{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0001","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 1]], [0, 1], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 1]], [0, 1], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177001,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0002","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 2]], [0, 2], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 2]], [0, 2], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177002,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0003","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 2], [0, 1, 2]], [0, 1, 2], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 1], [0, 2], [0, 1, 2]], [0, 1, 2], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177003,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0004","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,3],[1,2,4,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 3]], [0, 3], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 3]], [0, 3], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177004,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,3],[1,2,4,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3],[1,2,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0005","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 3], [0, 1, 3]], [0, 1, 3], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 1], [0, 3], [0, 1, 3]], [0, 1, 3], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177005,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0006","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 2], [0, 3], [0, 2, 3]], [0, 2, 3], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 2], [0, 3], [0, 2, 3]], [0, 2, 3], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177006,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0007","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]], [0, 1, 2, 3], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]], [0, 1, 2, 3], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177007,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0008","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,4],[1,2,3,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 4]], [0, 4], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 4]], [0, 4], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177008,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,4],[1,2,3,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,4],[1,2,3,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0009","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 4], [0, 1, 4]], [0, 1, 4], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 1], [0, 4], [0, 1, 4]], [0, 1, 4], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177009,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0010","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 2], [0, 4], [0, 2, 4]], [0, 2, 4], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 2], [0, 4], [0, 2, 4]], [0, 2, 4], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177010,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0011","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4]], [0, 1, 2, 4], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4]], [0, 1, 2, 4], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177011,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0012","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,3,4],[1,2,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 3], [0, 4], [0, 3, 4]], [0, 3, 4], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 3], [0, 4], [0, 3, 4]], [0, 3, 4], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177012,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,3,4],[1,2,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3,4],[1,2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0013","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 3], [0, 1, 3], [0, 4], [0, 1, 4], [0, 3, 4], [0, 1, 3, 4]], [0, 1, 3, 4], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 1], [0, 3], [0, 1, 3], [0, 4], [0, 1, 4], [0, 3, 4], [0, 1, 3, 4]], [0, 1, 3, 4], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177013,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0014","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 2], [0, 3], [0, 2, 3], [0, 4], [0, 2, 4], [0, 3, 4], [0, 2, 3, 4]], [0, 2, 3, 4], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 2], [0, 3], [0, 2, 3], [0, 4], [0, 2, 4], [0, 3, 4], [0, 2, 3, 4]], [0, 2, 3, 4], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177014,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0015","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4], [0, 3, 4], [0, 1, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], [0, 1, 2, 3, 4], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4], [0, 3, 4], [0, 1, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], [0, 1, 2, 3, 4], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177015,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0016","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,5],[1,2,3,4]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 5]], [0, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 5]], [0, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177016,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,5],[1,2,3,4]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,5],[1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0017","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 5], [0, 1, 5]], [0, 1, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 1], [0, 5], [0, 1, 5]], [0, 1, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177017,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0018","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 2], [0, 5], [0, 2, 5]], [0, 2, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 2], [0, 5], [0, 2, 5]], [0, 2, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177018,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0019","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5]], [0, 1, 2, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5]], [0, 1, 2, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177019,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0020","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,3,5],[1,2,4]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 3], [0, 5], [0, 3, 5]], [0, 3, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 3], [0, 5], [0, 3, 5]], [0, 3, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177020,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,3,5],[1,2,4]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3,5],[1,2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0021","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 3], [0, 1, 3], [0, 5], [0, 1, 5], [0, 3, 5], [0, 1, 3, 5]], [0, 1, 3, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 1], [0, 3], [0, 1, 3], [0, 5], [0, 1, 5], [0, 3, 5], [0, 1, 3, 5]], [0, 1, 3, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177021,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0022","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 2], [0, 3], [0, 2, 3], [0, 5], [0, 2, 5], [0, 3, 5], [0, 2, 3, 5]], [0, 2, 3, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 2], [0, 3], [0, 2, 3], [0, 5], [0, 2, 5], [0, 3, 5], [0, 2, 3, 5]], [0, 2, 3, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177022,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0023","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5], [0, 3, 5], [0, 1, 3, 5], [0, 2, 3, 5], [0, 1, 2, 3, 5]], [0, 1, 2, 3, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5], [0, 3, 5], [0, 1, 3, 5], [0, 2, 3, 5], [0, 1, 2, 3, 5]], [0, 1, 2, 3, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177023,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0024","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,4,5],[1,2,3]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 4], [0, 5], [0, 4, 5]], [0, 4, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 4], [0, 5], [0, 4, 5]], [0, 4, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177024,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,4,5],[1,2,3]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,4,5],[1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0025","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-connected-subsets；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 4], [0, 1, 4], [0, 5], [0, 1, 5], [0, 4, 5], [0, 1, 4, 5]], [0, 1, 4, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation_zh":"Wave6第十一層核對=(0, [[0], [0, 1], [0, 4], [0, 1, 4], [0, 5], [0, 1, 5], [0, 4, 5], [0, 1, 4, 5]], [0, 1, 4, 5], 是, union_of_connected_subsets_sharing_basepoint_is_connected)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-connected-subsets","seed":202611177025,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-connected-subsets","semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0026","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1], [[0], [0, 1]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 1], [[0], [0, 1]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177026,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0027","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 2], [[0], [0, 2]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 2], [[0], [0, 2]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177027,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0028","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 2], [[0], [0, 1], [0, 2], [0, 1, 2]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 1, 2], [[0], [0, 1], [0, 2], [0, 1, 2]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177028,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0029","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,3],[1,2,4,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 3], [[0], [0, 3]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 3], [[0], [0, 3]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177029,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,3],[1,2,4,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3],[1,2,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0030","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 3], [[0], [0, 1], [0, 3], [0, 1, 3]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 1, 3], [[0], [0, 1], [0, 3], [0, 1, 3]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177030,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0031","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 2, 3], [[0], [0, 2], [0, 3], [0, 2, 3]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 2, 3], [[0], [0, 2], [0, 3], [0, 2, 3]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177031,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0032","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 1, 2, 3], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177032,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0033","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,4],[1,2,3,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 4], [[0], [0, 4]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 4], [[0], [0, 4]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177033,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,4],[1,2,3,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,4],[1,2,3,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0034","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 4], [[0], [0, 1], [0, 4], [0, 1, 4]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 1, 4], [[0], [0, 1], [0, 4], [0, 1, 4]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177034,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0035","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 2, 4], [[0], [0, 2], [0, 4], [0, 2, 4]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 2, 4], [[0], [0, 2], [0, 4], [0, 2, 4]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177035,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0036","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 2, 4], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 1, 2, 4], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177036,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0037","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,3,4],[1,2,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 3, 4], [[0], [0, 3], [0, 4], [0, 3, 4]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 3, 4], [[0], [0, 3], [0, 4], [0, 3, 4]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177037,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,3,4],[1,2,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3,4],[1,2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0038","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 3, 4], [[0], [0, 1], [0, 