{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0001","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a0\",\"b0\",\"c0\"],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\",\"c0\"],[\"a0\"],[\"b0\",\"c0\"],[\"b0\"],[\"c0\"],[]],\"x\":\"a0\",\"y\":\"b0\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a0, b0, [[a0, c0], [b0]], 是, 是)","explanation_zh":"Wave6第五層核對=(a0, b0, [[a0, c0], [b0]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113001,"parameters":{"points":["a0","b0","c0"],"topology":[["a0","b0","c0"],["a0","b0"],["a0","c0"],["a0"],["b0","c0"],["b0"],["c0"],[]],"x":"a0","y":"b0","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a0\",\"b0\",\"c0\"],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\",\"c0\"],[\"a0\"],[\"b0\",\"c0\"],[\"b0\"],[\"c0\"],[]],\"x\":\"a0\",\"y\":\"b0\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0002","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a1\",\"b1\",\"c1\"],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\",\"c1\"],[\"a1\"],[\"b1\",\"c1\"],[\"b1\"],[\"c1\"],[]],\"x\":\"a1\",\"y\":\"b1\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a1, b1, [[a1, c1], [b1]], 是, 是)","explanation_zh":"Wave6第五層核對=(a1, b1, [[a1, c1], [b1]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113002,"parameters":{"points":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["b1","c1"],["b1"],["c1"],[]],"x":"a1","y":"b1","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a1\",\"b1\",\"c1\"],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\",\"c1\"],[\"a1\"],[\"b1\",\"c1\"],[\"b1\"],[\"c1\"],[]],\"x\":\"a1\",\"y\":\"b1\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0003","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a2\",\"b2\",\"c2\"],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\",\"c2\"],[\"a2\"],[\"b2\",\"c2\"],[\"b2\"],[\"c2\"],[]],\"x\":\"a2\",\"y\":\"b2\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a2, b2, [[a2, c2], [b2]], 是, 是)","explanation_zh":"Wave6第五層核對=(a2, b2, [[a2, c2], [b2]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113003,"parameters":{"points":["a2","b2","c2"],"topology":[["a2","b2","c2"],["a2","b2"],["a2","c2"],["a2"],["b2","c2"],["b2"],["c2"],[]],"x":"a2","y":"b2","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a2\",\"b2\",\"c2\"],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\",\"c2\"],[\"a2\"],[\"b2\",\"c2\"],[\"b2\"],[\"c2\"],[]],\"x\":\"a2\",\"y\":\"b2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0004","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a3\",\"b3\",\"c3\"],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\",\"c3\"],[\"a3\"],[\"b3\",\"c3\"],[\"b3\"],[\"c3\"],[]],\"x\":\"a3\",\"y\":\"b3\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a3, b3, [[a3, c3], [b3]], 是, 是)","explanation_zh":"Wave6第五層核對=(a3, b3, [[a3, c3], [b3]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113004,"parameters":{"points":["a3","b3","c3"],"topology":[["a3","b3","c3"],["a3","b3"],["a3","c3"],["a3"],["b3","c3"],["b3"],["c3"],[]],"x":"a3","y":"b3","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a3\",\"b3\",\"c3\"],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\",\"c3\"],[\"a3\"],[\"b3\",\"c3\"],[\"b3\"],[\"c3\"],[]],\"x\":\"a3\",\"y\":\"b3\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0005","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a4\",\"b4\",\"c4\"],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\",\"c4\"],[\"a4\"],[\"b4\",\"c4\"],[\"b4\"],[\"c4\"],[]],\"x\":\"a4\",\"y\":\"b4\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a4, b4, [[a4, c4], [b4]], 是, 是)","explanation_zh":"Wave6第五層核對=(a4, b4, [[a4, c4], [b4]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113005,"parameters":{"points":["a4","b4","c4"],"topology":[["a4","b4","c4"],["a4","b4"],["a4","c4"],["a4"],["b4","c4"],["b4"],["c4"],[]],"x":"a4","y":"b4","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a4\",\"b4\",\"c4\"],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\",\"c4\"],[\"a4\"],[\"b4\",\"c4\"],[\"b4\"],[\"c4\"],[]],\"x\":\"a4\",\"y\":\"b4\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0006","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a5\",\"b5\",\"c5\"],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\",\"c5\"],[\"a5\"],[\"b5\",\"c5\"],[\"b5\"],[\"c5\"],[]],\"x\":\"a5\",\"y\":\"b5\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a5, b5, [[a5, c5], [b5]], 是, 是)","explanation_zh":"Wave6第五層核對=(a5, b5, [[a5, c5], [b5]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113006,"parameters":{"points":["a5","b5","c5"],"topology":[["a5","b5","c5"],["a5","b5"],["a5","c5"],["a5"],["b5","c5"],["b5"],["c5"],[]],"x":"a5","y":"b5","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a5\",\"b5\",\"c5\"],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\",\"c5\"],[\"a5\"],[\"b5\",\"c5\"],[\"b5\"],[\"c5\"],[]],\"x\":\"a5\",\"y\":\"b5\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0007","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a6\",\"b6\",\"c6\"],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\",\"c6\"],[\"a6\"],[\"b6\",\"c6\"],[\"b6\"],[\"c6\"],[]],\"x\":\"a6\",\"y\":\"b6\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a6, b6, [[a6, c6], [b6]], 是, 是)","explanation_zh":"Wave6第五層核對=(a6, b6, [[a6, c6], [b6]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113007,"parameters":{"points":["a6","b6","c6"],"topology":[["a6","b6","c6"],["a6","b6"],["a6","c6"],["a6"],["b6","c6"],["b6"],["c6"],[]],"x":"a6","y":"b6","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a6\",\"b6\",\"c6\"],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\",\"c6\"],[\"a6\"],[\"b6\",\"c6\"],[\"b6\"],[\"c6\"],[]],\"x\":\"a6\",\"y\":\"b6\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0008","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a7\",\"b7\",\"c7\"],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\",\"c7\"],[\"a7\"],[\"b7\",\"c7\"],[\"b7\"],[\"c7\"],[]],\"x\":\"a7\",\"y\":\"b7\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a7, b7, [[a7, c7], [b7]], 是, 是)","explanation_zh":"Wave6第五層核對=(a7, b7, [[a7, c7], [b7]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113008,"parameters":{"points":["a7","b7","c7"],"topology":[["a7","b7","c7"],["a7","b7"],["a7","c7"],["a7"],["b7","c7"],["b7"],["c7"],[]],"x":"a7","y":"b7","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a7\",\"b7\",\"c7\"],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\",\"c7\"],[\"a7\"],[\"b7\",\"c7\"],[\"b7\"],[\"c7\"],[]],\"x\":\"a7\",\"y\":\"b7\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0009","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a8\",\"b8\",\"c8\"],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\",\"c8\"],[\"a8\"],[\"b8\",\"c8\"],[\"b8\"],[\"c8\"],[]],\"x\":\"a8\",\"y\":\"b8\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a8, b8, [[a8, c8], [b8]], 是, 是)","explanation_zh":"Wave6第五層核對=(a8, b8, [[a8, c8], [b8]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113009,"parameters":{"points":["a8","b8","c8"],"topology":[["a8","b8","c8"],["a8","b8"],["a8","c8"],["a8"],["b8","c8"],["b8"],["c8"],[]],"x":"a8","y":"b8","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a8\",\"b8\",\"c8\"],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\",\"c8\"],[\"a8\"],[\"b8\",\"c8\"],[\"b8\"],[\"c8\"],[]],\"x\":\"a8\",\"y\":\"b8\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0010","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a9\",\"b9\",\"c9\"],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\",\"c9\"],[\"a9\"],[\"b9\",\"c9\"],[\"b9\"],[\"c9\"],[]],\"x\":\"a9\",\"y\":\"b9\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a9, b9, [[a9, c9], [b9]], 是, 是)","explanation_zh":"Wave6第五層核對=(a9, b9, [[a9, c9], [b9]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113010,"parameters":{"points":["a9","b9","c9"],"topology":[["a9","b9","c9"],["a9","b9"],["a9","c9"],["a9"],["b9","c9"],["b9"],["c9"],[]],"x":"a9","y":"b9","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a9\",\"b9\",\"c9\"],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\",\"c9\"],[\"a9\"],[\"b9\",\"c9\"],[\"b9\"],[\"c9\"],[]],\"x\":\"a9\",\"y\":\"b9\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0011","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a10\",\"b10\",\"c10\"],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\",\"c10\"],[\"a10\"],[\"b10\",\"c10\"],[\"b10\"],[\"c10\"],[]],\"x\":\"a10\",\"y\":\"b10\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a10, b10, [[a10, c10], [b10]], 是, 是)","explanation_zh":"Wave6第五層核對=(a10, b10, [[a10, c10], [b10]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113011,"parameters":{"points":["a10","b10","c10"],"topology":[["a10","b10","c10"],["a10","b10"],["a10","c10"],["a10"],["b10","c10"],["b10"],["c10"],[]],"x":"a10","y":"b10","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a10\",\"b10\",\"c10\"],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\",\"c10\"],[\"a10\"],[\"b10\",\"c10\"],[\"b10\"],[\"c10\"],[]],\"x\":\"a10\",\"y\":\"b10\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0012","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a11\",\"b11\",\"c11\"],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\",\"c11\"],[\"a11\"],[\"b11\",\"c11\"],[\"b11\"],[\"c11\"],[]],\"x\":\"a11\",\"y\":\"b11\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a11, b11, [[a11, c11], [b11]], 是, 是)","explanation_zh":"Wave6第五層核對=(a11, b11, [[a11, c11], [b11]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113012,"parameters":{"points":["a11","b11","c11"],"topology":[["a11","b11","c11"],["a11","b11"],["a11","c11"],["a11"],["b11","c11"],["b11"],["c11"],[]],"x":"a11","y":"b11","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a11\",\"b11\",\"c11\"],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\",\"c11\"],[\"a11\"],[\"b11\",\"c11\"],[\"b11\"],[\"c11\"],[]],\"x\":\"a11\",\"y\":\"b11\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0013","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a12\",\"b12\",\"c12\"],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\",\"c12\"],[\"a12\"],[\"b12\",\"c12\"],[\"b12\"],[\"c12\"],[]],\"x\":\"a12\",\"y\":\"b12\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a12, b12, [[a12, c12], [b12]], 是, 