{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0001","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,0,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [0, 0, 0], [0], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [0, 0, 0], [0], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103001,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,0,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0002","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,0,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [1, 0, 0], [0, 1], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [1, 0, 0], [0, 1], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103002,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,0,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[1,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0003","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,0,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [2, 0, 0], [0, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [2, 0, 0], [0, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103003,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,0,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[2,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0004","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,1,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [0, 1, 0], [0, 1], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [0, 1, 0], [0, 1], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103004,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,1,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0005","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,1,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [1, 1, 0], [0, 1], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [1, 1, 0], [0, 1], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103005,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,1,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[1,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0006","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,1,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [2, 1, 0], [0, 1, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [2, 1, 0], [0, 1, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103006,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,1,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[2,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0007","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,2,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [0, 2, 0], [0, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [0, 2, 0], [0, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103007,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,2,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0008","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,2,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [1, 2, 0], [0, 1, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [1, 2, 0], [0, 1, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103008,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,2,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[1,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0009","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,2,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [2, 2, 0], [0, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [2, 2, 0], [0, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103009,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,2,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[2,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0010","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,0,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [0, 0, 1], [0, 1], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [0, 0, 1], [0, 1], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103010,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,0,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0011","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,0,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [1, 0, 1], [0, 1], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [1, 0, 1], [0, 1], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103011,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,0,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[1,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0012","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,0,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [2, 0, 1], [0, 1, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [2, 0, 1], [0, 1, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103012,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,0,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[2,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0013","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,1,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [0, 1, 1], [0, 1], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [0, 1, 1], [0, 1], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103013,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,1,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0014","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,1,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [1, 1, 1], [1], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [1, 1, 1], [1], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103014,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,1,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[1,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0015","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,1,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [2, 1, 1], [1, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [2, 1, 1], [1, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103015,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,1,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[2,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0016","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,2,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [0, 2, 1], [0, 1, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [0, 2, 1], [0, 1, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103016,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,2,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0017","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,2,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [1, 2, 1], [1, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [1, 2, 1], [1, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103017,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,2,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[1,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0018","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,2,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [2, 2, 1], [1, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [2, 2, 1], [1, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103018,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,2,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[2,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0019","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,0,2],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [0, 0, 2], [0, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [0, 0, 2], [0, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103019,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,0,2],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0020","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,0,2],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [1, 0, 2], [0, 1, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [1, 0, 2], [0, 1, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103020,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,0,2],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[1,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0021","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,0,2],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [2, 0, 2], [0, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [2, 0, 2], [0, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103021,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,0,2],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[2,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0022","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,1,2],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [0, 1, 2], [0, 1, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [0, 1, 2], [0, 1, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103022,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,1,2],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0023","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,1,2],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [1, 1, 2], [1, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [1, 1, 2], [1, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103023,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,1,2],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[1,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0024","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,1,2],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [2, 1, 2], [1, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [2, 1, 