,2],[0,1,2,3]],"x":2,"candidate":[0,2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,2],\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0004","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"x\":3,\"candidate\":[1,2]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, [1, 2], [], 否)","explanation_zh":"第二批擴展核對=(4, 3, 是, [1, 2], [], 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192004,"parameters":{"n":4,"opens":[[],[3],[0,3],[0,1,3],[0,1,2,3]],"x":3,"candidate":[1,2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[1,2],\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0005","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"x\":-1,\"candidate\":[0,1,2]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, -1, 否, [0, 1, 2], 不求值, 不求值)","explanation_zh":"第二批擴展核對=(4, -1, 否, [0, 1, 2], 不求值, 不求值)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192005,"parameters":{"n":4,"opens":[[],[0],[0,2],[0,1,2],[0,1,2,3]],"x":-1,"candidate":[0,1,2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,1,2],\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0006","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"x\":1,\"candidate\":[3]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [3], [], 否)","explanation_zh":"第二批擴展核對=(4, 1, 是, [3], [], 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192006,"parameters":{"n":4,"opens":[[],[1],[1,2],[0,1,2],[0,1,2,3]],"x":1,"candidate":[3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[3],\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0007","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"x\":2,\"candidate\":[0,3]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 是, [0, 3], [], 否)","explanation_zh":"第二批擴展核對=(4, 2, 是, [0, 3], [], 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192007,"parameters":{"n":4,"opens":[[],[2],[1,2],[0,1,2],[0,1,2,3]],"x":2,"candidate":[0,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,3],\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0008","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"x\":3,\"candidate\":[1,3]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, [1, 3], [[3], [1, 3]], 是)","explanation_zh":"第二批擴展核對=(4, 3, 是, [1, 3], [[3], [1, 3]], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192008,"parameters":{"n":4,"opens":[[],[3],[1,3],[0,1,3],[0,1,2,3]],"x":3,"candidate":[1,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[1,3],\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0009","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"x\":0,\"candidate\":[0,1,3]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, [0, 1, 3], [[0], [0, 3], [0, 1, 3]], 是)","explanation_zh":"第二批擴展核對=(4, 0, 是, [0, 1, 3], [[0], [0, 3], [0, 1, 3]], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192009,"parameters":{"n":4,"opens":[[],[0],[0,3],[0,1,3],[0,1,2,3]],"x":0,"candidate":[0,1,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,1,3],\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0010","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"x\":-1,\"candidate\":[2,3]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, -1, 否, [2, 3], 不求值, 不求值)","explanation_zh":"第二批擴展核對=(4, -1, 否, [2, 3], 不求值, 不求值)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192010,"parameters":{"n":4,"opens":[[],[1],[1,3],[0,1,3],[0,1,2,3]],"x":-1,"candidate":[2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[2,3],\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0011","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"x\":2,\"candidate\":[0,2,3]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 是, [0, 2, 3], [[2], [2, 3], [0, 2, 3]], 是)","explanation_zh":"第二批擴展核對=(4, 2, 是, [0, 2, 3], [[2], [2, 3], [0, 2, 3]], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192011,"parameters":{"n":4,"opens":[[],[2],[2,3],[0,2,3],[0,1,2,3]],"x":2,"candidate":[0,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,2,3],\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0012","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"x\":3,\"candidate\":[1,2,3]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, [1, 2, 3], [[3], [2, 3]], 是)","explanation_zh":"第二批擴展核對=(4, 3, 是, [1, 2, 3], [[3], [2, 3]], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192012,"parameters":{"n":4,"opens":[[],[3],[2,3],[0,2,3],[0,1,2,3]],"x":3,"candidate":[1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[1,2,3],\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0013","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"x\":0,\"candidate\":[0,1,2,3]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, [0, 1, 2, 3], [[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], 是)","explanation_zh":"第二批擴展核對=(4, 0, 是, [0, 1, 2, 3], [[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192013,"parameters":{"n":4,"opens":[[],[0],[0,1],[0,1,3],[0,1,2,3]],"x":0,"candidate":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,1,2,3],\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0014","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"x\":1,\"candidate\":[]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [], [], 否)","explanation_zh":"第二批擴展核對=(4, 1, 是, [], [], 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192014,"parameters":{"n":4,"opens":[[],[1],[0,1],[0,1,3],[0,1,2,3]],"x":1,"candidate":[],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[],\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0015","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"x\":-1,\"candidate\":[0]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, -1, 