{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0001","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"left\":[0,1],\"right\":[2,3,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 1], [2, 3, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035001,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"left":[0,1],"right":[2,3,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,1],\"n\":5,\"right\":[2,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0002","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,4],[0,1,2,3,4]],\"left\":[0,1],\"right\":[2,3,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 1], [2, 3, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035002,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"left":[0,1],"right":[2,3,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,1],\"n\":5,\"right\":[2,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0003","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"left\":[0,1],\"right\":[2,3,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 1], [2, 3, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035003,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"left":[0,1],"right":[2,3,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,1],\"n\":5,\"right\":[2,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0004","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"left\":[0,1],\"right\":[2,3,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 1], [2, 3, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035004,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"left":[0,1],"right":[2,3,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,1],\"n\":5,\"right\":[2,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0005","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"left\":[0,1],\"right\":[2,3,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 1], [2, 3, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035005,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"left":[0,1],"right":[2,3,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,1],\"n\":5,\"right\":[2,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0006","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"left\":[0,1],\"right\":[2,3,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 1], [2, 3, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035006,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"left":[0,1],"right":[2,3,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,1],\"n\":5,\"right\":[2,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0007","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"left\":[0,2],\"right\":[1,3,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 2], [1, 3, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035007,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"left":[0,2],"right":[1,3,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,2],\"n\":5,\"right\":[1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0008","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],\"left\":[0,2],\"right\":[1,3,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 2], [1, 3, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035008,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"left":[0,2],"right":[1,3,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,2],\"n\":5,\"right\":[1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0009","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,1,2,3],[0,1,2,3,4]],\"left\":[0,2],\"right\":[1,3,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 2], [1, 3, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035009,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"left":[0,2],"right":[1,3,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,2],\"n\":5,\"right\":[1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0010","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"left\":[0,2],\"right\":[1,3,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 2], [1, 3, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035010,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"left":[0,2],"right":[1,3,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,2],\"n\":5,\"right\":[1,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0011","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"left\":[0,2],\"right\":[1,3,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 2], [1, 3, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035011,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"left":[0,2],"right":[1,3,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,2],\"n\":5,\"right\":[1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0012","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"left\":[0,2],\"right\":[1,3,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 2], [1, 3, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035012,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"left":[0,2],"right":[1,3,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,2],\"n\":5,\"right\":[1,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0013","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"left\":[0,3],\"right\":[1,2,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 3], [1, 2, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035013,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"left":[0,3],"right":[1,2,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,3],\"n\":5,\"right\":[1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0014","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"left\":[0,3],\"right\":[1,2,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 3], [1, 2, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035014,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"left":[0,3],"right":[1,2,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,3],\"n\":5,\"right\":[1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0015","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],\"left\":[0,3],\"right\":[1,2,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 3], [1, 2, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035015,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"left":[0,3],"right":[1,2,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,3],\"n\":5,\"right\":[1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0016","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"left\":[0,3],\"right\":[1,2,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 3], [1, 2, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035016,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"left":[0,3],"right":[1,2,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,3],\"n\":5,\"right\":[1,2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0017","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"left\":[0,3],\"right\":[1,2,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 3], [1, 2, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035017,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"left":[0,3],"right":[1,2,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,3],\"n\":5,\"right\":[1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0018","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"left\":[0,3],\"right\":[1,2,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 3], [1, 2, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035018,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"left":[0,3],"right":[1,2,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,3],\"n\":5,\"right\":[1,2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0019","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"left\":[0,4],\"right\":[1,2,3]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 4], [1, 2, 3], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035019,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"left":[0,4],"right":[1,2,3],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,4],\"n\":5,\"right\":[1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0020","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"left\":[0,4],\"right\":[1,2,3]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 4], [1, 2, 3], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035020,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"left":[0,4],"right":[1,2,3],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,4],\"n\":5,\"right\":[1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0021","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"left\":[0,4],\"right\":[1,2,3]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 4], [1, 2, 3], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035021,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"left":[0,4],"right":[1,2,3],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,4],\"n\":5,\"right\":[1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0022","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"left\":[0,4],\"right\":[1,2,3]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 