3], [0, 1, 3], [0, 4], [0, 1, 4], [0, 3, 4], [0, 1, 3, 4]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 1, 3, 4], [[0], [0, 1], [0, 3], [0, 1, 3], [0, 4], [0, 1, 4], [0, 3, 4], [0, 1, 3, 4]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177038,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0039","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 2, 3, 4], [[0], [0, 2], [0, 3], [0, 2, 3], [0, 4], [0, 2, 4], [0, 3, 4], [0, 2, 3, 4]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 2, 3, 4], [[0], [0, 2], [0, 3], [0, 2, 3], [0, 4], [0, 2, 4], [0, 3, 4], [0, 2, 3, 4]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177039,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0040","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3, 4], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4], [0, 3, 4], [0, 1, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 1, 2, 3, 4], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4], [0, 3, 4], [0, 1, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177040,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0041","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,5],[1,2,3,4]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 5], [[0], [0, 5]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 5], [[0], [0, 5]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177041,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,5],[1,2,3,4]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,5],[1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0042","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 5], [[0], [0, 1], [0, 5], [0, 1, 5]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 1, 5], [[0], [0, 1], [0, 5], [0, 1, 5]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177042,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0043","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 2, 5], [[0], [0, 2], [0, 5], [0, 2, 5]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 2, 5], [[0], [0, 2], [0, 5], [0, 2, 5]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177043,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0044","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 2, 5], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 1, 2, 5], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177044,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0045","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,3,5],[1,2,4]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 3, 5], [[0], [0, 3], [0, 5], [0, 3, 5]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 3, 5], [[0], [0, 3], [0, 5], [0, 3, 5]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177045,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,3,5],[1,2,4]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3,5],[1,2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0046","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 3, 5], [[0], [0, 1], [0, 3], [0, 1, 3], [0, 5], [0, 1, 5], [0, 3, 5], [0, 1, 3, 5]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 1, 3, 5], [[0], [0, 1], [0, 3], [0, 1, 3], [0, 5], [0, 1, 5], [0, 3, 5], [0, 1, 3, 5]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177046,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0047","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 2, 3, 5], [[0], [0, 2], [0, 3], [0, 2, 3], [0, 5], [0, 2, 5], [0, 3, 5], [0, 2, 3, 5]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 2, 3, 5], [[0], [0, 2], [0, 3], [0, 2, 3], [0, 5], [0, 2, 5], [0, 3, 5], [0, 2, 3, 5]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177047,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0048","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3, 5], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5], [0, 3, 5], [0, 1, 3, 5], [0, 2, 3, 5], [0, 1, 2, 3, 5]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 1, 2, 3, 5], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5], [0, 3, 5], [0, 1, 3, 5], [0, 2, 3, 5], [0, 1, 2, 3, 5]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177048,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0049","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,4,5],[1,2,3]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 4, 5], [[0], [0, 4], [0, 5], [0, 4, 5]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 4, 5], [[0], [0, 4], [0, 5], [0, 4, 5]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177049,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,4,5],[1,2,3]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,4,5],[1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0050","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-maximality；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]],\"basepoint\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [0, 1, 4, 5], [[0], [0, 1], [0, 4], [0, 1, 4], [0, 5], [0, 1, 5], [0, 4, 5], [0, 1, 4, 5]], 是, 是, unique_inclusion_maximal_connected_subset)","explanation_zh":"Wave6第十一層核對=(0, [0, 1, 4, 5], [[0], [0, 1], [0, 4], [0, 1, 4], [0, 5], [0, 1, 5], [0, 4, 5], [0, 1, 4, 5]], 是, 是, unique_inclusion_maximal_connected_subset)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-maximality","seed":202611177050,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]],"basepoint":0,"domain":"Complete_finite_topological_components","operation":"component-maximality","semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0051","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3, 4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 1], [2, 3, 4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177051,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0052","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 2], [1, 3, 4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 2], [1, 3, 4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177052,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0053","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2], [3, 4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 1, 2], [3, 4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177053,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0054","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,3],[1,2,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3], [1, 2, 4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 3], [1, 2, 