是)","explanation_zh":"Wave6第五層核對=(a12, b12, [[a12, c12], [b12]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113013,"parameters":{"points":["a12","b12","c12"],"topology":[["a12","b12","c12"],["a12","b12"],["a12","c12"],["a12"],["b12","c12"],["b12"],["c12"],[]],"x":"a12","y":"b12","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a12\",\"b12\",\"c12\"],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\",\"c12\"],[\"a12\"],[\"b12\",\"c12\"],[\"b12\"],[\"c12\"],[]],\"x\":\"a12\",\"y\":\"b12\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0014","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a13\",\"b13\",\"c13\"],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\",\"c13\"],[\"a13\"],[\"b13\",\"c13\"],[\"b13\"],[\"c13\"],[]],\"x\":\"a13\",\"y\":\"b13\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a13, b13, [[a13, c13], [b13]], 是, 是)","explanation_zh":"Wave6第五層核對=(a13, b13, [[a13, c13], [b13]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113014,"parameters":{"points":["a13","b13","c13"],"topology":[["a13","b13","c13"],["a13","b13"],["a13","c13"],["a13"],["b13","c13"],["b13"],["c13"],[]],"x":"a13","y":"b13","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a13\",\"b13\",\"c13\"],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\",\"c13\"],[\"a13\"],[\"b13\",\"c13\"],[\"b13\"],[\"c13\"],[]],\"x\":\"a13\",\"y\":\"b13\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0015","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a14\",\"b14\",\"c14\"],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\",\"c14\"],[\"a14\"],[\"b14\",\"c14\"],[\"b14\"],[\"c14\"],[]],\"x\":\"a14\",\"y\":\"b14\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a14, b14, [[a14, c14], [b14]], 是, 是)","explanation_zh":"Wave6第五層核對=(a14, b14, [[a14, c14], [b14]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113015,"parameters":{"points":["a14","b14","c14"],"topology":[["a14","b14","c14"],["a14","b14"],["a14","c14"],["a14"],["b14","c14"],["b14"],["c14"],[]],"x":"a14","y":"b14","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a14\",\"b14\",\"c14\"],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\",\"c14\"],[\"a14\"],[\"b14\",\"c14\"],[\"b14\"],[\"c14\"],[]],\"x\":\"a14\",\"y\":\"b14\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0016","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a15\",\"b15\",\"c15\"],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\",\"c15\"],[\"a15\"],[\"b15\",\"c15\"],[\"b15\"],[\"c15\"],[]],\"x\":\"a15\",\"y\":\"b15\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a15, b15, [[a15, c15], [b15]], 是, 是)","explanation_zh":"Wave6第五層核對=(a15, b15, [[a15, c15], [b15]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113016,"parameters":{"points":["a15","b15","c15"],"topology":[["a15","b15","c15"],["a15","b15"],["a15","c15"],["a15"],["b15","c15"],["b15"],["c15"],[]],"x":"a15","y":"b15","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a15\",\"b15\",\"c15\"],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\",\"c15\"],[\"a15\"],[\"b15\",\"c15\"],[\"b15\"],[\"c15\"],[]],\"x\":\"a15\",\"y\":\"b15\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0017","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a16\",\"b16\",\"c16\"],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\",\"c16\"],[\"a16\"],[\"b16\",\"c16\"],[\"b16\"],[\"c16\"],[]],\"x\":\"a16\",\"y\":\"b16\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a16, b16, [[a16, c16], [b16]], 是, 是)","explanation_zh":"Wave6第五層核對=(a16, b16, [[a16, c16], [b16]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113017,"parameters":{"points":["a16","b16","c16"],"topology":[["a16","b16","c16"],["a16","b16"],["a16","c16"],["a16"],["b16","c16"],["b16"],["c16"],[]],"x":"a16","y":"b16","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a16\",\"b16\",\"c16\"],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\",\"c16\"],[\"a16\"],[\"b16\",\"c16\"],[\"b16\"],[\"c16\"],[]],\"x\":\"a16\",\"y\":\"b16\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0018","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a17\",\"b17\",\"c17\"],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\",\"c17\"],[\"a17\"],[\"b17\",\"c17\"],[\"b17\"],[\"c17\"],[]],\"x\":\"a17\",\"y\":\"b17\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a17, b17, [[a17, c17], [b17]], 是, 是)","explanation_zh":"Wave6第五層核對=(a17, b17, [[a17, c17], [b17]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113018,"parameters":{"points":["a17","b17","c17"],"topology":[["a17","b17","c17"],["a17","b17"],["a17","c17"],["a17"],["b17","c17"],["b17"],["c17"],[]],"x":"a17","y":"b17","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a17\",\"b17\",\"c17\"],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\",\"c17\"],[\"a17\"],[\"b17\",\"c17\"],[\"b17\"],[\"c17\"],[]],\"x\":\"a17\",\"y\":\"b17\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0019","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a18\",\"b18\",\"c18\"],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\",\"c18\"],[\"a18\"],[\"b18\",\"c18\"],[\"b18\"],[\"c18\"],[]],\"x\":\"a18\",\"y\":\"b18\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a18, b18, [[a18, c18], [b18]], 是, 是)","explanation_zh":"Wave6第五層核對=(a18, b18, [[a18, c18], [b18]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113019,"parameters":{"points":["a18","b18","c18"],"topology":[["a18","b18","c18"],["a18","b18"],["a18","c18"],["a18"],["b18","c18"],["b18"],["c18"],[]],"x":"a18","y":"b18","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a18\",\"b18\",\"c18\"],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\",\"c18\"],[\"a18\"],[\"b18\",\"c18\"],[\"b18\"],[\"c18\"],[]],\"x\":\"a18\",\"y\":\"b18\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0020","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a19\",\"b19\",\"c19\"],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\",\"c19\"],[\"a19\"],[\"b19\",\"c19\"],[\"b19\"],[\"c19\"],[]],\"x\":\"a19\",\"y\":\"b19\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a19, b19, [[a19, c19], [b19]], 是, 是)","explanation_zh":"Wave6第五層核對=(a19, b19, [[a19, c19], [b19]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113020,"parameters":{"points":["a19","b19","c19"],"topology":[["a19","b19","c19"],["a19","b19"],["a19","c19"],["a19"],["b19","c19"],["b19"],["c19"],[]],"x":"a19","y":"b19","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a19\",\"b19\",\"c19\"],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\",\"c19\"],[\"a19\"],[\"b19\",\"c19\"],[\"b19\"],[\"c19\"],[]],\"x\":\"a19\",\"y\":\"b19\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0021","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"b20\",\"c20\"],[\"b20\"],[\"c20\"],[]],\"x\":\"a20\",\"y\":\"b20\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a20, b20, [[a20, c20], [b20]], 是, 是)","explanation_zh":"Wave6第五層核對=(a20, b20, [[a20, c20], [b20]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113021,"parameters":{"points":["a20","b20","c20"],"topology":[["a20","b20","c20"],["a20","b20"],["a20","c20"],["a20"],["b20","c20"],["b20"],["c20"],[]],"x":"a20","y":"b20","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"b20\",\"c20\"],[\"b20\"],[\"c20\"],[]],\"x\":\"a20\",\"y\":\"b20\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0022","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a21\",\"b21\",\"c21\"],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\",\"c21\"],[\"a21\"],[\"b21\",\"c21\"],[\"b21\"],[\"c21\"],[]],\"x\":\"a21\",\"y\":\"b21\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a21, b21, [[a21, c21], [b21]], 是, 是)","explanation_zh":"Wave6第五層核對=(a21, b21, [[a21, c21], [b21]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113022,"parameters":{"points":["a21","b21","c21"],"topology":[["a21","b21","c21"],["a21","b21"],["a21","c21"],["a21"],["b21","c21"],["b21"],["c21"],[]],"x":"a21","y":"b21","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a21\",\"b21\",\"c21\"],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\",\"c21\"],[\"a21\"],[\"b21\",\"c21\"],[\"b21\"],[\"c21\"],[]],\"x\":\"a21\",\"y\":\"b21\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0023","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a22\",\"b22\",\"c22\"],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\",\"c22\"],[\"a22\"],[\"b22\",\"c22\"],[\"b22\"],[\"c22\"],[]],\"x\":\"a22\",\"y\":\"b22\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a22, b22, [[a22, c22], [b22]], 是, 是)","explanation_zh":"Wave6第五層核對=(a22, b22, [[a22, c22], [b22]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113023,"parameters":{"points":["a22","b22","c22"],"topology":[["a22","b22","c22"],["a22","b22"],["a22","c22"],["a22"],["b22","c22"],["b22"],["c22"],[]],"x":"a22","y":"b22","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a22\",\"b22\",\"c22\"],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\",\"c22\"],[\"a22\"],[\"b22\",\"c22\"],[\"b22\"],[\"c22\"],[]],\"x\":\"a22\",\"y\":\"b22\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0024","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a23\",\"b23\",\"c23\"],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\",\"c23\"],[\"a23\"],[\"b23\",\"c23\"],[\"b23\"],[\"c23\"],[]],\"x\":\"a23\",\"y\":\"b23\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a23, b23, [[a23, c23], [b23]], 是, 是)","explanation_zh":"Wave6第五層核對=(a23, b23, [[a23, c23], [b23]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113024,"parameters":{"points":["a23","b23","c23"],"topology":[["a23","b23","c23"],["a23","b23"],["a23","c23"],["a23"],["b23","c23"],["b23"],["c23"],[]],"x":"a23","y":"b23","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a23\",\"b23\",\"c23\"],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\",\"c23\"],[\"a23\"],[\"b23\",\"c23\"],[\"b23\"],[\"c23\"],[]],\"x\":\"a23\",\"y\":\"b23\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0025","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-pair-separator；資料={\"points\":[\"a24\",\"b24\",\"c24\"],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\",\"c24\"],[\"a24\"],[\"b24\",\"c24\"],[\"b24\"],[\"c24\"],[]],\"x\":\"a24\",\"y\":\"b24\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a24, b24, [[a24, c24], [b24]], 是, 是)","explanation_zh":"Wave6第五層核對=(a24, b24, [[a24, c24], [b24]], 是, 