4], [1, 2, 3], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035022,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"left":[0,4],"right":[1,2,3],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,4],\"n\":5,\"right\":[1,2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0023","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"left\":[0,4],\"right\":[1,2,3]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 4], [1, 2, 3], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035023,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"left":[0,4],"right":[1,2,3],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,4],\"n\":5,\"right\":[1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0024","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"left\":[0,4],\"right\":[1,2,3]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 4], [1, 2, 3], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035024,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"left":[0,4],"right":[1,2,3],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,4],\"n\":5,\"right\":[1,2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0025","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-separation-audit；資料={\"n\":5,\"topology\":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"left\":[0,1],\"right\":[2,3,4]}。請依 [left,right,valid_separation,classification] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], 否, not_a_separation)","explanation_zh":"Wave5第三層核對=([0, 1], [2, 3, 4], 否, not_a_separation)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-separation-audit","seed":202611035025,"parameters":{"n":5,"topology":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"left":[0,1],"right":[2,3,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-separation-audit","semantic_template":"qf-mod34-connected-separation-audit/{\"left\":[0,1],\"n\":5,\"right\":[2,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0026","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"subset\":[0,1]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 1], [2, 3, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035026,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"subset":[0,1],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,1],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0027","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,4],[0,1,2,3,4]],\"subset\":[0,1]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 1], [2, 3, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035027,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"subset":[0,1],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,1],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0028","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"subset\":[0,1]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 1], [2, 3, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035028,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"subset":[0,1],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,1],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0029","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"subset\":[0,1]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 1], [2, 3, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035029,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"subset":[0,1],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,1],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0030","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"subset\":[0,1]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 1], [2, 3, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035030,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"subset":[0,1],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,1],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0031","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"subset\":[0,1]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 1], [2, 3, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035031,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"subset":[0,1],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,1],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0032","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"subset\":[0,2]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 2], [1, 3, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035032,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"subset":[0,2],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0033","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],\"subset\":[0,2]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 2], [1, 3, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035033,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"subset":[0,2],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0034","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,1,2,3],[0,1,2,3,4]],\"subset\":[0,2]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 2], [1, 3, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035034,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"subset":[0,2],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0035","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"subset\":[0,2]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 2], [1, 3, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035035,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"subset":[0,2],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0036","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"subset\":[0,2]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 2], [1, 3, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035036,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"subset":[0,2],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0037","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"subset\":[0,2]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 2], [1, 3, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035037,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"subset":[0,2],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0038","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"subset\":[0,3]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 3], [1, 2, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035038,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"subset":[0,3],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0039","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"subset\":[0,3]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 3], [1, 2, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035039,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"subset":[0,3],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0040","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],\"subset\":[0,3]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 3], [1, 2, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035040,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"subset":[0,3],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0041","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"subset\":[0,3]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 3], [1, 2, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035041,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"subset":[0,3],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0042","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"subset\":[0,3]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 3], [1, 2, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035042,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"subset":[0,3],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0043","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"subset\":[0,3]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 3], [1, 2, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035043,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"subset":[0,3],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0044","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"subset\":[0,4]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 4], [1, 2, 3], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035044,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"subset":[0,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0045","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"subset\":[0,4]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 4], [1, 2, 3], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035045,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"subset":[0,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0046","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"subset\":[0,4]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 4], [1, 2, 3], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035046,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"subset":[0,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0047","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"subset\":[0,4]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 4], [1, 2, 3], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035047,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"subset":[0,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0048","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"subset\":[0,4]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 4], [1, 2, 3], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035048,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"subset":[0,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0049","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"subset\":[0,4]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 4], [1, 2, 3], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035049,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"subset":[0,4],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0050","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-clopen-witness；資料={\"n\":5,\"topology\":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"subset\":[0,1]}。