否, [0], 不求值, 不求值)","explanation_zh":"第二批擴展核對=(4, -1, 否, [0], 不求值, 不求值)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192015,"parameters":{"n":4,"opens":[[],[2],[0,2],[0,2,3],[0,1,2,3]],"x":-1,"candidate":[0],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0],\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0016","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"x\":3,\"candidate\":[1]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, [1], [], 否)","explanation_zh":"第二批擴展核對=(4, 3, 是, [1], [], 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192016,"parameters":{"n":4,"opens":[[],[3],[0,3],[0,2,3],[0,1,2,3]],"x":3,"candidate":[1],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[1],\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0017","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"x\":0,\"candidate\":[0,1]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, [0, 1], [[0]], 是)","explanation_zh":"第二批擴展核對=(4, 0, 是, [0, 1], [[0]], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192017,"parameters":{"n":4,"opens":[[],[0],[0,2],[0,2,3],[0,1,2,3]],"x":0,"candidate":[0,1],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,1],\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0018","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"x\":1,\"candidate\":[2]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [2], [], 否)","explanation_zh":"第二批擴展核對=(4, 1, 是, [2], [], 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192018,"parameters":{"n":4,"opens":[[],[1],[1,2],[1,2,3],[0,1,2,3]],"x":1,"candidate":[2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[2],\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0019","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"x\":2,\"candidate\":[0,2]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 是, [0, 2], [[2]], 是)","explanation_zh":"第二批擴展核對=(4, 2, 是, [0, 2], [[2]], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192019,"parameters":{"n":4,"opens":[[],[2],[1,2],[1,2,3],[0,1,2,3]],"x":2,"candidate":[0,2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,2],\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0020","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"x\":-1,\"candidate\":[1,2]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, -1, 否, [1, 2], 不求值, 不求值)","explanation_zh":"第二批擴展核對=(4, -1, 否, [1, 2], 不求值, 不求值)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192020,"parameters":{"n":4,"opens":[[],[3],[1,3],[1,2,3],[0,1,2,3]],"x":-1,"candidate":[1,2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[1,2],\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0021","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"x\":0,\"candidate\":[0,1,2]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, [0, 1, 2], [[0]], 是)","explanation_zh":"第二批擴展核對=(4, 0, 是, [0, 1, 2], [[0]], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192021,"parameters":{"n":4,"opens":[[],[0],[0,3],[0,2,3],[0,1,2,3]],"x":0,"candidate":[0,1,2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,1,2],\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0022","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"x\":1,\"candidate\":[3]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [3], [], 否)","explanation_zh":"第二批擴展核對=(4, 1, 是, [3], [], 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192022,"parameters":{"n":4,"opens":[[],[1],[1,3],[1,2,3],[0,1,2,3]],"x":1,"candidate":[3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[3],\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0023","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"x\":2,\"candidate\":[0,3]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 是, [0, 3], [], 否)","explanation_zh":"第二批擴展核對=(4, 2, 是, [0, 3], [], 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192023,"parameters":{"n":4,"opens":[[],[2],[2,3],[1,2,3],[0,1,2,3]],"x":2,"candidate":[0,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,3],\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0024","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"x\":3,\"candidate\":[1,3]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, [1, 3], [[3]], 是)","explanation_zh":"第二批擴展核對=(4, 3, 是, [1, 3], [[3]], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192024,"parameters":{"n":4,"opens":[[],[3],[2,3],[1,2,3],[0,1,2,3]],"x":3,"candidate":[1,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[1,3],\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0025","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"x\":-1,\"candidate\":[0,1]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。","answer":{"kind":"expression","value":"(3, -1, 否, [0, 1], 不求值, 不求值)","explanation_zh":"第二批擴展核對=(3, -1, 否, [0, 1], 不求值, 不求值)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-witness","seed":202610192025,"parameters":{"n":3,"opens":[[],[0],[0,1],[0,1,2]],"x":-1,"candidate":[0,1],"domain":"Fin_topological_neighborhood","operation":"neighborhood-witness","semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,1],\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0026","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"x\":0,\"candidate\":[0,1]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, [0, 1], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 0, 是, [0, 1], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192026,"parameters":{"n":4,"opens":[[],[0],[0,1],[0,1,2],[0,1,2,3]],"x":0,"candidate":[0,1],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,1],\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0027","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"x\":1,\"candidate\":[2]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [2], 