是)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-pair-separator","seed":202611113025,"parameters":{"points":["a24","b24","c24"],"topology":[["a24","b24","c24"],["a24","b24"],["a24","c24"],["a24"],["b24","c24"],["b24"],["c24"],[]],"x":"a24","y":"b24","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-pair-separator","semantic_template":"qf-mod42-hausdorff-pair-separator/{\"points\":[\"a24\",\"b24\",\"c24\"],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\",\"c24\"],[\"a24\"],[\"b24\",\"c24\"],[\"b24\"],[\"c24\"],[]],\"x\":\"a24\",\"y\":\"b24\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0026","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a0\",\"b0\",\"c0\"],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\",\"c0\"],[\"a0\"],[\"b0\",\"c0\"],[\"b0\"],[\"c0\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a0, b0, c0], [[a0, b0, [[a0, c0], [b0]]], [a0, c0, [[a0, b0], [c0]]], [b0, c0, [[a0, b0], [c0]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a0, b0, c0], [[a0, b0, [[a0, c0], [b0]]], [a0, c0, [[a0, b0], [c0]]], [b0, c0, [[a0, b0], [c0]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113026,"parameters":{"points":["a0","b0","c0"],"topology":[["a0","b0","c0"],["a0","b0"],["a0","c0"],["a0"],["b0","c0"],["b0"],["c0"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a0\",\"b0\",\"c0\"],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\",\"c0\"],[\"a0\"],[\"b0\",\"c0\"],[\"b0\"],[\"c0\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0027","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a1\",\"b1\",\"c1\"],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\",\"c1\"],[\"a1\"],[\"b1\",\"c1\"],[\"b1\"],[\"c1\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a1, b1, c1], [[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a1, b1, c1], [[a1, b1, [[a1, c1], [b1]]], [a1, c1, [[a1, b1], [c1]]], [b1, c1, [[a1, b1], [c1]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113027,"parameters":{"points":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["b1","c1"],["b1"],["c1"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a1\",\"b1\",\"c1\"],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\",\"c1\"],[\"a1\"],[\"b1\",\"c1\"],[\"b1\"],[\"c1\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0028","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a2\",\"b2\",\"c2\"],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\",\"c2\"],[\"a2\"],[\"b2\",\"c2\"],[\"b2\"],[\"c2\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a2, b2, c2], [[a2, b2, [[a2, c2], [b2]]], [a2, c2, [[a2, b2], [c2]]], [b2, c2, [[a2, b2], [c2]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a2, b2, c2], [[a2, b2, [[a2, c2], [b2]]], [a2, c2, [[a2, b2], [c2]]], [b2, c2, [[a2, b2], [c2]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113028,"parameters":{"points":["a2","b2","c2"],"topology":[["a2","b2","c2"],["a2","b2"],["a2","c2"],["a2"],["b2","c2"],["b2"],["c2"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a2\",\"b2\",\"c2\"],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\",\"c2\"],[\"a2\"],[\"b2\",\"c2\"],[\"b2\"],[\"c2\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0029","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a3\",\"b3\",\"c3\"],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\",\"c3\"],[\"a3\"],[\"b3\",\"c3\"],[\"b3\"],[\"c3\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a3, b3, c3], [[a3, b3, [[a3, c3], [b3]]], [a3, c3, [[a3, b3], [c3]]], [b3, c3, [[a3, b3], [c3]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a3, b3, c3], [[a3, b3, [[a3, c3], [b3]]], [a3, c3, [[a3, b3], [c3]]], [b3, c3, [[a3, b3], [c3]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113029,"parameters":{"points":["a3","b3","c3"],"topology":[["a3","b3","c3"],["a3","b3"],["a3","c3"],["a3"],["b3","c3"],["b3"],["c3"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a3\",\"b3\",\"c3\"],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\",\"c3\"],[\"a3\"],[\"b3\",\"c3\"],[\"b3\"],[\"c3\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0030","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a4\",\"b4\",\"c4\"],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\",\"c4\"],[\"a4\"],[\"b4\",\"c4\"],[\"b4\"],[\"c4\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a4, b4, c4], [[a4, b4, [[a4, c4], [b4]]], [a4, c4, [[a4, b4], [c4]]], [b4, c4, [[a4, b4], [c4]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a4, b4, c4], [[a4, b4, [[a4, c4], [b4]]], [a4, c4, [[a4, b4], [c4]]], [b4, c4, [[a4, b4], [c4]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113030,"parameters":{"points":["a4","b4","c4"],"topology":[["a4","b4","c4"],["a4","b4"],["a4","c4"],["a4"],["b4","c4"],["b4"],["c4"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a4\",\"b4\",\"c4\"],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\",\"c4\"],[\"a4\"],[\"b4\",\"c4\"],[\"b4\"],[\"c4\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0031","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a5\",\"b5\",\"c5\"],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\",\"c5\"],[\"a5\"],[\"b5\",\"c5\"],[\"b5\"],[\"c5\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a5, b5, c5], [[a5, b5, [[a5, c5], [b5]]], [a5, c5, [[a5, b5], [c5]]], [b5, c5, [[a5, b5], [c5]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a5, b5, c5], [[a5, b5, [[a5, c5], [b5]]], [a5, c5, [[a5, b5], [c5]]], [b5, c5, [[a5, b5], [c5]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113031,"parameters":{"points":["a5","b5","c5"],"topology":[["a5","b5","c5"],["a5","b5"],["a5","c5"],["a5"],["b5","c5"],["b5"],["c5"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a5\",\"b5\",\"c5\"],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\",\"c5\"],[\"a5\"],[\"b5\",\"c5\"],[\"b5\"],[\"c5\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0032","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a6\",\"b6\",\"c6\"],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\",\"c6\"],[\"a6\"],[\"b6\",\"c6\"],[\"b6\"],[\"c6\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a6, b6, c6], [[a6, b6, [[a6, c6], [b6]]], [a6, c6, [[a6, b6], [c6]]], [b6, c6, [[a6, b6], [c6]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a6, b6, c6], [[a6, b6, [[a6, c6], [b6]]], [a6, c6, [[a6, b6], [c6]]], [b6, c6, [[a6, b6], [c6]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113032,"parameters":{"points":["a6","b6","c6"],"topology":[["a6","b6","c6"],["a6","b6"],["a6","c6"],["a6"],["b6","c6"],["b6"],["c6"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a6\",\"b6\",\"c6\"],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\",\"c6\"],[\"a6\"],[\"b6\",\"c6\"],[\"b6\"],[\"c6\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0033","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a7\",\"b7\",\"c7\"],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\",\"c7\"],[\"a7\"],[\"b7\",\"c7\"],[\"b7\"],[\"c7\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a7, b7, c7], [[a7, b7, [[a7, c7], [b7]]], [a7, c7, [[a7, b7], [c7]]], [b7, c7, [[a7, b7], [c7]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a7, b7, c7], [[a7, b7, [[a7, c7], [b7]]], [a7, c7, [[a7, b7], [c7]]], [b7, c7, [[a7, b7], [c7]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113033,"parameters":{"points":["a7","b7","c7"],"topology":[["a7","b7","c7"],["a7","b7"],["a7","c7"],["a7"],["b7","c7"],["b7"],["c7"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a7\",\"b7\",\"c7\"],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\",\"c7\"],[\"a7\"],[\"b7\",\"c7\"],[\"b7\"],[\"c7\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0034","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a8\",\"b8\",\"c8\"],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\",\"c8\"],[\"a8\"],[\"b8\",\"c8\"],[\"b8\"],[\"c8\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a8, b8, c8], [[a8, b8, [[a8, c8], [b8]]], [a8, c8, [[a8, b8], [c8]]], [b8, c8, [[a8, b8], [c8]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a8, b8, c8], [[a8, b8, [[a8, c8], [b8]]], [a8, c8, [[a8, b8], [c8]]], [b8, c8, [[a8, b8], [c8]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113034,"parameters":{"points":["a8","b8","c8"],"topology":[["a8","b8","c8"],["a8","b8"],["a8","c8"],["a8"],["b8","c8"],["b8"],["c8"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a8\",\"b8\",\"c8\"],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\",\"c8\"],[\"a8\"],[\"b8\",\"c8\"],[\"b8\"],[\"c8\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0035","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a9\",\"b9\",\"c9\"],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\",\"c9\"],[\"a9\"],[\"b9\",\"c9\"],[\"b9\"],[\"c9\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a9, b9, c9], [[a9, b9, [[a9, c9], [b9]]], [a9, c9, [[a9, b9], [c9]]], [b9, c9, [[a9, b9], [c9]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a9, b9, c9], [[a9, b9, [[a9, c9], [b9]]], [a9, c9, [[a9, b9], [c9]]], [b9, c9, [[a9, b9], [c9]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113035,"parameters":{"points":["a9","b9","c9"],"topology":[["a9","b9","c9"],["a9","b9"],["a9","c9"],["a9"],["b9","c9"],["b9"],["c9"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a9\",\"b9\",\"c9\"],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\",\"c9\"],[\"a9\"],[\"b9\",\"c9\"],[\"b9\"],[\"c9\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0036","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a10\",\"b10\",\"c10\"],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\",\"c10\"],[\"a10\"],[\"b10\",\"c10\"],[\"b10\"],[\"c10\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a10, b10, c10], [[a10, b10, [[a10, c10], [b10]]], [a10, c10, [[a10, b10], [c10]]], [b10, c10, [[a10, b10], [c10]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a10, b10, c10], [[a10, b10, [[a10, c10], [b10]]], [a10, c10, [[a10, b10], [c10]]], [b10, c10, [[a10, b10], [c10]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113036,"parameters":{"points":["a10","b10","c10"],"topology":[["a10","b10","c10"],["a10","b10"],["a10","c10"],["a10"],["b10","c10"],["b10"],["c10"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a10\",\"b10\",\"c10\"],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\",\"c10\"],[\"a10\"],[\"b10\",\"c10\"],[\"b10\"],[\"c10\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