4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177054,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,3],[1,2,4,5]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3],[1,2,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0055","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 3], [2, 4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 1, 3], [2, 4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177055,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0056","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 2, 3], [1, 4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 2, 3], [1, 4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177056,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0057","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3], [4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 1, 2, 3], [4, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177057,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0058","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,4],[1,2,3,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 4], [1, 2, 3, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 4], [1, 2, 3, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177058,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,4],[1,2,3,5]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,4],[1,2,3,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0059","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 4], [2, 3, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 1, 4], [2, 3, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177059,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0060","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 2, 4], [1, 3, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 2, 4], [1, 3, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177060,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0061","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 4], [3, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 1, 2, 4], [3, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177061,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0062","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,3,4],[1,2,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3, 4], [1, 2, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 3, 4], [1, 2, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177062,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,3,4],[1,2,5]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3,4],[1,2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0063","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 3, 4], [2, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 1, 3, 4], [2, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177063,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0064","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 2, 3, 4], [1, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 2, 3, 4], [1, 5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177064,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0065","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4], [5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 1, 2, 3, 4], [5]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177065,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0066","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,5],[1,2,3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 5], [1, 2, 3, 4]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 5], [1, 2, 3, 4]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177066,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,5],[1,2,3,4]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,5],[1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0067","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 5], [2, 3, 4]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 1, 5], [2, 3, 4]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177067,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0068","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 2, 5], [1, 3, 4]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 2, 5], [1, 3, 4]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177068,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0069","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 5], [3, 4]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 1, 2, 5], [3, 4]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177069,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0070","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,3,5],[1,2,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3, 5], [1, 2, 4]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 3, 5], [1, 2, 4]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177070,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,3,5],[1,2,4]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3,5],[1,2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0071","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 3, 5], [2, 4]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 1, 3, 5], [2, 4]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177071,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0072","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 2, 3, 5], [1, 4]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 2, 3, 5], [1, 4]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177072,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0073","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 5], [4]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 1, 2, 3, 5], [4]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177073,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0074","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,4,5],[1,2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 4, 5], [1, 2, 3]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 4, 5], [1, 2, 3]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177074,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,4,5],[1,2,3]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,4,5],[1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0075","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-partition；資料={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 4, 5], [2, 3]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)","explanation_zh":"Wave6第十一層核對=([[0, 1, 4, 5], [2, 3]], 是, 是, [是, 是], [是, 是], 