否, 否, 否)","explanation_zh":"第二批擴展核對=(4, 1, 是, [2], 否, 否, 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192027,"parameters":{"n":4,"opens":[[],[1],[0,1],[0,1,2],[0,1,2,3]],"x":1,"candidate":[2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[2],\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0028","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"x\":2,\"candidate\":[0,2]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 是, [0, 2], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 2, 是, [0, 2], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192028,"parameters":{"n":4,"opens":[[],[2],[0,2],[0,1,2],[0,1,2,3]],"x":2,"candidate":[0,2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,2],\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0029","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"x\":3,\"candidate\":[1,2]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, [1, 2], 否, 否, 否)","explanation_zh":"第二批擴展核對=(4, 3, 是, [1, 2], 否, 否, 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192029,"parameters":{"n":4,"opens":[[],[3],[0,3],[0,1,3],[0,1,2,3]],"x":3,"candidate":[1,2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[1,2],\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0030","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"x\":-1,\"candidate\":[0,1,2]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, -1, 否, [0, 1, 2], 不求值, 不求值, 不求值)","explanation_zh":"第二批擴展核對=(4, -1, 否, [0, 1, 2], 不求值, 不求值, 不求值)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192030,"parameters":{"n":4,"opens":[[],[0],[0,2],[0,1,2],[0,1,2,3]],"x":-1,"candidate":[0,1,2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,1,2],\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0031","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"x\":1,\"candidate\":[3]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [3], 否, 否, 否)","explanation_zh":"第二批擴展核對=(4, 1, 是, [3], 否, 否, 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192031,"parameters":{"n":4,"opens":[[],[1],[1,2],[0,1,2],[0,1,2,3]],"x":1,"candidate":[3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[3],\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0032","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"x\":2,\"candidate\":[0,3]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 是, [0, 3], 否, 否, 否)","explanation_zh":"第二批擴展核對=(4, 2, 是, [0, 3], 否, 否, 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192032,"parameters":{"n":4,"opens":[[],[2],[1,2],[0,1,2],[0,1,2,3]],"x":2,"candidate":[0,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,3],\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0033","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"x\":3,\"candidate\":[1,3]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, [1, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 3, 是, [1, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192033,"parameters":{"n":4,"opens":[[],[3],[1,3],[0,1,3],[0,1,2,3]],"x":3,"candidate":[1,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[1,3],\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0034","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"x\":0,\"candidate\":[0,1,3]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, [0, 1, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 0, 是, [0, 1, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192034,"parameters":{"n":4,"opens":[[],[0],[0,3],[0,1,3],[0,1,2,3]],"x":0,"candidate":[0,1,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,1,3],\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0035","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"x\":-1,\"candidate\":[2,3]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, -1, 否, [2, 3], 不求值, 不求值, 不求值)","explanation_zh":"第二批擴展核對=(4, -1, 否, [2, 3], 不求值, 不求值, 不求值)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192035,"parameters":{"n":4,"opens":[[],[1],[1,3],[0,1,3],[0,1,2,3]],"x":-1,"candidate":[2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[2,3],\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0036","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"x\":2,\"candidate\":[0,2,3]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 是, [0, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 2, 是, [0, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192036,"parameters":{"n":4,"opens":[[],[2],[2,3],[0,2,3],[0,1,2,3]],"x":2,"candidate":[0,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,2,3],\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0037","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"x\":3,\"candidate\":[1,2,3]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, [1, 2, 3], 是, 否, 是)","explanation_zh":"第二批擴展核對=(4, 3, 是, [1, 2, 3], 是, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192037,"parameters":{"n":4,"opens":[[],[3],[2,3],[0,2,3],[0,1,2,3]],"x":3,"candidate":[1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[1,2,3],\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0038","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"x\":0,\"candidate\":[0,1,2,3]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 0, 是, [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192038,"parameters":{"n":4,"opens":[[],[0],[0,1],[0,1,3],[0,1,2,3]],"x":0,"candidate":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,1,2,3],\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0039","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"x\":1,\"candidate\":[]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [], 