0037","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a11\",\"b11\",\"c11\"],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\",\"c11\"],[\"a11\"],[\"b11\",\"c11\"],[\"b11\"],[\"c11\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a11, b11, c11], [[a11, b11, [[a11, c11], [b11]]], [a11, c11, [[a11, b11], [c11]]], [b11, c11, [[a11, b11], [c11]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a11, b11, c11], [[a11, b11, [[a11, c11], [b11]]], [a11, c11, [[a11, b11], [c11]]], [b11, c11, [[a11, b11], [c11]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113037,"parameters":{"points":["a11","b11","c11"],"topology":[["a11","b11","c11"],["a11","b11"],["a11","c11"],["a11"],["b11","c11"],["b11"],["c11"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a11\",\"b11\",\"c11\"],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\",\"c11\"],[\"a11\"],[\"b11\",\"c11\"],[\"b11\"],[\"c11\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0038","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a12\",\"b12\",\"c12\"],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\",\"c12\"],[\"a12\"],[\"b12\",\"c12\"],[\"b12\"],[\"c12\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a12, b12, c12], [[a12, b12, [[a12, c12], [b12]]], [a12, c12, [[a12, b12], [c12]]], [b12, c12, [[a12, b12], [c12]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a12, b12, c12], [[a12, b12, [[a12, c12], [b12]]], [a12, c12, [[a12, b12], [c12]]], [b12, c12, [[a12, b12], [c12]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113038,"parameters":{"points":["a12","b12","c12"],"topology":[["a12","b12","c12"],["a12","b12"],["a12","c12"],["a12"],["b12","c12"],["b12"],["c12"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a12\",\"b12\",\"c12\"],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\",\"c12\"],[\"a12\"],[\"b12\",\"c12\"],[\"b12\"],[\"c12\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0039","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a13\",\"b13\",\"c13\"],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\",\"c13\"],[\"a13\"],[\"b13\",\"c13\"],[\"b13\"],[\"c13\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a13, b13, c13], [[a13, b13, [[a13, c13], [b13]]], [a13, c13, [[a13, b13], [c13]]], [b13, c13, [[a13, b13], [c13]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a13, b13, c13], [[a13, b13, [[a13, c13], [b13]]], [a13, c13, [[a13, b13], [c13]]], [b13, c13, [[a13, b13], [c13]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113039,"parameters":{"points":["a13","b13","c13"],"topology":[["a13","b13","c13"],["a13","b13"],["a13","c13"],["a13"],["b13","c13"],["b13"],["c13"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a13\",\"b13\",\"c13\"],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\",\"c13\"],[\"a13\"],[\"b13\",\"c13\"],[\"b13\"],[\"c13\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0040","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a14\",\"b14\",\"c14\"],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\",\"c14\"],[\"a14\"],[\"b14\",\"c14\"],[\"b14\"],[\"c14\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a14, b14, c14], [[a14, b14, [[a14, c14], [b14]]], [a14, c14, [[a14, b14], [c14]]], [b14, c14, [[a14, b14], [c14]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a14, b14, c14], [[a14, b14, [[a14, c14], [b14]]], [a14, c14, [[a14, b14], [c14]]], [b14, c14, [[a14, b14], [c14]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113040,"parameters":{"points":["a14","b14","c14"],"topology":[["a14","b14","c14"],["a14","b14"],["a14","c14"],["a14"],["b14","c14"],["b14"],["c14"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a14\",\"b14\",\"c14\"],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\",\"c14\"],[\"a14\"],[\"b14\",\"c14\"],[\"b14\"],[\"c14\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0041","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a15\",\"b15\",\"c15\"],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\",\"c15\"],[\"a15\"],[\"b15\",\"c15\"],[\"b15\"],[\"c15\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a15, b15, c15], [[a15, b15, [[a15, c15], [b15]]], [a15, c15, [[a15, b15], [c15]]], [b15, c15, [[a15, b15], [c15]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a15, b15, c15], [[a15, b15, [[a15, c15], [b15]]], [a15, c15, [[a15, b15], [c15]]], [b15, c15, [[a15, b15], [c15]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113041,"parameters":{"points":["a15","b15","c15"],"topology":[["a15","b15","c15"],["a15","b15"],["a15","c15"],["a15"],["b15","c15"],["b15"],["c15"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a15\",\"b15\",\"c15\"],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\",\"c15\"],[\"a15\"],[\"b15\",\"c15\"],[\"b15\"],[\"c15\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0042","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a16\",\"b16\",\"c16\"],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\",\"c16\"],[\"a16\"],[\"b16\",\"c16\"],[\"b16\"],[\"c16\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a16, b16, c16], [[a16, b16, [[a16, c16], [b16]]], [a16, c16, [[a16, b16], [c16]]], [b16, c16, [[a16, b16], [c16]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a16, b16, c16], [[a16, b16, [[a16, c16], [b16]]], [a16, c16, [[a16, b16], [c16]]], [b16, c16, [[a16, b16], [c16]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113042,"parameters":{"points":["a16","b16","c16"],"topology":[["a16","b16","c16"],["a16","b16"],["a16","c16"],["a16"],["b16","c16"],["b16"],["c16"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a16\",\"b16\",\"c16\"],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\",\"c16\"],[\"a16\"],[\"b16\",\"c16\"],[\"b16\"],[\"c16\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0043","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a17\",\"b17\",\"c17\"],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\",\"c17\"],[\"a17\"],[\"b17\",\"c17\"],[\"b17\"],[\"c17\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a17, b17, c17], [[a17, b17, [[a17, c17], [b17]]], [a17, c17, [[a17, b17], [c17]]], [b17, c17, [[a17, b17], [c17]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a17, b17, c17], [[a17, b17, [[a17, c17], [b17]]], [a17, c17, [[a17, b17], [c17]]], [b17, c17, [[a17, b17], [c17]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113043,"parameters":{"points":["a17","b17","c17"],"topology":[["a17","b17","c17"],["a17","b17"],["a17","c17"],["a17"],["b17","c17"],["b17"],["c17"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a17\",\"b17\",\"c17\"],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\",\"c17\"],[\"a17\"],[\"b17\",\"c17\"],[\"b17\"],[\"c17\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0044","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a18\",\"b18\",\"c18\"],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\",\"c18\"],[\"a18\"],[\"b18\",\"c18\"],[\"b18\"],[\"c18\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a18, b18, c18], [[a18, b18, [[a18, c18], [b18]]], [a18, c18, [[a18, b18], [c18]]], [b18, c18, [[a18, b18], [c18]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a18, b18, c18], [[a18, b18, [[a18, c18], [b18]]], [a18, c18, [[a18, b18], [c18]]], [b18, c18, [[a18, b18], [c18]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113044,"parameters":{"points":["a18","b18","c18"],"topology":[["a18","b18","c18"],["a18","b18"],["a18","c18"],["a18"],["b18","c18"],["b18"],["c18"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a18\",\"b18\",\"c18\"],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\",\"c18\"],[\"a18\"],[\"b18\",\"c18\"],[\"b18\"],[\"c18\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0045","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a19\",\"b19\",\"c19\"],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\",\"c19\"],[\"a19\"],[\"b19\",\"c19\"],[\"b19\"],[\"c19\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a19, b19, c19], [[a19, b19, [[a19, c19], [b19]]], [a19, c19, [[a19, b19], [c19]]], [b19, c19, [[a19, b19], [c19]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a19, b19, c19], [[a19, b19, [[a19, c19], [b19]]], [a19, c19, [[a19, b19], [c19]]], [b19, c19, [[a19, b19], [c19]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113045,"parameters":{"points":["a19","b19","c19"],"topology":[["a19","b19","c19"],["a19","b19"],["a19","c19"],["a19"],["b19","c19"],["b19"],["c19"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a19\",\"b19\",\"c19\"],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\",\"c19\"],[\"a19\"],[\"b19\",\"c19\"],[\"b19\"],[\"c19\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0046","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"b20\",\"c20\"],[\"b20\"],[\"c20\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a20, b20, c20], [[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a20, b20, c20], [[a20, b20, [[a20, c20], [b20]]], [a20, c20, [[a20, b20], [c20]]], [b20, c20, [[a20, b20], [c20]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113046,"parameters":{"points":["a20","b20","c20"],"topology":[["a20","b20","c20"],["a20","b20"],["a20","c20"],["a20"],["b20","c20"],["b20"],["c20"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"b20\",\"c20\"],[\"b20\"],[\"c20\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0047","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a21\",\"b21\",\"c21\"],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\",\"c21\"],[\"a21\"],[\"b21\",\"c21\"],[\"b21\"],[\"c21\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a21, b21, c21], [[a21, b21, [[a21, c21], [b21]]], [a21, c21, [[a21, b21], [c21]]], [b21, c21, [[a21, b21], [c21]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a21, b21, c21], [[a21, b21, [[a21, c21], [b21]]], [a21, c21, [[a21, b21], [c21]]], [b21, c21, [[a21, b21], [c21]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113047,"parameters":{"points":["a21","b21","c21"],"topology":[["a21","b21","c21"],["a21","b21"],["a21","c21"],["a21"],["b21","c21"],["b21"],["c21"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a21\",\"b21\",\"c21\"],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\",\"c21\"],[\"a21\"],[\"b21\",\"c21\"],[\"b21\"],[\"c21\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0048","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