是, finite_components_partition_and_are_clopen_finite_specific)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-partition","seed":202611177075,"parameters":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]],"domain":"Complete_finite_topological_components","operation":"component-partition","semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0076","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"finite_clopen\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_clopen, [[0, 1], [2, 3, 4, 5]], [是, 是], [是, 是], 是, finite_number_of_closed_components_makes_each_open)","explanation_zh":"Wave6第十一層核對=(finite_clopen, [[0, 1], [2, 3, 4, 5]], [是, 是], [是, 是], 是, finite_number_of_closed_components_makes_each_open)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177076,"parameters":{"mode":"finite_clopen","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"finite_clopen\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0077","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"finite_no_nonopen_witness\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_no_nonopen_witness, [[0, 2], [1, 3, 4, 5]], 是, 否, finite_models_cannot_witness_a_general_nonopen_component)","explanation_zh":"Wave6第十一層核對=(finite_no_nonopen_witness, [[0, 2], [1, 3, 4, 5]], 是, 否, finite_models_cannot_witness_a_general_nonopen_component)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177077,"parameters":{"mode":"finite_no_nonopen_witness","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"finite_no_nonopen_witness\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0078","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"graph_confusion\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]],\"graph_edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(graph_confusion, [[0, 1, 2], [3, 4, 5]], [[0, 1, 2, 3, 4, 5]], 是, 否, same_carrier_different_structures_give_different_components)","explanation_zh":"Wave6第十一層核對=(graph_confusion, [[0, 1, 2], [3, 4, 5]], [[0, 1, 2, 3, 4, 5]], 是, 否, same_carrier_different_structures_give_different_components)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177078,"parameters":{"mode":"graph_confusion","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]],"graph_edges":[[0,1],[1,2],[2,3],[3,4],[4,5]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"graph_confusion\",model:{graph_edges:[[0,1],[1,2],[2,3],[3,4],[4,5]],points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0079","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"cardinality_not_definition\",\"model\":{\"points\":[0,1,2],\"topology\":[[],[0,1,2],[0],[1,2]],\"basepoint\":0}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(cardinality_not_definition, 0, [0], [1, 2], 1, 2, 否, component_is_inclusion_maximal_not_global_maximum_cardinality)","explanation_zh":"Wave6第十一層核對=(cardinality_not_definition, 0, [0], [1, 2], 1, 2, 否, component_is_inclusion_maximal_not_global_maximum_cardinality)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177079,"parameters":{"mode":"cardinality_not_definition","model":{"points":[0,1,2],"topology":[[],[0,1,2],[0],[1,2]],"basepoint":0},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"cardinality_not_definition\",model:{basepoint:0,points:[0,1,2],topology:[[],[0,1,2],[0],[1,2]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0080","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"finite_clopen\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_clopen, [[0, 1, 3], [2, 4, 5]], [是, 是], [是, 是], 是, finite_number_of_closed_components_makes_each_open)","explanation_zh":"Wave6第十一層核對=(finite_clopen, [[0, 1, 3], [2, 4, 5]], [是, 是], [是, 是], 是, finite_number_of_closed_components_makes_each_open)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177080,"parameters":{"mode":"finite_clopen","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"finite_clopen\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0081","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"finite_no_nonopen_witness\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_no_nonopen_witness, [[0, 2, 3], [1, 4, 5]], 是, 否, finite_models_cannot_witness_a_general_nonopen_component)","explanation_zh":"Wave6第十一層核對=(finite_no_nonopen_witness, [[0, 2, 3], [1, 4, 5]], 是, 否, finite_models_cannot_witness_a_general_nonopen_component)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177081,"parameters":{"mode":"finite_no_nonopen_witness","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"finite_no_nonopen_witness\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0082","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"graph_confusion\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]],\"graph_edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(graph_confusion, [[0, 1, 2, 3], [4, 5]], [[0, 1, 2, 3, 4, 5]], 是, 否, same_carrier_different_structures_give_different_components)","explanation_zh":"Wave6第十一層核對=(graph_confusion, [[0, 1, 2, 3], [4, 5]], [[0, 1, 2, 3, 4, 5]], 是, 否, same_carrier_different_structures_give_different_components)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177082,"parameters":{"mode":"graph_confusion","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]],"graph_edges":[[0,1],[1,2],[2,3],[3,4],[4,5]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"graph_confusion\",model:{graph_edges:[[0,1],[1,2],[2,3],[3,4],[4,5]],points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0083","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"cardinality_not_definition\",\"model\":{\"points\":[0,1,2,3],\"topology\":[[],[0,1,2,3],[0],[1,2,3]],\"basepoint\":0}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(cardinality_not_definition, 0, [0], [1, 2, 3], 1, 3, 否, component_is_inclusion_maximal_not_global_maximum_cardinality)","explanation_zh":"Wave6第十一層核對=(cardinality_not_definition, 0, [0], [1, 2, 3], 1, 3, 否, component_is_inclusion_maximal_not_global_maximum_cardinality)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177083,"parameters":{"mode":"cardinality_not_definition","model":{"points":[0,1,2,3],"topology":[[],[0,1,2,3],[0],[1,2,3]],"basepoint":0},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"cardinality_not_definition\",model:{basepoint:0,points:[0,1,2,3],topology:[[],[0,1,2,3],[0],[1,2,3]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0084","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"finite_clopen\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_clopen, [[0, 1, 4], [2, 3, 5]], [是, 是], [是, 是], 是, finite_number_of_closed_components_makes_each_open)","explanation_zh":"Wave6第十一層核對=(finite_clopen, [[0, 