否, 是, 否)","explanation_zh":"第二批擴展核對=(4, 1, 是, [], 否, 是, 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192039,"parameters":{"n":4,"opens":[[],[1],[0,1],[0,1,3],[0,1,2,3]],"x":1,"candidate":[],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[],\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0040","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"x\":-1,\"candidate\":[0]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, -1, 否, [0], 不求值, 不求值, 不求值)","explanation_zh":"第二批擴展核對=(4, -1, 否, [0], 不求值, 不求值, 不求值)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192040,"parameters":{"n":4,"opens":[[],[2],[0,2],[0,2,3],[0,1,2,3]],"x":-1,"candidate":[0],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0],\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0041","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"x\":3,\"candidate\":[1]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, [1], 否, 否, 否)","explanation_zh":"第二批擴展核對=(4, 3, 是, [1], 否, 否, 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192041,"parameters":{"n":4,"opens":[[],[3],[0,3],[0,2,3],[0,1,2,3]],"x":3,"candidate":[1],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[1],\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0042","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"x\":0,\"candidate\":[0,1]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, [0, 1], 是, 否, 是)","explanation_zh":"第二批擴展核對=(4, 0, 是, [0, 1], 是, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192042,"parameters":{"n":4,"opens":[[],[0],[0,2],[0,2,3],[0,1,2,3]],"x":0,"candidate":[0,1],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,1],\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0043","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"x\":1,\"candidate\":[2]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [2], 否, 否, 否)","explanation_zh":"第二批擴展核對=(4, 1, 是, [2], 否, 否, 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192043,"parameters":{"n":4,"opens":[[],[1],[1,2],[1,2,3],[0,1,2,3]],"x":1,"candidate":[2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[2],\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0044","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"x\":2,\"candidate\":[0,2]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 是, [0, 2], 是, 否, 是)","explanation_zh":"第二批擴展核對=(4, 2, 是, [0, 2], 是, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192044,"parameters":{"n":4,"opens":[[],[2],[1,2],[1,2,3],[0,1,2,3]],"x":2,"candidate":[0,2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,2],\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0045","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"x\":-1,\"candidate\":[1,2]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, -1, 否, [1, 2], 不求值, 不求值, 不求值)","explanation_zh":"第二批擴展核對=(4, -1, 否, [1, 2], 不求值, 不求值, 不求值)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192045,"parameters":{"n":4,"opens":[[],[3],[1,3],[1,2,3],[0,1,2,3]],"x":-1,"candidate":[1,2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[1,2],\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0046","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"x\":0,\"candidate\":[0,1,2]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, [0, 1, 2], 是, 否, 是)","explanation_zh":"第二批擴展核對=(4, 0, 是, [0, 1, 2], 是, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192046,"parameters":{"n":4,"opens":[[],[0],[0,3],[0,2,3],[0,1,2,3]],"x":0,"candidate":[0,1,2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,1,2],\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0047","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"x\":1,\"candidate\":[3]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [3], 否, 否, 否)","explanation_zh":"第二批擴展核對=(4, 1, 是, [3], 否, 否, 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192047,"parameters":{"n":4,"opens":[[],[1],[1,3],[1,2,3],[0,1,2,3]],"x":1,"candidate":[3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[3],\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0048","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"x\":2,\"candidate\":[0,3]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 是, [0, 3], 否, 否, 否)","explanation_zh":"第二批擴展核對=(4, 2, 是, [0, 3], 否, 否, 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192048,"parameters":{"n":4,"opens":[[],[2],[2,3],[1,2,3],[0,1,2,3]],"x":2,"candidate":[0,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,3],\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0049","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"x\":3,\"candidate\":[1,3]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, [1, 3], 是, 否, 是)","explanation_zh":"第二批擴展核對=(4, 3, 是, [1, 3], 是, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192049,"parameters":{"n":4,"opens":[[],[3],[2,3],[1,2,3],[0,1,2,3]],"x":3,"candidate":[1,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[1,3],\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0050","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-qualification；資料={\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"x\":-1,\"candidate\":[0,1]}。