a22\",\"b22\",\"c22\"],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\",\"c22\"],[\"a22\"],[\"b22\",\"c22\"],[\"b22\"],[\"c22\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a22, b22, c22], [[a22, b22, [[a22, c22], [b22]]], [a22, c22, [[a22, b22], [c22]]], [b22, c22, [[a22, b22], [c22]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a22, b22, c22], [[a22, b22, [[a22, c22], [b22]]], [a22, c22, [[a22, b22], [c22]]], [b22, c22, [[a22, b22], [c22]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113048,"parameters":{"points":["a22","b22","c22"],"topology":[["a22","b22","c22"],["a22","b22"],["a22","c22"],["a22"],["b22","c22"],["b22"],["c22"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a22\",\"b22\",\"c22\"],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\",\"c22\"],[\"a22\"],[\"b22\",\"c22\"],[\"b22\"],[\"c22\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0049","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a23\",\"b23\",\"c23\"],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\",\"c23\"],[\"a23\"],[\"b23\",\"c23\"],[\"b23\"],[\"c23\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a23, b23, c23], [[a23, b23, [[a23, c23], [b23]]], [a23, c23, [[a23, b23], [c23]]], [b23, c23, [[a23, b23], [c23]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a23, b23, c23], [[a23, b23, [[a23, c23], [b23]]], [a23, c23, [[a23, b23], [c23]]], [b23, c23, [[a23, b23], [c23]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113049,"parameters":{"points":["a23","b23","c23"],"topology":[["a23","b23","c23"],["a23","b23"],["a23","c23"],["a23"],["b23","c23"],["b23"],["c23"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a23\",\"b23\",\"c23\"],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\",\"c23\"],[\"a23\"],[\"b23\",\"c23\"],[\"b23\"],[\"c23\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0050","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-full-audit；資料={\"points\":[\"a24\",\"b24\",\"c24\"],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\",\"c24\"],[\"a24\"],[\"b24\",\"c24\"],[\"b24\"],[\"c24\"],[]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([a24, b24, c24], [[a24, b24, [[a24, c24], [b24]]], [a24, c24, [[a24, b24], [c24]]], [b24, c24, [[a24, b24], [c24]]]], 是, all_distinct_pairs_exhaustively_separated)","explanation_zh":"Wave6第五層核對=([a24, b24, c24], [[a24, b24, [[a24, c24], [b24]]], [a24, c24, [[a24, b24], [c24]]], [b24, c24, [[a24, b24], [c24]]]], 是, all_distinct_pairs_exhaustively_separated)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-full-audit","seed":202611113050,"parameters":{"points":["a24","b24","c24"],"topology":[["a24","b24","c24"],["a24","b24"],["a24","c24"],["a24"],["b24","c24"],["b24"],["c24"],[]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-full-audit","semantic_template":"qf-mod42-hausdorff-full-audit/{\"points\":[\"a24\",\"b24\",\"c24\"],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\",\"c24\"],[\"a24\"],[\"b24\",\"c24\"],[\"b24\"],[\"c24\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0051","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a0\",\"b0\",\"c0\"],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\"],[]],\"x\":\"a0\",\"y\":\"b0\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a0, b0, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a0, b0, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113051,"parameters":{"points":["a0","b0","c0"],"topology":[["a0","b0","c0"],["a0","b0"],["a0"],[]],"x":"a0","y":"b0","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a0\",\"b0\",\"c0\"],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\"],[]],\"x\":\"a0\",\"y\":\"b0\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0052","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a1\",\"b1\",\"c1\"],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\"],[]],\"x\":\"a1\",\"y\":\"b1\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a1, b1, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a1, b1, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113052,"parameters":{"points":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1"],[]],"x":"a1","y":"b1","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a1\",\"b1\",\"c1\"],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\"],[]],\"x\":\"a1\",\"y\":\"b1\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0053","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a2\",\"b2\",\"c2\"],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\"],[]],\"x\":\"a2\",\"y\":\"b2\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a2, b2, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a2, b2, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113053,"parameters":{"points":["a2","b2","c2"],"topology":[["a2","b2","c2"],["a2","b2"],["a2"],[]],"x":"a2","y":"b2","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a2\",\"b2\",\"c2\"],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\"],[]],\"x\":\"a2\",\"y\":\"b2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0054","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a3\",\"b3\",\"c3\"],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\"],[]],\"x\":\"a3\",\"y\":\"b3\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a3, b3, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a3, b3, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113054,"parameters":{"points":["a3","b3","c3"],"topology":[["a3","b3","c3"],["a3","b3"],["a3"],[]],"x":"a3","y":"b3","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a3\",\"b3\",\"c3\"],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\"],[]],\"x\":\"a3\",\"y\":\"b3\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0055","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a4\",\"b4\",\"c4\"],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\"],[]],\"x\":\"a4\",\"y\":\"b4\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a4, b4, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a4, b4, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113055,"parameters":{"points":["a4","b4","c4"],"topology":[["a4","b4","c4"],["a4","b4"],["a4"],[]],"x":"a4","y":"b4","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a4\",\"b4\",\"c4\"],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\"],[]],\"x\":\"a4\",\"y\":\"b4\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0056","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a5\",\"b5\",\"c5\"],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\"],[]],\"x\":\"a5\",\"y\":\"b5\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a5, b5, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a5, b5, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113056,"parameters":{"points":["a5","b5","c5"],"topology":[["a5","b5","c5"],["a5","b5"],["a5"],[]],"x":"a5","y":"b5","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a5\",\"b5\",\"c5\"],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\"],[]],\"x\":\"a5\",\"y\":\"b5\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0057","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a6\",\"b6\",\"c6\"],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\"],[]],\"x\":\"a6\",\"y\":\"b6\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a6, b6, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a6, b6, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113057,"parameters":{"points":["a6","b6","c6"],"topology":[["a6","b6","c6"],["a6","b6"],["a6"],[]],"x":"a6","y":"b6","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a6\",\"b6\",\"c6\"],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\"],[]],\"x\":\"a6\",\"y\":\"b6\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0058","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a7\",\"b7\",\"c7\"],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\"],[]],\"x\":\"a7\",\"y\":\"b7\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a7, b7, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a7, b7, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113058,"parameters":{"points":["a7","b7","c7"],"topology":[["a7","b7","c7"],["a7","b7"],["a7"],[]],"x":"a7","y":"b7","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a7\",\"b7\",\"c7\"],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\"],[]],\"x\":\"a7\",\"y\":\"b7\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0059","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a8\",\"b8\",\"c8\"],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\"],[]],\"x\":\"a8\",\"y\":\"b8\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a8, b8, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a8, b8, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113059,"parameters":{"points":["a8","b8","c8"],"topology":[["a8","b8","c8"],["a8","b8"],["a8"],[]],"x":"a8","y":"b8","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a8\",\"b8\",\"c8\"],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\"],[]],\"x\":\"a8\",\"y\":\"b8\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0060","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a9\",\"b9\",\"c9\"],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\"],[]],\"x\":\"a9\",\"y\":\"b9\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a9, b9, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a9, b9, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113060,"parameters":{"points":["a9","b9","c9"],"topology":[["a9","b9","c9"],["a9","b9"],["a9"],[]],"x":"a9","y":"b9","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a9\",\"b9\",\"c9\"],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\"],[]],\"x\":\"a9\",\"y\":\"b9\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0061","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a10\",\"b10\",\"c10\"],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\"],[]],\"x\":\"a10\",\"y\":\"b10\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a10, b10, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a10, b10, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113061,"parameters":{"points":["a10","b10","c10"],"topology":[["a10","b10","c10"],["a10","b10"],["a10"],[]],"x":"a10","y":"b10","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a10\",\"b10\",\"c10\"],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\"],[]],\"x\":\"a10\",\"y\":\"b10\