請依 [subset,complement,clopen,nontrivial,disconnects,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], 否, 是, 否, nontrivial_clopen_witness)","explanation_zh":"Wave5第三層核對=([0, 1], [2, 3, 4], 否, 是, 否, nontrivial_clopen_witness)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-clopen-witness","seed":202611035050,"parameters":{"n":5,"topology":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"subset":[0,1],"domain":"Complete_finite_topology_connectedness","operation":"connected-clopen-witness","semantic_template":"qf-mod34-connected-clopen-witness/{\"n\":5,\"subset\":[0,1],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0051","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035051,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0052","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035052,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0053","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035053,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0054","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035054,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0055","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035055,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0056","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035056,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0057","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035057,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0058","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035058,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0059","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,1,2,3],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035059,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0060","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035060,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0061","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035061,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0062","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035062,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0063","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035063,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0064","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035064,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0065","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035065,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0066","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035066,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0067","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035067,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0068","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035068,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0069","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035069,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0070","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035070,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0071","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035071,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0072","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035072,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0073","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035073,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0074","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035074,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0075","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選。操作=connected-exhaustive；資料={\"n\":5,\"topology\":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]]}。請依 [clopen_sets,nontrivial_clopen,connected,certificate] 欄序回答。","answer":{"kind":"expression","value":"([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)","explanation_zh":"Wave5第三層核對=([[], [0, 1, 2, 3, 4]], [], 是, exhaustive_complete_finite_topology)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-exhaustive","seed":202611035075,"parameters":{"n":5,"topology":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topology_connectedness","operation":"connected-exhaustive","semantic_template":"qf-mod34-connected-exhaustive/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0076","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"which\":\"empty\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)","explanation_zh":"Wave5第三層核對=(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035076,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"which":"empty","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]],\"which\":\"empty\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0077","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,4],[0,1,2,3,4]],\"which\":\"singleton\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(singleton, 5, 是, singleton_connected)","explanation_zh":"Wave5第三層核對=(singleton, 5, 是, singleton_connected)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035077,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"which":"singleton","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]],\"which\":\"singleton\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0078","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"which\":\"empty\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)","explanation_zh":"Wave5第三層核對=(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035078,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"which":"empty","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]],\"which\":\"empty\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0079","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"which\":\"singleton\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(singleton, 5, 是, singleton_connected)","explanation_zh":"Wave5第三層核對=(singleton, 5, 是, singleton_connected)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035079,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"which":"singleton","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]],\"which\":\"singleton\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0080","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"which\":\"empty\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)","explanation_zh":"Wave5第三層核對=(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035080,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"which":"empty","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]],\"which\":\"empty\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0081","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"which\":\"singleton\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(singleton, 5, 是, singleton_connected)","explanation_zh":"Wave5第三層核對=(singleton, 5, 是, singleton_connected)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035081,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"which":"singleton","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]],\"which\":\"singleton\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0082","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"which\":\"empty\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)","explanation_zh":"Wave5第三層核對=(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035082,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"which":"empty","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]],\"which\":\"empty\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0083","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],\"which\":\"singleton\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(singleton, 5, 是, singleton_connected)","explanation_zh":"Wave5第三層核對=(singleton, 5, 