請依 [n,x,point_qualified,candidate,x_in_candidate,candidate_open,neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(3, -1, 否, [0, 1], 不求值, 不求值, 不求值)","explanation_zh":"第二批擴展核對=(3, -1, 否, [0, 1], 不求值, 不求值, 不求值)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-qualification","seed":202610192050,"parameters":{"n":3,"opens":[[],[0],[0,1],[0,1,2]],"x":-1,"candidate":[0,1],"domain":"Fin_topological_neighborhood","operation":"neighborhood-qualification","semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,1],\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0051","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"x\":0,\"small\":[0],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, [0], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 0, 是, [0], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192051,"parameters":{"n":4,"opens":[[],[0],[0,1],[0,1,2],[0,1,2,3]],"x":0,"small":[0],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"small\":[0],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0052","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"x\":1,\"small\":[1],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [1], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 1, 是, [1], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192052,"parameters":{"n":4,"opens":[[],[1],[0,1],[0,1,2],[0,1,2,3]],"x":1,"small":[1],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"small\":[1],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0053","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"x\":2,\"small\":[2],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 是, [2], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 2, 是, [2], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192053,"parameters":{"n":4,"opens":[[],[2],[0,2],[0,1,2],[0,1,2,3]],"x":2,"small":[2],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"small\":[2],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0054","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"x\":3,\"small\":[3],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, [3], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 3, 是, [3], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192054,"parameters":{"n":4,"opens":[[],[3],[0,3],[0,1,3],[0,1,2,3]],"x":3,"small":[3],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"small\":[3],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0055","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"x\":-1,\"small\":[0],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, -1, 否, [0], [0, 1, 2, 3], 是, 不求值, 不求值)","explanation_zh":"第二批擴展核對=(4, -1, 否, [0], [0, 1, 2, 3], 是, 不求值, 不求值)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192055,"parameters":{"n":4,"opens":[[],[0],[0,2],[0,1,2],[0,1,2,3]],"x":-1,"small":[0],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"small\":[0],\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0056","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"x\":1,\"small\":[1],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [1], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 1, 是, [1], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192056,"parameters":{"n":4,"opens":[[],[1],[1,2],[0,1,2],[0,1,2,3]],"x":1,"small":[1],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"small\":[1],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0057","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"x\":2,\"small\":[2],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 是, [2], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 2, 是, [2], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192057,"parameters":{"n":4,"opens":[[],[2],[1,2],[0,1,2],[0,1,2,3]],"x":2,"small":[2],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"small\":[2],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0058","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"x\":3,\"small\":[3],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, [3], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 3, 是, [3], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192058,"parameters":{"n":4,"opens":[[],[3],[1,3],[0,1,3],[0,1,2,3]],"x":3,"small":[3],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"small\":[3],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0059","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"x\":0,\"small\":[0],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, [0], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 0, 是, [0], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192059,"parameters":{"n":4,"opens":[[],[0],[0,3],[0,1,3],[0,1,2,3]],"x":0,"small":[0],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"small\":[0],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0060","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"x\":-1,\"small\":[0,1,3],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, -1, 否, [0, 1, 3], [0, 1, 2, 3], 是, 不求值, 不求值)","explanation_zh":"第二批擴展核對=(4, -1, 否, [0, 1, 3], [0, 1, 2, 3], 是, 不求值, 不求值)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192060,"parameters":{"n":4,"opens":[[],[1],[1,3],[0,1,3],[0,1,2,3]],"x":-1,"small":[0,1,3],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"small\":[0,1,3],\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0061","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"x\":2,\"small\":[2],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 是, [2], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 2, 是, [2], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192061,"parameters":{"n":4,"opens":[[],[2],[2,3],[0,2,3],[0,1,2,3]],"x":2,"small":[2],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"small\":[2],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0062","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"x\":3,\"small\":[3],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, [3], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 