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0062","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a11\",\"b11\",\"c11\"],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\"],[]],\"x\":\"a11\",\"y\":\"b11\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a11, b11, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a11, b11, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113062,"parameters":{"points":["a11","b11","c11"],"topology":[["a11","b11","c11"],["a11","b11"],["a11"],[]],"x":"a11","y":"b11","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a11\",\"b11\",\"c11\"],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\"],[]],\"x\":\"a11\",\"y\":\"b11\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0063","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a12\",\"b12\",\"c12\"],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\"],[]],\"x\":\"a12\",\"y\":\"b12\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a12, b12, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a12, b12, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113063,"parameters":{"points":["a12","b12","c12"],"topology":[["a12","b12","c12"],["a12","b12"],["a12"],[]],"x":"a12","y":"b12","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a12\",\"b12\",\"c12\"],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\"],[]],\"x\":\"a12\",\"y\":\"b12\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0064","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a13\",\"b13\",\"c13\"],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\"],[]],\"x\":\"a13\",\"y\":\"b13\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a13, b13, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a13, b13, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113064,"parameters":{"points":["a13","b13","c13"],"topology":[["a13","b13","c13"],["a13","b13"],["a13"],[]],"x":"a13","y":"b13","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a13\",\"b13\",\"c13\"],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\"],[]],\"x\":\"a13\",\"y\":\"b13\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0065","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a14\",\"b14\",\"c14\"],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\"],[]],\"x\":\"a14\",\"y\":\"b14\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a14, b14, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a14, b14, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113065,"parameters":{"points":["a14","b14","c14"],"topology":[["a14","b14","c14"],["a14","b14"],["a14"],[]],"x":"a14","y":"b14","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a14\",\"b14\",\"c14\"],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\"],[]],\"x\":\"a14\",\"y\":\"b14\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0066","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a15\",\"b15\",\"c15\"],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\"],[]],\"x\":\"a15\",\"y\":\"b15\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a15, b15, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a15, b15, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113066,"parameters":{"points":["a15","b15","c15"],"topology":[["a15","b15","c15"],["a15","b15"],["a15"],[]],"x":"a15","y":"b15","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a15\",\"b15\",\"c15\"],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\"],[]],\"x\":\"a15\",\"y\":\"b15\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0067","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a16\",\"b16\",\"c16\"],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\"],[]],\"x\":\"a16\",\"y\":\"b16\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a16, b16, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a16, b16, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113067,"parameters":{"points":["a16","b16","c16"],"topology":[["a16","b16","c16"],["a16","b16"],["a16"],[]],"x":"a16","y":"b16","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a16\",\"b16\",\"c16\"],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\"],[]],\"x\":\"a16\",\"y\":\"b16\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0068","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a17\",\"b17\",\"c17\"],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\"],[]],\"x\":\"a17\",\"y\":\"b17\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a17, b17, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a17, b17, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113068,"parameters":{"points":["a17","b17","c17"],"topology":[["a17","b17","c17"],["a17","b17"],["a17"],[]],"x":"a17","y":"b17","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a17\",\"b17\",\"c17\"],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\"],[]],\"x\":\"a17\",\"y\":\"b17\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0069","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a18\",\"b18\",\"c18\"],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\"],[]],\"x\":\"a18\",\"y\":\"b18\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a18, b18, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a18, b18, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113069,"parameters":{"points":["a18","b18","c18"],"topology":[["a18","b18","c18"],["a18","b18"],["a18"],[]],"x":"a18","y":"b18","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a18\",\"b18\",\"c18\"],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\"],[]],\"x\":\"a18\",\"y\":\"b18\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0070","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a19\",\"b19\",\"c19\"],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\"],[]],\"x\":\"a19\",\"y\":\"b19\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a19, b19, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a19, b19, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113070,"parameters":{"points":["a19","b19","c19"],"topology":[["a19","b19","c19"],["a19","b19"],["a19"],[]],"x":"a19","y":"b19","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a19\",\"b19\",\"c19\"],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\"],[]],\"x\":\"a19\",\"y\":\"b19\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0071","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\"],[]],\"x\":\"a20\",\"y\":\"b20\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a20, b20, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a20, b20, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113071,"parameters":{"points":["a20","b20","c20"],"topology":[["a20","b20","c20"],["a20","b20"],["a20"],[]],"x":"a20","y":"b20","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\"],[]],\"x\":\"a20\",\"y\":\"b20\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0072","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a21\",\"b21\",\"c21\"],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\"],[]],\"x\":\"a21\",\"y\":\"b21\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a21, b21, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a21, b21, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113072,"parameters":{"points":["a21","b21","c21"],"topology":[["a21","b21","c21"],["a21","b21"],["a21"],[]],"x":"a21","y":"b21","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a21\",\"b21\",\"c21\"],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\"],[]],\"x\":\"a21\",\"y\":\"b21\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0073","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a22\",\"b22\",\"c22\"],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\"],[]],\"x\":\"a22\",\"y\":\"b22\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a22, b22, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a22, b22, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113073,"parameters":{"points":["a22","b22","c22"],"topology":[["a22","b22","c22"],["a22","b22"],["a22"],[]],"x":"a22","y":"b22","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a22\",\"b22\",\"c22\"],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\"],[]],\"x\":\"a22\",\"y\":\"b22\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0074","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a23\",\"b23\",\"c23\"],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\"],[]],\"x\":\"a23\",\"y\":\"b23\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a23, b23, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a23, b23, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113074,"parameters":{"points":["a23","b23","c23"],"topology":[["a23","b23","c23"],["a23","b23"],["a23"],[]],"x":"a23","y":"b23","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a23\",\"b23\",\"c23\"],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\"],[]],\"x\":\"a23\",\"y\":\"b23\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0075","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-inseparable-pair；資料={\"points\":[\"a24\",\"b24\",\"c24\"],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\"],[]],\"x\":\"a24\",\"y\":\"b24\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(a24, b24, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)","explanation_zh":"Wave6第五層核對=(a24, b24, null, 