1, 4], [2, 3, 5]], [是, 是], [是, 是], 是, finite_number_of_closed_components_makes_each_open)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177084,"parameters":{"mode":"finite_clopen","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"finite_clopen\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0085","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"finite_no_nonopen_witness\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_no_nonopen_witness, [[0, 2, 4], [1, 3, 5]], 是, 否, finite_models_cannot_witness_a_general_nonopen_component)","explanation_zh":"Wave6第十一層核對=(finite_no_nonopen_witness, [[0, 2, 4], [1, 3, 5]], 是, 否, finite_models_cannot_witness_a_general_nonopen_component)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177085,"parameters":{"mode":"finite_no_nonopen_witness","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"finite_no_nonopen_witness\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0086","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"graph_confusion\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]],\"graph_edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(graph_confusion, [[0, 1, 2, 4], [3, 5]], [[0, 1, 2, 3, 4, 5]], 是, 否, same_carrier_different_structures_give_different_components)","explanation_zh":"Wave6第十一層核對=(graph_confusion, [[0, 1, 2, 4], [3, 5]], [[0, 1, 2, 3, 4, 5]], 是, 否, same_carrier_different_structures_give_different_components)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177086,"parameters":{"mode":"graph_confusion","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]],"graph_edges":[[0,1],[1,2],[2,3],[3,4],[4,5]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"graph_confusion\",model:{graph_edges:[[0,1],[1,2],[2,3],[3,4],[4,5]],points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0087","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"cardinality_not_definition\",\"model\":{\"points\":[0,1,2,3,4],\"topology\":[[],[0,1,2,3,4],[0],[1,2,3,4]],\"basepoint\":0}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(cardinality_not_definition, 0, [0], [1, 2, 3, 4], 1, 4, 否, component_is_inclusion_maximal_not_global_maximum_cardinality)","explanation_zh":"Wave6第十一層核對=(cardinality_not_definition, 0, [0], [1, 2, 3, 4], 1, 4, 否, component_is_inclusion_maximal_not_global_maximum_cardinality)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177087,"parameters":{"mode":"cardinality_not_definition","model":{"points":[0,1,2,3,4],"topology":[[],[0,1,2,3,4],[0],[1,2,3,4]],"basepoint":0},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"cardinality_not_definition\",model:{basepoint:0,points:[0,1,2,3,4],topology:[[],[0,1,2,3,4],[0],[1,2,3,4]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0088","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"finite_clopen\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_clopen, [[0, 1, 3, 4], [2, 5]], [是, 是], [是, 是], 是, finite_number_of_closed_components_makes_each_open)","explanation_zh":"Wave6第十一層核對=(finite_clopen, [[0, 1, 3, 4], [2, 5]], [是, 是], [是, 是], 是, finite_number_of_closed_components_makes_each_open)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177088,"parameters":{"mode":"finite_clopen","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"finite_clopen\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0089","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"finite_no_nonopen_witness\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_no_nonopen_witness, [[0, 2, 3, 4], [1, 5]], 是, 否, finite_models_cannot_witness_a_general_nonopen_component)","explanation_zh":"Wave6第十一層核對=(finite_no_nonopen_witness, [[0, 2, 3, 4], [1, 5]], 是, 否, finite_models_cannot_witness_a_general_nonopen_component)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177089,"parameters":{"mode":"finite_no_nonopen_witness","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"finite_no_nonopen_witness\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0090","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"graph_confusion\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]],\"graph_edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(graph_confusion, [[0, 1, 2, 3, 4], [5]], [[0, 1, 2, 3, 4, 5]], 是, 否, same_carrier_different_structures_give_different_components)","explanation_zh":"Wave6第十一層核對=(graph_confusion, [[0, 1, 2, 3, 4], [5]], [[0, 1, 2, 3, 4, 5]], 是, 否, same_carrier_different_structures_give_different_components)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177090,"parameters":{"mode":"graph_confusion","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]],"graph_edges":[[0,1],[1,2],[2,3],[3,4],[4,5]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"graph_confusion\",model:{graph_edges:[[0,1],[1,2],[2,3],[3,4],[4,5]],points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0091","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"cardinality_not_definition\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0],[1,2,3,4,5]],\"basepoint\":0}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5], 1, 5, 否, component_is_inclusion_maximal_not_global_maximum_cardinality)","explanation_zh":"Wave6第十一層核對=(cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5], 1, 5, 否, component_is_inclusion_maximal_not_global_maximum_cardinality)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177091,"parameters":{"mode":"cardinality_not_definition","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0],[1,2,3,4,5]],"basepoint":0},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"cardinality_not_definition\",model:{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0],[1,2,3,4,5]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0092","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"finite_clopen\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_clopen, [[0, 1, 5], [2, 3, 4]], [是, 是], [是, 是], 是, finite_number_of_closed_components_makes_each_open)","explanation_zh":"Wave6第十一層核對=(finite_clopen, [[0, 1, 5], [2, 3, 4]], [是, 是], [是, 是], 是, finite_number_of_closed_components_makes_each_open)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177092,"parameters":{"mode":"finite_clopen","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"finite_clopen\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0093","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"finite_no_nonopen_witness\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_no_nonopen_witness, [[0, 2, 5], [1, 3, 4]], 是, 