3, 是, [3], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192062,"parameters":{"n":4,"opens":[[],[3],[2,3],[0,2,3],[0,1,2,3]],"x":3,"small":[3],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"small\":[3],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0063","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"x\":0,\"small\":[0],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, [0], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 0, 是, [0], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192063,"parameters":{"n":4,"opens":[[],[0],[0,1],[0,1,3],[0,1,2,3]],"x":0,"small":[0],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"small\":[0],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0064","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"x\":1,\"small\":[1],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [1], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 1, 是, [1], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192064,"parameters":{"n":4,"opens":[[],[1],[0,1],[0,1,3],[0,1,2,3]],"x":1,"small":[1],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"small\":[1],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0065","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"x\":-1,\"small\":[0,2],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, -1, 否, [0, 2], [0, 1, 2, 3], 是, 不求值, 不求值)","explanation_zh":"第二批擴展核對=(4, -1, 否, [0, 2], [0, 1, 2, 3], 是, 不求值, 不求值)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192065,"parameters":{"n":4,"opens":[[],[2],[0,2],[0,2,3],[0,1,2,3]],"x":-1,"small":[0,2],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"small\":[0,2],\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0066","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"x\":3,\"small\":[3],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, [3], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 3, 是, [3], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192066,"parameters":{"n":4,"opens":[[],[3],[0,3],[0,2,3],[0,1,2,3]],"x":3,"small":[3],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"small\":[3],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0067","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"x\":0,\"small\":[0],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, [0], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 0, 是, [0], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192067,"parameters":{"n":4,"opens":[[],[0],[0,2],[0,2,3],[0,1,2,3]],"x":0,"small":[0],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"small\":[0],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0068","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"x\":1,\"small\":[1],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [1], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 1, 是, [1], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192068,"parameters":{"n":4,"opens":[[],[1],[1,2],[1,2,3],[0,1,2,3]],"x":1,"small":[1],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"small\":[1],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0069","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"x\":2,\"small\":[2],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 是, [2], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 2, 是, [2], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192069,"parameters":{"n":4,"opens":[[],[2],[1,2],[1,2,3],[0,1,2,3]],"x":2,"small":[2],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"small\":[2],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0070","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"x\":-1,\"small\":[0,1,2,3],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, -1, 否, [0, 1, 2, 3], [0, 1, 2, 3], 是, 不求值, 不求值)","explanation_zh":"第二批擴展核對=(4, -1, 否, [0, 1, 2, 3], [0, 1, 2, 3], 是, 不求值, 不求值)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192070,"parameters":{"n":4,"opens":[[],[3],[1,3],[1,2,3],[0,1,2,3]],"x":-1,"small":[0,1,2,3],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"small\":[0,1,2,3],\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0071","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"x\":0,\"small\":[0],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, [0], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 0, 是, [0], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192071,"parameters":{"n":4,"opens":[[],[0],[0,3],[0,2,3],[0,1,2,3]],"x":0,"small":[0],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"small\":[0],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0072","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"x\":1,\"small\":[1],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, [1], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 1, 是, [1], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192072,"parameters":{"n":4,"opens":[[],[1],[1,3],[1,2,3],[0,1,2,3]],"x":1,"small":[1],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"small\":[1],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0073","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"x\":2,\"small\":[2],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 是, [2], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 2, 是, [2], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192073,"parameters":{"n":4,"opens":[[],[2],[2,3],[1,2,3],[0,1,2,3]],"x":2,"small":[2],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"small\":[2],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0074","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"x\":3,\"small\":[3],\"large\":[0,1,2,3]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, [3], [0, 1, 2, 3], 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, 3, 是, [3], [0, 1, 2, 3], 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192074,"parameters":{"n":4,"opens":[[],[3],[2,3],[1,2,3],[0,1,2,3]],"x":3,"small":[3],"large":[0,1,2,3],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"small\":[3],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0075","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-upward-closure；資料={\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"x\":-1,\"small\":[0],\"large\":[0,1,2]}。