否, one_failed_distinct_pair_proves_non_Hausdorff)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-inseparable-pair","seed":202611113075,"parameters":{"points":["a24","b24","c24"],"topology":[["a24","b24","c24"],["a24","b24"],["a24"],[]],"x":"a24","y":"b24","domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-inseparable-pair","semantic_template":"qf-mod42-hausdorff-inseparable-pair/{\"points\":[\"a24\",\"b24\",\"c24\"],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\"],[]],\"x\":\"a24\",\"y\":\"b24\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0076","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a0\",\"b0\",\"c0\"],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\",\"c0\"],[\"a0\"],[\"c0\"],[]],\"sampled_pairs\":[[\"c0\",\"a0\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c0, a0, [[c0], [a0, b0]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c0, a0, [[c0], [a0, b0]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113076,"parameters":{"points":["a0","b0","c0"],"topology":[["a0","b0","c0"],["a0","b0"],["a0","c0"],["a0"],["c0"],[]],"sampled_pairs":[["c0","a0"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a0\",\"b0\",\"c0\"],\"sampled_pairs\":[[\"c0\",\"a0\"]],\"topology\":[[\"a0\",\"b0\",\"c0\"],[\"a0\",\"b0\"],[\"a0\",\"c0\"],[\"a0\"],[\"c0\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0077","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a1\",\"b1\",\"c1\"],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\",\"c1\"],[\"a1\"],[\"c1\"],[]],\"sampled_pairs\":[[\"c1\",\"a1\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c1, a1, [[c1], [a1, b1]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c1, a1, [[c1], [a1, b1]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113077,"parameters":{"points":["a1","b1","c1"],"topology":[["a1","b1","c1"],["a1","b1"],["a1","c1"],["a1"],["c1"],[]],"sampled_pairs":[["c1","a1"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a1\",\"b1\",\"c1\"],\"sampled_pairs\":[[\"c1\",\"a1\"]],\"topology\":[[\"a1\",\"b1\",\"c1\"],[\"a1\",\"b1\"],[\"a1\",\"c1\"],[\"a1\"],[\"c1\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0078","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a2\",\"b2\",\"c2\"],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\",\"c2\"],[\"a2\"],[\"c2\"],[]],\"sampled_pairs\":[[\"c2\",\"a2\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c2, a2, [[c2], [a2, b2]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c2, a2, [[c2], [a2, b2]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113078,"parameters":{"points":["a2","b2","c2"],"topology":[["a2","b2","c2"],["a2","b2"],["a2","c2"],["a2"],["c2"],[]],"sampled_pairs":[["c2","a2"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a2\",\"b2\",\"c2\"],\"sampled_pairs\":[[\"c2\",\"a2\"]],\"topology\":[[\"a2\",\"b2\",\"c2\"],[\"a2\",\"b2\"],[\"a2\",\"c2\"],[\"a2\"],[\"c2\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0079","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a3\",\"b3\",\"c3\"],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\",\"c3\"],[\"a3\"],[\"c3\"],[]],\"sampled_pairs\":[[\"c3\",\"a3\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c3, a3, [[c3], [a3, b3]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c3, a3, [[c3], [a3, b3]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113079,"parameters":{"points":["a3","b3","c3"],"topology":[["a3","b3","c3"],["a3","b3"],["a3","c3"],["a3"],["c3"],[]],"sampled_pairs":[["c3","a3"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a3\",\"b3\",\"c3\"],\"sampled_pairs\":[[\"c3\",\"a3\"]],\"topology\":[[\"a3\",\"b3\",\"c3\"],[\"a3\",\"b3\"],[\"a3\",\"c3\"],[\"a3\"],[\"c3\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0080","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a4\",\"b4\",\"c4\"],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\",\"c4\"],[\"a4\"],[\"c4\"],[]],\"sampled_pairs\":[[\"c4\",\"a4\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c4, a4, [[c4], [a4, b4]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c4, a4, [[c4], [a4, b4]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113080,"parameters":{"points":["a4","b4","c4"],"topology":[["a4","b4","c4"],["a4","b4"],["a4","c4"],["a4"],["c4"],[]],"sampled_pairs":[["c4","a4"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a4\",\"b4\",\"c4\"],\"sampled_pairs\":[[\"c4\",\"a4\"]],\"topology\":[[\"a4\",\"b4\",\"c4\"],[\"a4\",\"b4\"],[\"a4\",\"c4\"],[\"a4\"],[\"c4\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0081","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a5\",\"b5\",\"c5\"],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\",\"c5\"],[\"a5\"],[\"c5\"],[]],\"sampled_pairs\":[[\"c5\",\"a5\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c5, a5, [[c5], [a5, b5]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c5, a5, [[c5], [a5, b5]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113081,"parameters":{"points":["a5","b5","c5"],"topology":[["a5","b5","c5"],["a5","b5"],["a5","c5"],["a5"],["c5"],[]],"sampled_pairs":[["c5","a5"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a5\",\"b5\",\"c5\"],\"sampled_pairs\":[[\"c5\",\"a5\"]],\"topology\":[[\"a5\",\"b5\",\"c5\"],[\"a5\",\"b5\"],[\"a5\",\"c5\"],[\"a5\"],[\"c5\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0082","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a6\",\"b6\",\"c6\"],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\",\"c6\"],[\"a6\"],[\"c6\"],[]],\"sampled_pairs\":[[\"c6\",\"a6\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c6, a6, [[c6], [a6, b6]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c6, a6, [[c6], [a6, b6]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113082,"parameters":{"points":["a6","b6","c6"],"topology":[["a6","b6","c6"],["a6","b6"],["a6","c6"],["a6"],["c6"],[]],"sampled_pairs":[["c6","a6"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a6\",\"b6\",\"c6\"],\"sampled_pairs\":[[\"c6\",\"a6\"]],\"topology\":[[\"a6\",\"b6\",\"c6\"],[\"a6\",\"b6\"],[\"a6\",\"c6\"],[\"a6\"],[\"c6\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0083","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a7\",\"b7\",\"c7\"],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\",\"c7\"],[\"a7\"],[\"c7\"],[]],\"sampled_pairs\":[[\"c7\",\"a7\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c7, a7, [[c7], [a7, b7]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c7, a7, [[c7], [a7, b7]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113083,"parameters":{"points":["a7","b7","c7"],"topology":[["a7","b7","c7"],["a7","b7"],["a7","c7"],["a7"],["c7"],[]],"sampled_pairs":[["c7","a7"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a7\",\"b7\",\"c7\"],\"sampled_pairs\":[[\"c7\",\"a7\"]],\"topology\":[[\"a7\",\"b7\",\"c7\"],[\"a7\",\"b7\"],[\"a7\",\"c7\"],[\"a7\"],[\"c7\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0084","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a8\",\"b8\",\"c8\"],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\",\"c8\"],[\"a8\"],[\"c8\"],[]],\"sampled_pairs\":[[\"c8\",\"a8\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c8, a8, [[c8], [a8, b8]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c8, a8, [[c8], [a8, b8]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113084,"parameters":{"points":["a8","b8","c8"],"topology":[["a8","b8","c8"],["a8","b8"],["a8","c8"],["a8"],["c8"],[]],"sampled_pairs":[["c8","a8"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a8\",\"b8\",\"c8\"],\"sampled_pairs\":[[\"c8\",\"a8\"]],\"topology\":[[\"a8\",\"b8\",\"c8\"],[\"a8\",\"b8\"],[\"a8\",\"c8\"],[\"a8\"],[\"c8\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0085","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a9\",\"b9\",\"c9\"],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\",\"c9\"],[\"a9\"],[\"c9\"],[]],\"sampled_pairs\":[[\"c9\",\"a9\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c9, a9, [[c9], [a9, b9]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c9, a9, [[c9], [a9, b9]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113085,"parameters":{"points":["a9","b9","c9"],"topology":[["a9","b9","c9"],["a9","b9"],["a9","c9"],["a9"],["c9"],[]],"sampled_pairs":[["c9","a9"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a9\",\"b9\",\"c9\"],\"sampled_pairs\":[[\"c9\",\"a9\"]],\"topology\":[[\"a9\",\"b9\",\"c9\"],[\"a9\",\"b9\"],[\"a9\",\"c9\"],[\"a9\"],[\"c9\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0086","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a10\",\"b10\",\"c10\"],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\",\"c10\"],[\"a10\"],[\"c10\"],[]],\"sampled_pairs\":[[\"c10\",\"a10\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c10, a10, [[c10], [a10, b10]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c10, a10, [[c10], [a10, b10]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113086,"parameters":{"points":["a10","b10","c10"],"topology":[["a10","b10","c10"],["a10","b10"],["a10","c10"],["a10"],["c10"],[]],"sampled_pairs":[["c10","a10"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a10\",\"b10\",\"c10\"],\"sampled_pairs\":[[\"c10\",\"a10\"]],\"topology\":[[\"a10\",\"b10\",\"c10\"],[\"a10\",\"b10\"],[\"a10\",\"c10\"],[\"a10\"],[\"c10\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0087","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a11\",\"b11\",\"c11\"],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\",\"c11\"],[\"a11\"],[\"c11\"],[]],\"sampled_pairs\":[[\"c11\",\"a11\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c11, a11, [[c11], [a11, b11]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c11, a11, [[c11], [a11, b11]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113087,"parameters":{"points":["a11","b11","c11"],"topology":[["a11","b11","c11"],["a11","b11"],["a11","c11"],["a11"],["c11"],[]],"sampled_pairs":[["c11","a11"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a11\",\"b11\",\"c11\"],\"sampled_pairs\":[[\"c11\",\"a11\"]],\"topology\":[[\"a11\",\"b11\",\"c11\"],[\"a11\",\"b11\"],[\"a11\",\"c11\"],[\"a11\"],[\"c11\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0088","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a12\",\"b12\",\"c12\"],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\",\"c12\"],[\"a12\"],[\"c12\"],[]],\"sampled_pairs\":[[\"c12\",\"a12\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c12, a12, [[c12], [a12, b12]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c12, a12, [[c12], [a12, b12]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113088,"parameters":{"points":["a12","b12","c12"],"topology":[["a12","b12","c12"],["a12","b12"],["a12","c12"],["a12"],["c12"],[]],"sampled_pairs":[["c12","a12"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a12\",\"b12\",\"c12\"],\"sampled_pairs\":[[\"c12\",\"a12\"]],\"topology\":[[\"a12\",\"b12\",\"c12\"],[\"a12\",\"b12\"],[\"a12\",\"c12\"],[\"a12\"],[\"c12\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0089","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a13\",\"b13\",\"c13\"],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\",\"c13\"],[\"a13\"],[\"c13\"],[]],\"sampled_pairs\":[[\"c13\",\"a13\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c13, a13, [[c13], [a13, b13]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c13, a13, [[c13], [a13, b13]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113