否, finite_models_cannot_witness_a_general_nonopen_component)","explanation_zh":"Wave6第十一層核對=(finite_no_nonopen_witness, [[0, 2, 5], [1, 3, 4]], 是, 否, finite_models_cannot_witness_a_general_nonopen_component)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177093,"parameters":{"mode":"finite_no_nonopen_witness","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"finite_no_nonopen_witness\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0094","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"graph_confusion\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]],\"graph_edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(graph_confusion, [[0, 1, 2, 5], [3, 4]], [[0, 1, 2, 3, 4, 5]], 是, 否, same_carrier_different_structures_give_different_components)","explanation_zh":"Wave6第十一層核對=(graph_confusion, [[0, 1, 2, 5], [3, 4]], [[0, 1, 2, 3, 4, 5]], 是, 否, same_carrier_different_structures_give_different_components)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177094,"parameters":{"mode":"graph_confusion","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]],"graph_edges":[[0,1],[1,2],[2,3],[3,4],[4,5]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"graph_confusion\",model:{graph_edges:[[0,1],[1,2],[2,3],[3,4],[4,5]],points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0095","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"cardinality_not_definition\",\"model\":{\"points\":[0,1,2,3,4,5,6],\"topology\":[[],[0,1,2,3,4,5,6],[0],[1,2,3,4,5,6]],\"basepoint\":0}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5, 6], 1, 6, 否, component_is_inclusion_maximal_not_global_maximum_cardinality)","explanation_zh":"Wave6第十一層核對=(cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5, 6], 1, 6, 否, component_is_inclusion_maximal_not_global_maximum_cardinality)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177095,"parameters":{"mode":"cardinality_not_definition","model":{"points":[0,1,2,3,4,5,6],"topology":[[],[0,1,2,3,4,5,6],[0],[1,2,3,4,5,6]],"basepoint":0},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"cardinality_not_definition\",model:{basepoint:0,points:[0,1,2,3,4,5,6],topology:[[],[0,1,2,3,4,5,6],[0],[1,2,3,4,5,6]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0096","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"finite_clopen\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_clopen, [[0, 1, 3, 5], [2, 4]], [是, 是], [是, 是], 是, finite_number_of_closed_components_makes_each_open)","explanation_zh":"Wave6第十一層核對=(finite_clopen, [[0, 1, 3, 5], [2, 4]], [是, 是], [是, 是], 是, finite_number_of_closed_components_makes_each_open)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177096,"parameters":{"mode":"finite_clopen","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"finite_clopen\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0097","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"finite_no_nonopen_witness\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_no_nonopen_witness, [[0, 2, 3, 5], [1, 4]], 是, 否, finite_models_cannot_witness_a_general_nonopen_component)","explanation_zh":"Wave6第十一層核對=(finite_no_nonopen_witness, [[0, 2, 3, 5], [1, 4]], 是, 否, finite_models_cannot_witness_a_general_nonopen_component)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177097,"parameters":{"mode":"finite_no_nonopen_witness","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"finite_no_nonopen_witness\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0098","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"graph_confusion\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]],\"graph_edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(graph_confusion, [[0, 1, 2, 3, 5], [4]], [[0, 1, 2, 3, 4, 5]], 是, 否, same_carrier_different_structures_give_different_components)","explanation_zh":"Wave6第十一層核對=(graph_confusion, [[0, 1, 2, 3, 5], [4]], [[0, 1, 2, 3, 4, 5]], 是, 否, same_carrier_different_structures_give_different_components)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177098,"parameters":{"mode":"graph_confusion","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]],"graph_edges":[[0,1],[1,2],[2,3],[3,4],[4,5]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"graph_confusion\",model:{graph_edges:[[0,1],[1,2],[2,3],[3,4],[4,5]],points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0099","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"cardinality_not_definition\",\"model\":{\"points\":[0,1,2,3,4,5,6,7],\"topology\":[[],[0,1,2,3,4,5,6,7],[0],[1,2,3,4,5,6,7]],\"basepoint\":0}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5, 6, 7], 1, 7, 否, component_is_inclusion_maximal_not_global_maximum_cardinality)","explanation_zh":"Wave6第十一層核對=(cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5, 6, 7], 1, 7, 否, component_is_inclusion_maximal_not_global_maximum_cardinality)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177099,"parameters":{"mode":"cardinality_not_definition","model":{"points":[0,1,2,3,4,5,6,7],"topology":[[],[0,1,2,3,4,5,6,7],[0],[1,2,3,4,5,6,7]],"basepoint":0},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"cardinality_not_definition\",model:{basepoint:0,points:[0,1,2,3,4,5,6,7],topology:[[],[0,1,2,3,4,5,6,7],[0],[1,2,3,4,5,6,7]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r07-topological-connected-component-0100","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","language":"zh-Hant","source_mko_ids":["mko-topological-connected-component"],"title_zh":"拓撲連通分支","stem_zh":"subspace connectedness、包含關係maximality與完整component partition責任。操作=component-boundary；資料={\"mode\":\"finite_clopen\",\"model\":{\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_clopen, [[0, 1, 4, 5], [2, 3]], [是, 是], [是, 是], 是, finite_number_of_closed_components_makes_each_open)","explanation_zh":"Wave6第十一層核對=(finite_clopen, [[0, 1, 4, 5], [2, 3]], [是, 是], [是, 是], 是, finite_number_of_closed_components_makes_each_open)；依完整明示資料獨立重算；finite topology以全部subsets與opens枚舉並核closedness；finite-space clopen現象不得外推；graph components與最大基數connected subsets另有定義。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["拓撲連通分支"]},"variant":{"family_id":"qf-mod48-component-boundary","seed":202611177100,"parameters":{"mode":"finite_clopen","model":{"points":[0,1,2,3,4,5],"topology":[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]]},"domain":"Complete_finite_topological_components","operation":"component-boundary","semantic_template":"component-boundary|{mode:\"finite_clopen\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}