是, singleton_connected)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035083,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"which":"singleton","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]],\"which\":\"singleton\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0084","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,1,2,3],[0,1,2,3,4]],\"which\":\"empty\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)","explanation_zh":"Wave5第三層核對=(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035084,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"which":"empty","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]],\"which\":\"empty\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0085","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"which\":\"singleton\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(singleton, 5, 是, singleton_connected)","explanation_zh":"Wave5第三層核對=(singleton, 5, 是, singleton_connected)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035085,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"which":"singleton","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]],\"which\":\"singleton\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0086","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"which\":\"empty\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)","explanation_zh":"Wave5第三層核對=(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035086,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"which":"empty","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]],\"which\":\"empty\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0087","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"which\":\"singleton\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(singleton, 5, 是, singleton_connected)","explanation_zh":"Wave5第三層核對=(singleton, 5, 是, singleton_connected)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035087,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"which":"singleton","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]],\"which\":\"singleton\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0088","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"which\":\"empty\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)","explanation_zh":"Wave5第三層核對=(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035088,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"which":"empty","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]],\"which\":\"empty\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0089","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"which\":\"singleton\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(singleton, 5, 是, singleton_connected)","explanation_zh":"Wave5第三層核對=(singleton, 5, 是, singleton_connected)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035089,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"which":"singleton","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]],\"which\":\"singleton\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0090","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],\"which\":\"empty\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)","explanation_zh":"Wave5第三層核對=(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035090,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"which":"empty","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]],\"which\":\"empty\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0091","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"which\":\"singleton\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(singleton, 5, 是, singleton_connected)","explanation_zh":"Wave5第三層核對=(singleton, 5, 是, singleton_connected)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035091,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"which":"singleton","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]],\"which\":\"singleton\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0092","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"which\":\"empty\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)","explanation_zh":"Wave5第三層核對=(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035092,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"which":"empty","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]],\"which\":\"empty\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0093","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"which\":\"singleton\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(singleton, 5, 是, singleton_connected)","explanation_zh":"Wave5第三層核對=(singleton, 5, 是, singleton_connected)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035093,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"which":"singleton","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]],\"which\":\"singleton\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0094","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"which\":\"empty\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)","explanation_zh":"Wave5第三層核對=(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035094,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"which":"empty","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]],\"which\":\"empty\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0095","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"which\":\"singleton\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(singleton, 5, 是, singleton_connected)","explanation_zh":"Wave5第三層核對=(singleton, 5, 是, singleton_connected)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035095,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"which":"singleton","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]],\"which\":\"singleton\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0096","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"which\":\"empty\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)","explanation_zh":"Wave5第三層核對=(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035096,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"which":"empty","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]],\"which\":\"empty\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0097","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"which\":\"singleton\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(singleton, 5, 是, singleton_connected)","explanation_zh":"Wave5第三層核對=(singleton, 5, 是, singleton_connected)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035097,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"which":"singleton","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]],\"which\":\"singleton\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0098","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"which\":\"empty\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)","explanation_zh":"Wave5第三層核對=(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035098,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"which":"empty","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]],\"which\":\"empty\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0099","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"which\":\"singleton\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(singleton, 5, 是, singleton_connected)","explanation_zh":"Wave5第三層核對=(singleton, 5, 是, singleton_connected)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035099,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"which":"singleton","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]],\"which\":\"singleton\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r05-connected-topological-space-0100","batch_id":"batch-20261014-advance-20261103-r05-connected-topological-space-foundations","language":"zh-Hant","source_mko_ids":["mko-connected-topological-space"],"title_zh":"連通拓撲空間","stem_zh":"完整有限拓撲與全部clopen/separation候選；which僅指定獨立的空或單點參考慣例；n與topology描述實際待判定空間，不聲稱實際載體就是參考空間。操作=connected-empty-singleton-boundary；資料={\"n\":5,\"topology\":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"which\":\"empty\"}。請依 [reference_kind,actual_carrier_size,actual_connected,reference_convention] 欄序回答。","answer":{"kind":"expression","value":"(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)","explanation_zh":"Wave5第三層核對=(empty, 5, 是, empty_space_connected_by_declared_vacuous_convention)；依完整明示資料重算；connected需exhaustive排除非平凡clopen且disconnected需witness；不混同graph connectivity；空空間慣例與singleton界線明示；actual_connected只由實際完整拓撲決定；reference_convention不得當成該空間的連通性證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["連通拓撲空間"]},"variant":{"family_id":"qf-mod34-connected-empty-singleton-boundary","seed":202611035100,"parameters":{"n":5,"topology":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"which":"empty","domain":"Complete_finite_topology_connectedness","operation":"connected-empty-singleton-boundary","semantic_template":"qf-mod34-connected-empty-singleton-boundary/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]],\"which\":\"empty\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}