2], [1, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103024,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,1,2],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[2,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0025","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-image-set；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,2,2],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [0, 2, 2], [0, 2], 是, finite_compact_source_subset_exact_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [0, 2, 2], [0, 2], 是, finite_compact_source_subset_exact_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-image-set","seed":202611103025,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,2,2],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-image-set","semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,2,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0026","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,0,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [0, 1, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], []], [[2], []]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [0, 1, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], []], [[2], []]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103026,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,0,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0027","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,0,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [1, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0]], [[2], []]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [1, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0]], [[2], []]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103027,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,0,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[1,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0028","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,0,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [1, 2]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0]], [[2], [0]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [1, 2]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0]], [[2], [0]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103028,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,0,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[2,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0029","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,1,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [0, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [1]], [[2], []]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [0, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [1]], [[2], []]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103029,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,1,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0030","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,1,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [2]], [[1], [0, 1]], [[1, 2], [0, 1]], [[2], []]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [2]], [[1], [0, 1]], [[1, 2], [0, 1]], [[2], []]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103030,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,1,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[1,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0031","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,1,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [2]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [0, 1]], [[2], [0]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [2]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [0, 1]], [[2], [0]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103031,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,1,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[2,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0032","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,2,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [0, 2]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [1]], [[2], [1]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [0, 2]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [1]], [[2], [1]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103032,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,2,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0033","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,2,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [2]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0, 1]], [[2], [1]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [2]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0, 1]], [[2], [1]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103033,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,2,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[1,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0034","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,2,0],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [2]], [[0, 1], [2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0, 1]], [[2], [0, 1]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [2]], [[0, 1], [2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0, 1]], [[2], [0, 1]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103034,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,2,0],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[2,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0035","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,0,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [0, 1]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [2]], [[2], []]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [0, 1]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [2]], [[2], []]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103035,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,0,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0036","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,0,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [1]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1]], [[1], [0, 2]], [[1, 2], [0, 2]], [[2], []]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [1]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1]], [[1], [0, 2]], [[1, 2], [0, 2]], [[2], []]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103036,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,0,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[1,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0037","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,0,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [1]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [0, 2]], [[2], [0]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [1]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [0, 2]], [[2], [0]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103037,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,0,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[2,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0038","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,1,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [0]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0]], [[1], [1, 2]], [[1, 2], [1, 2]], [[2], []]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [0]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0]], [[1], [1, 2]], [[1, 2], [1, 2]], [[2], []]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103038,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,1,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0039","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,1,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], []], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], []], [[1], [0, 1, 2]], [[1, 2], [0, 1, 2]], [[2], []]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], []], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], []], [[1], [0, 1, 2]], [[1, 2], [0, 1, 2]], [[2], []]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103039,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,1,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[1,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0040","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,1,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], []], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0]], [[1], [1, 2]], [[1, 2], [0, 1, 2]], [[2], [0]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], []], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0]], [[1], [1, 2]], [[1, 2], [0, 1, 2]], [[2], [0]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103040,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,1,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[2,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0041","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,2,1],\"subset\":[0,1,2]}。