請依 [n,x,point_qualified,small,large,small_subset_large,small_neighborhood,large_neighborhood] 欄序回答。","answer":{"kind":"expression","value":"(3, -1, 否, [0], [0, 1, 2], 是, 不求值, 不求值)","explanation_zh":"第二批擴展核對=(3, -1, 否, [0], [0, 1, 2], 是, 不求值, 不求值)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192075,"parameters":{"n":3,"opens":[[],[0],[0,1],[0,1,2]],"x":-1,"small":[0],"large":[0,1,2],"domain":"Fin_topological_neighborhood","operation":"neighborhood-upward-closure","semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2],\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"small\":[0],\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0076","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"x\":0}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, [[0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 0, [[0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192076,"parameters":{"n":4,"opens":[[],[0],[0,1],[0,1,2],[0,1,2,3]],"x":0,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0077","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"x\":1}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, [[1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 1, [[1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192077,"parameters":{"n":4,"opens":[[],[1],[0,1],[0,1,2],[0,1,2,3]],"x":1,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0078","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"x\":2}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, [[2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 2, [[2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192078,"parameters":{"n":4,"opens":[[],[2],[0,2],[0,1,2],[0,1,2,3]],"x":2,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0079","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"x\":3}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, [[3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 3, [[3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192079,"parameters":{"n":4,"opens":[[],[3],[0,3],[0,1,3],[0,1,2,3]],"x":3,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0080","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"x\":0}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, [[0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 0, [[0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192080,"parameters":{"n":4,"opens":[[],[0],[0,2],[0,1,2],[0,1,2,3]],"x":0,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0081","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"x\":1}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, [[1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 1, [[1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192081,"parameters":{"n":4,"opens":[[],[1],[1,2],[0,1,2],[0,1,2,3]],"x":1,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0082","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"x\":2}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, [[2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 2, [[2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192082,"parameters":{"n":4,"opens":[[],[2],[1,2],[0,1,2],[0,1,2,3]],"x":2,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0083","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"x\":3}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, [[3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 3, [[3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192083,"parameters":{"n":4,"opens":[[],[3],[1,3],[0,1,3],[0,1,2,3]],"x":3,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0084","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"x\":0}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, [[0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 0, [[0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192084,"parameters":{"n":4,"opens":[[],[0],[0,3],[0,1,3],[0,1,2,3]],"x":0,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0085","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"x\":0}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, [[0, 1, 3], [0, 1, 2, 3]], [[0, 1, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 0, [[0, 1, 3], [0, 1, 2, 3]], [[0, 1, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192085,"parameters":{"n":4,"opens":[[],[1],[1,3],[0,1,3],[0,1,2,3]],"x":0,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0086","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"x\":2}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, [[2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 2, [[2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192086,"parameters":{"n":4,"opens":[[],[2],[2,3],[0,2,3],[0,1,2,3]],"x":2,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0087","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"x\":3}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, [[3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 