089,"parameters":{"points":["a13","b13","c13"],"topology":[["a13","b13","c13"],["a13","b13"],["a13","c13"],["a13"],["c13"],[]],"sampled_pairs":[["c13","a13"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a13\",\"b13\",\"c13\"],\"sampled_pairs\":[[\"c13\",\"a13\"]],\"topology\":[[\"a13\",\"b13\",\"c13\"],[\"a13\",\"b13\"],[\"a13\",\"c13\"],[\"a13\"],[\"c13\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0090","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a14\",\"b14\",\"c14\"],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\",\"c14\"],[\"a14\"],[\"c14\"],[]],\"sampled_pairs\":[[\"c14\",\"a14\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c14, a14, [[c14], [a14, b14]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c14, a14, [[c14], [a14, b14]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113090,"parameters":{"points":["a14","b14","c14"],"topology":[["a14","b14","c14"],["a14","b14"],["a14","c14"],["a14"],["c14"],[]],"sampled_pairs":[["c14","a14"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a14\",\"b14\",\"c14\"],\"sampled_pairs\":[[\"c14\",\"a14\"]],\"topology\":[[\"a14\",\"b14\",\"c14\"],[\"a14\",\"b14\"],[\"a14\",\"c14\"],[\"a14\"],[\"c14\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0091","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a15\",\"b15\",\"c15\"],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\",\"c15\"],[\"a15\"],[\"c15\"],[]],\"sampled_pairs\":[[\"c15\",\"a15\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c15, a15, [[c15], [a15, b15]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c15, a15, [[c15], [a15, b15]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113091,"parameters":{"points":["a15","b15","c15"],"topology":[["a15","b15","c15"],["a15","b15"],["a15","c15"],["a15"],["c15"],[]],"sampled_pairs":[["c15","a15"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a15\",\"b15\",\"c15\"],\"sampled_pairs\":[[\"c15\",\"a15\"]],\"topology\":[[\"a15\",\"b15\",\"c15\"],[\"a15\",\"b15\"],[\"a15\",\"c15\"],[\"a15\"],[\"c15\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0092","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a16\",\"b16\",\"c16\"],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\",\"c16\"],[\"a16\"],[\"c16\"],[]],\"sampled_pairs\":[[\"c16\",\"a16\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c16, a16, [[c16], [a16, b16]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c16, a16, [[c16], [a16, b16]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113092,"parameters":{"points":["a16","b16","c16"],"topology":[["a16","b16","c16"],["a16","b16"],["a16","c16"],["a16"],["c16"],[]],"sampled_pairs":[["c16","a16"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a16\",\"b16\",\"c16\"],\"sampled_pairs\":[[\"c16\",\"a16\"]],\"topology\":[[\"a16\",\"b16\",\"c16\"],[\"a16\",\"b16\"],[\"a16\",\"c16\"],[\"a16\"],[\"c16\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0093","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a17\",\"b17\",\"c17\"],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\",\"c17\"],[\"a17\"],[\"c17\"],[]],\"sampled_pairs\":[[\"c17\",\"a17\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c17, a17, [[c17], [a17, b17]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c17, a17, [[c17], [a17, b17]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113093,"parameters":{"points":["a17","b17","c17"],"topology":[["a17","b17","c17"],["a17","b17"],["a17","c17"],["a17"],["c17"],[]],"sampled_pairs":[["c17","a17"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a17\",\"b17\",\"c17\"],\"sampled_pairs\":[[\"c17\",\"a17\"]],\"topology\":[[\"a17\",\"b17\",\"c17\"],[\"a17\",\"b17\"],[\"a17\",\"c17\"],[\"a17\"],[\"c17\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0094","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a18\",\"b18\",\"c18\"],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\",\"c18\"],[\"a18\"],[\"c18\"],[]],\"sampled_pairs\":[[\"c18\",\"a18\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c18, a18, [[c18], [a18, b18]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c18, a18, [[c18], [a18, b18]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113094,"parameters":{"points":["a18","b18","c18"],"topology":[["a18","b18","c18"],["a18","b18"],["a18","c18"],["a18"],["c18"],[]],"sampled_pairs":[["c18","a18"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a18\",\"b18\",\"c18\"],\"sampled_pairs\":[[\"c18\",\"a18\"]],\"topology\":[[\"a18\",\"b18\",\"c18\"],[\"a18\",\"b18\"],[\"a18\",\"c18\"],[\"a18\"],[\"c18\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0095","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a19\",\"b19\",\"c19\"],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\",\"c19\"],[\"a19\"],[\"c19\"],[]],\"sampled_pairs\":[[\"c19\",\"a19\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c19, a19, [[c19], [a19, b19]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c19, a19, [[c19], [a19, b19]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113095,"parameters":{"points":["a19","b19","c19"],"topology":[["a19","b19","c19"],["a19","b19"],["a19","c19"],["a19"],["c19"],[]],"sampled_pairs":[["c19","a19"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a19\",\"b19\",\"c19\"],\"sampled_pairs\":[[\"c19\",\"a19\"]],\"topology\":[[\"a19\",\"b19\",\"c19\"],[\"a19\",\"b19\"],[\"a19\",\"c19\"],[\"a19\"],[\"c19\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0096","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a20\",\"b20\",\"c20\"],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"c20\"],[]],\"sampled_pairs\":[[\"c20\",\"a20\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c20, a20, [[c20], [a20, b20]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c20, a20, [[c20], [a20, b20]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113096,"parameters":{"points":["a20","b20","c20"],"topology":[["a20","b20","c20"],["a20","b20"],["a20","c20"],["a20"],["c20"],[]],"sampled_pairs":[["c20","a20"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a20\",\"b20\",\"c20\"],\"sampled_pairs\":[[\"c20\",\"a20\"]],\"topology\":[[\"a20\",\"b20\",\"c20\"],[\"a20\",\"b20\"],[\"a20\",\"c20\"],[\"a20\"],[\"c20\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0097","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a21\",\"b21\",\"c21\"],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\",\"c21\"],[\"a21\"],[\"c21\"],[]],\"sampled_pairs\":[[\"c21\",\"a21\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c21, a21, [[c21], [a21, b21]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c21, a21, [[c21], [a21, b21]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113097,"parameters":{"points":["a21","b21","c21"],"topology":[["a21","b21","c21"],["a21","b21"],["a21","c21"],["a21"],["c21"],[]],"sampled_pairs":[["c21","a21"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a21\",\"b21\",\"c21\"],\"sampled_pairs\":[[\"c21\",\"a21\"]],\"topology\":[[\"a21\",\"b21\",\"c21\"],[\"a21\",\"b21\"],[\"a21\",\"c21\"],[\"a21\"],[\"c21\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0098","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a22\",\"b22\",\"c22\"],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\",\"c22\"],[\"a22\"],[\"c22\"],[]],\"sampled_pairs\":[[\"c22\",\"a22\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c22, a22, [[c22], [a22, b22]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c22, a22, [[c22], [a22, b22]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113098,"parameters":{"points":["a22","b22","c22"],"topology":[["a22","b22","c22"],["a22","b22"],["a22","c22"],["a22"],["c22"],[]],"sampled_pairs":[["c22","a22"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a22\",\"b22\",\"c22\"],\"sampled_pairs\":[[\"c22\",\"a22\"]],\"topology\":[[\"a22\",\"b22\",\"c22\"],[\"a22\",\"b22\"],[\"a22\",\"c22\"],[\"a22\"],[\"c22\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0099","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a23\",\"b23\",\"c23\"],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\",\"c23\"],[\"a23\"],[\"c23\"],[]],\"sampled_pairs\":[[\"c23\",\"a23\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c23, a23, [[c23], [a23, b23]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c23, a23, [[c23], [a23, b23]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113099,"parameters":{"points":["a23","b23","c23"],"topology":[["a23","b23","c23"],["a23","b23"],["a23","c23"],["a23"],["c23"],[]],"sampled_pairs":[["c23","a23"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a23\",\"b23\",\"c23\"],\"sampled_pairs\":[[\"c23\",\"a23\"]],\"topology\":[[\"a23\",\"b23\",\"c23\"],[\"a23\",\"b23\"],[\"a23\",\"c23\"],[\"a23\"],[\"c23\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r03-hausdorff-space-0100","batch_id":"batch-20261014-advance-20261111-r03-hausdorff-space-foundations","language":"zh-Hant","source_mko_ids":["mko-hausdorff-space"],"title_zh":"Hausdorff空間","stem_zh":"完整有限拓撲上每一對distinct points的disjoint open-neighborhood witnesses。操作=hausdorff-sampled-boundary；資料={\"points\":[\"a24\",\"b24\",\"c24\"],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\",\"c24\"],[\"a24\"],[\"c24\"],[]],\"sampled_pairs\":[[\"c24\",\"a24\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[c24, a24, [[c24], [a24, b24]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)","explanation_zh":"Wave6第五層核對=([[c24, a24, [[c24], [a24, b24]]]], 是, 1, 3, 否, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)；依完整明示資料獨立重算；一個失敗pair即可證non-Hausdorff；empty與singleton vacuously Hausdorff；sampled成功pairs不足以證unchecked或infinite spaces；finite Hausdorff implies discrete不可外推。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Hausdorff空間"]},"variant":{"family_id":"qf-mod42-hausdorff-sampled-boundary","seed":202611113100,"parameters":{"points":["a24","b24","c24"],"topology":[["a24","b24","c24"],["a24","b24"],["a24","c24"],["a24"],["c24"],[]],"sampled_pairs":[["c24","a24"]],"domain":"Complete_finite_topology_Hausdorff_audit","operation":"hausdorff-sampled-boundary","semantic_template":"qf-mod42-hausdorff-sampled-boundary/{\"points\":[\"a24\",\"b24\",\"c24\"],\"sampled_pairs\":[[\"c24\",\"a24\"]],\"topology\":[[\"a24\",\"b24\",\"c24\"],[\"a24\",\"b24\"],[\"a24\",\"c24\"],[\"a24\"],[\"c24\"],[]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}