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]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [1, 2]], [[2], [1]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [0]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [1, 2]], [[2], [1]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103041,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,2,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0042","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,2,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], []], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1]], [[1], [0, 2]], [[1, 2], [0, 1, 2]], [[2], [1]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], []], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1]], [[1], [0, 2]], [[1, 2], [0, 1, 2]], [[2], [1]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103042,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,2,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[1,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0043","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,2,1],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], []], [[0, 1], [2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [0, 1, 2]], [[2], [0, 1]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], []], [[0, 1], [2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [0, 1, 2]], [[2], [0, 1]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103043,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,2,1],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[2,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0044","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,0,2],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [0, 1]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [2]], [[2], [2]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [0, 1]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [2]], [[2], [2]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103044,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,0,2],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0045","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,0,2],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [1]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0, 2]], [[2], [2]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [1]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0, 2]], [[2], [2]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103045,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,0,2],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[1,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0046","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,0,2],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [1]], [[0, 1], [1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0, 2]], [[2], [0, 2]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [1]], [[0, 1], [1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0, 2]], [[2], [0, 2]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103046,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,0,2],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[2,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0047","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,1,2],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [0]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [1, 2]], [[2], [2]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [0]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [1, 2]], [[2], [2]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103047,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,1,2],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0048","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,1,2],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], []], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [2]], [[1], [0, 1]], [[1, 2], [0, 1, 2]], [[2], [2]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], []], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [2]], [[1], [0, 1]], [[1, 2], [0, 1, 2]], [[2], [2]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103048,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,1,2],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[1,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0049","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,1,2],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], []], [[0, 1], [1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [0, 1, 2]], [[2], [0, 2]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], []], [[0, 1], [1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [0, 1, 2]], [[2], [0, 2]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103049,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,1,2],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[2,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0050","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-continuity-audit；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,2,2],\"subset\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[], []], [[0], [0]], [[0, 1], [0]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [1, 2]], [[2], [1, 2]]], 是, every_target_open_has_source_open_preimage)","explanation_zh":"Wave6第四層核對=([[[], []], [[0], [0]], [[0, 1], [0]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [1, 2]], [[2], [1, 2]]], 是, every_target_open_has_source_open_preimage)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103050,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,2,2],"subset":[0,1,2],"domain":"Finite_topological_compact_image","operation":"compactimg-continuity-audit","semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,2,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0051","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,0,0],\"subset\":[0,1,2],\"cover\":[[0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0], [[0]], [[0, 1, 2]], [0], [[0]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0], [[0]], [[0, 1, 2]], [0], [[0]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103051,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,0,0],"subset":[0,1,2],"cover":[[0]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0]],\"map\":[0,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0052","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,0,0],\"subset\":[0,1,2],\"cover\":[[0],[1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [[0], [1]], [[1, 2], [0]], [0, 1], [[0], [1]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 1], [[0], [1]], [[1, 2], [0]], [0, 1], [[0], [1]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103052,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,0,0],"subset":[0,1,2],"cover":[[0],[1]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1]],\"map\":[1,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0053","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,0,0],\"subset\":[0,1,2],\"cover\":[[0],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [[0], [2]], [[1, 2], [0]], [0, 1], [[0], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 2], [[0], [2]], [[1, 2], [0]], [0, 1], [[0], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103053,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,0,0],"subset":[0,1,2],"cover":[[0],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[2]],\"map\":[2,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0054","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,1,0],\"subset\":[0,1,2],\"cover\":[[0],[1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [[0], [1]], [[0, 2], [1]], [0, 1], [[0], [1]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 1], [[0], [1]], [[0, 2], [1]], [0, 1], [[0], [1]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103054,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,1,0],"subset":[0,1,2],"cover":[[0],[1]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1]],\"map\":[0,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0055","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,1,0],\"subset\":[0,1,2],\"cover\":[[0],[1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [[0], [1]], [[2], [0, 1]], [0, 1], [[0], [1]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 1], [[0], [1]], [[2], [0, 1]], [0, 1], [[0], [1]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103055,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,1,0],"subset":[0,1,2],"cover":[[0],[1]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1]],\"map\":[1,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0056","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,1,0],\"subset\":[0,1,2],\"cover\":[[0],[1],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0], [1], [2]], [[2], [1], [0]], [0, 1, 2], [[0], [1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [[0], [1], [2]], [[2], [1], [0]], [0, 1, 2], [[0], [1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103056,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,1,0],"subset":[0,1,2],"cover":[[0],[1],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1],[2]],\"map\":[2,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0057","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,2,0],\"subset\":[0,1,2],\"cover\":[[0],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [[0], [2]], [[0, 2], [1]], [0, 1], [[0], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 2], [[0], [2]], [[0, 2], [1]], [0, 1], [[0], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103057,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,2,0],"subset":[0,1,2],"cover":[[0],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[2]],\"map\":[0,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0058","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,2,0],\"subset\":[0,1,2],\"cover\":[[0],[1],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0], [1], [2]], [[2], [0], [1]], [0, 1, 2], [[0], [1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [[0], [1], [2]], [[2], [0], [1]], [0, 1, 2], [[0], [1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103058,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,2,0],"subset":[0,1,2],"cover":[[0],[1],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1],[2]],\"map\":[1,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0059","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,2,0],\"subset\":[0,1,2],\"cover\":[[0],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [[0], [2]], [[2], [0, 