3, [[3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192087,"parameters":{"n":4,"opens":[[],[3],[2,3],[0,2,3],[0,1,2,3]],"x":3,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0088","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"x\":0}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, [[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 0, [[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192088,"parameters":{"n":4,"opens":[[],[0],[0,1],[0,1,3],[0,1,2,3]],"x":0,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0089","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"x\":1}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, [[1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 1, [[1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192089,"parameters":{"n":4,"opens":[[],[1],[0,1],[0,1,3],[0,1,2,3]],"x":1,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0090","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"x\":0}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, [[0, 2], [0, 2, 3], [0, 1, 2, 3]], [[0, 2], [0, 1, 2], [0, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 0, [[0, 2], [0, 2, 3], [0, 1, 2, 3]], [[0, 2], [0, 1, 2], [0, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192090,"parameters":{"n":4,"opens":[[],[2],[0,2],[0,2,3],[0,1,2,3]],"x":0,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0091","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"x\":3}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, [[3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 3, [[3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192091,"parameters":{"n":4,"opens":[[],[3],[0,3],[0,2,3],[0,1,2,3]],"x":3,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0092","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"x\":0}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, [[0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 0, [[0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192092,"parameters":{"n":4,"opens":[[],[0],[0,2],[0,2,3],[0,1,2,3]],"x":0,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0093","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"x\":1}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, [[1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 1, [[1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192093,"parameters":{"n":4,"opens":[[],[1],[1,2],[1,2,3],[0,1,2,3]],"x":1,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0094","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"x\":2}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, [[2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 2, [[2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192094,"parameters":{"n":4,"opens":[[],[2],[1,2],[1,2,3],[0,1,2,3]],"x":2,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0095","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"x\":0}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, [[0, 1, 2, 3]], [[0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 0, [[0, 1, 2, 3]], [[0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192095,"parameters":{"n":4,"opens":[[],[3],[1,3],[1,2,3],[0,1,2,3]],"x":0,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0096","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"x\":0}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, [[0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 0, [[0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192096,"parameters":{"n":4,"opens":[[],[0],[0,3],[0,2,3],[0,1,2,3]],"x":0,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0097","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"x\":1}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, [[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 1, [[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192097,"parameters":{"n":4,"opens":[[],[1],[1,3],[1,2,3],[0,1,2,3]],"x":1,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0098","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"x\":2}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 2, [[2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 2, [[2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192098,"parameters":{"n":4,"opens":[[],[2],[2,3],[1,2,3],[0,1,2,3]],"x":2,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0099","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"x\":3}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(4, 3, [[3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, 3, [[3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192099,"parameters":{"n":4,"opens":[[],[3],[2,3],[1,2,3],[0,1,2,3]],"x":3,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r02-neighborhood-0100","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","language":"zh-Hant","source_mko_ids":["mko-neighborhood"],"title_zh":"鄰域與開集見證","stem_zh":"完整有限拓撲與指定點。操作=neighborhood-complete-inventory；資料={\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"x\":0}。請依 [n,x,open_basis,all_neighborhoods] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, [[0], [0, 1], [0, 1, 2]], [[0], [0, 1], [0, 2], [0, 1, 2]])","explanation_zh":"第二批擴展核對=(3, 0, [[0], [0, 1], [0, 1, 2]], [[0], [0, 1], [0, 2], [0, 1, 2]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["鄰域與開集見證"]},"variant":{"family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192100,"parameters":{"n":3,"opens":[[],[0],[0,1],[0,1,2]],"x":0,"domain":"Fin_topological_neighborhood","operation":"neighborhood-complete-inventory","semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}