1]], [0, 1], [[0], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 2], [[0], [2]], [[2], [0, 1]], [0, 1], [[0], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103059,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,2,0],"subset":[0,1,2],"cover":[[0],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[2]],\"map\":[2,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0060","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,0,1],\"subset\":[0,1,2],\"cover\":[[0],[1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [[0], [1]], [[0, 1], [2]], [0, 1], [[0], [1]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 1], [[0], [1]], [[0, 1], [2]], [0, 1], [[0], [1]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103060,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,0,1],"subset":[0,1,2],"cover":[[0],[1]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1]],\"map\":[0,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0061","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,0,1],\"subset\":[0,1,2],\"cover\":[[0],[1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [[0], [1]], [[1], [0, 2]], [0, 1], [[0], [1]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 1], [[0], [1]], [[1], [0, 2]], [0, 1], [[0], [1]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103061,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,0,1],"subset":[0,1,2],"cover":[[0],[1]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1]],\"map\":[1,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0062","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,0,1],\"subset\":[0,1,2],\"cover\":[[0],[1],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0], [1], [2]], [[1], [2], [0]], [0, 1, 2], [[0], [1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [[0], [1], [2]], [[1], [2], [0]], [0, 1, 2], [[0], [1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103062,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,0,1],"subset":[0,1,2],"cover":[[0],[1],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1],[2]],\"map\":[2,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0063","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,1,1],\"subset\":[0,1,2],\"cover\":[[0],[1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [[0], [1]], [[0], [1, 2]], [0, 1], [[0], [1]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 1], [[0], [1]], [[0], [1, 2]], [0, 1], [[0], [1]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103063,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,1,1],"subset":[0,1,2],"cover":[[0],[1]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1]],\"map\":[0,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0064","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,1,1],\"subset\":[0,1,2],\"cover\":[[1]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1], [[1]], [[0, 1, 2]], [0], [[1]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([1], [[1]], [[0, 1, 2]], [0], [[1]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103064,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,1,1],"subset":[0,1,2],"cover":[[1]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[1]],\"map\":[1,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0065","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,1,1],\"subset\":[0,1,2],\"cover\":[[1],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [[1], [2]], [[1, 2], [0]], [0, 1], [[1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([1, 2], [[1], [2]], [[1, 2], [0]], [0, 1], [[1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103065,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,1,1],"subset":[0,1,2],"cover":[[1],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[1],[2]],\"map\":[2,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0066","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,2,1],\"subset\":[0,1,2],\"cover\":[[0],[1],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0], [1], [2]], [[0], [2], [1]], [0, 1, 2], [[0], [1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [[0], [1], [2]], [[0], [2], [1]], [0, 1, 2], [[0], [1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103066,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,2,1],"subset":[0,1,2],"cover":[[0],[1],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1],[2]],\"map\":[0,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0067","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,2,1],\"subset\":[0,1,2],\"cover\":[[1],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [[1], [2]], [[0, 2], [1]], [0, 1], [[1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([1, 2], [[1], [2]], [[0, 2], [1]], [0, 1], [[1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103067,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,2,1],"subset":[0,1,2],"cover":[[1],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[1],[2]],\"map\":[1,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0068","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,2,1],\"subset\":[0,1,2],\"cover\":[[1],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [[1], [2]], [[2], [0, 1]], [0, 1], [[1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([1, 2], [[1], [2]], [[2], [0, 1]], [0, 1], [[1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103068,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,2,1],"subset":[0,1,2],"cover":[[1],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[1],[2]],\"map\":[2,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0069","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,0,2],\"subset\":[0,1,2],\"cover\":[[0],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [[0], [2]], [[0, 1], [2]], [0, 1], [[0], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 2], [[0], [2]], [[0, 1], [2]], [0, 1], [[0], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103069,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,0,2],"subset":[0,1,2],"cover":[[0],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[2]],\"map\":[0,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0070","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,0,2],\"subset\":[0,1,2],\"cover\":[[0],[1],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0], [1], [2]], [[1], [0], [2]], [0, 1, 2], [[0], [1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [[0], [1], [2]], [[1], [0], [2]], [0, 1, 2], [[0], [1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103070,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,0,2],"subset":[0,1,2],"cover":[[0],[1],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1],[2]],\"map\":[1,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0071","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,0,2],\"subset\":[0,1,2],\"cover\":[[0],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [[0], [2]], [[1], [0, 2]], [0, 1], [[0], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 2], [[0], [2]], [[1], [0, 2]], [0, 1], [[0], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103071,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,0,2],"subset":[0,1,2],"cover":[[0],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[2]],\"map\":[2,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0072","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,1,2],\"subset\":[0,1,2],\"cover\":[[0],[1],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], [[0], [1], [2]], [[0], [1], [2]], [0, 1, 2], [[0], [1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 1, 2], [[0], [1], [2]], [[0], [1], [2]], [0, 1, 2], [[0], [1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103072,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,1,2],"subset":[0,1,2],"cover":[[0],[1],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1],[2]],\"map\":[0,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0073","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[1,1,2],\"subset\":[0,1,2],\"cover\":[[1],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [[1], [2]], [[0, 1], [2]], [0, 1], [[1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([1, 2], [[1], [2]], [[0, 1], [2]], [0, 1], [[1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103073,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[1,1,2],"subset":[0,1,2],"cover":[[1],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[1],[2]],\"map\":[1,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0074","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[2,1,2],\"subset\":[0,1,2],\"cover\":[[1],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [[1], [2]], [[1], [0, 2]], [0, 1], [[1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([1, 2], [[1], [2]], [[1], [0, 2]], [0, 1], [[1], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103074,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[2,1,2],"subset":[0,1,2],"cover":[[1],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[1],[2]],\"map\":[2,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0075","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-cover-pullback；資料={\"source_n\":3,\"target_n\":3,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"target_topology\":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],\"map\":[0,2,2],\"subset\":[0,1,2],\"cover\":[[0],[2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 2], [[0], [2]], [[0], [1, 2]], [0, 1], [[0], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation_zh":"Wave6第四層核對=([0, 2], [[0], [2]], [[0], [1, 2]], [0, 1], [[0], [2]], 是, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103075,"parameters":{"source_n":3,"target_n":3,"source_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"target_topology":[[],[0],[0,1],[0,1,2],[0,2],[1],[1,2],[2]],"map":[0,2,2],"subset":[0,1,2],"cover":[[0],[2]],"domain":"Finite_topological_compact_image","operation":"compactimg-cover-pullback","semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[2]],\"map\":[0,2,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0076","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-2,\"right\":2,\"value\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 2, 0, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-2, 2, 0, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103076,"parameters":{"left":-2,"right":2,"value":0,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-2,\"right\":2,\"value\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0077","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-3,\"right\":3,\"value\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 3, 1, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-3, 3, 1, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103077,"parameters":{"left":-3,"right":3,"value":1,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-3,\"right\":3,\"value\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0078","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-4,\"right\":4,\"value\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, 4, 2, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-4, 4, 2, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103078,"parameters":{"left":-4,"right":4,"value":2,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-4,\"right\":4,\"value\":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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0079","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-5,\"right\":5,\"value\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, 5, 3, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-5, 5, 3, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103079,"parameters":{"left":-5,"right":5,"value":3,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-5,\"right\":5,\"value\":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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0080","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-6,\"right\":6,\"value\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, 6, 4, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-6, 6, 4, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103080,"parameters":{"left":-6,"right":6,"value":4,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-6,\"right\":6,\"value\":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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0081","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-7,\"right\":7,\"value\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, 7, 5, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-7, 7, 5, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103081,"parameters":{"left":-7,"right":7,"value":5,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-7,\"right\":7,\"value\":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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0082","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-8,\"right\":8,\"value\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, 8, 6, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-8, 8, 6, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103082,"parameters":{"left":-8,"right":8,"value":6,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-8,\"right\":8,\"value\":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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0083","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-9,\"right\":9,\"value\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, 9, 7, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-9, 9, 7, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103083,"parameters":{"left":-9,"right":9,"value":7,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-9,\"right\":9,\"value\":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；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0084","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-10,\"right\":10,\"value\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, 10, 8, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-10, 10, 8, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103084,"parameters":{"left":-10,"right":10,"value":8,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-10,\"right\":10,\"value\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0085","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-11,\"right\":11,\"value\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, 11, 9, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-11, 11, 9, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103085,"parameters":{"left":-11,"right":11,"value":9,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-11,\"right\":11,\"value\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0086","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-12,\"right\":12,\"value\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, 12, 10, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-12, 12, 10, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103086,"parameters":{"left":-12,"right":12,"value":10,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-12,\"right\":12,\"value\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0087","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-13,\"right\":13,\"value\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-13, 13, 11, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-13, 13, 11, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103087,"parameters":{"left":-13,"right":13,"value":11,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-13,\"right\":13,\"value\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0088","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-14,\"right\":14,\"value\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-14, 14, 12, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-14, 14, 12, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103088,"parameters":{"left":-14,"right":14,"value":12,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-14,\"right\":14,\"value\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0089","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-15,\"right\":15,\"value\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-15, 15, 13, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-15, 15, 13, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103089,"parameters":{"left":-15,"right":15,"value":13,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-15,\"right\":15,\"value\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0090","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-16,\"right\":16,\"value\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-16, 16, 14, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-16, 16, 14, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103090,"parameters":{"left":-16,"right":16,"value":14,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-16,\"right\":16,\"value\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0091","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-17,\"right\":17,\"value\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-17, 17, 15, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-17, 17, 15, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103091,"parameters":{"left":-17,"right":17,"value":15,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-17,\"right\":17,\"value\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0092","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-18,\"right\":18,\"value\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-18, 18, 16, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-18, 18, 16, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103092,"parameters":{"left":-18,"right":18,"value":16,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-18,\"right\":18,\"value\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0093","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-19,\"right\":19,\"value\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-19, 19, 17, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-19, 19, 17, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103093,"parameters":{"left":-19,"right":19,"value":17,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-19,\"right\":19,\"value\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0094","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-20,\"right\":20,\"value\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-20, 20, 18, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-20, 20, 18, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103094,"parameters":{"left":-20,"right":20,"value":18,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-20,\"right\":20,\"value\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0095","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-21,\"right\":21,\"value\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-21, 21, 19, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-21, 21, 19, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103095,"parameters":{"left":-21,"right":21,"value":19,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-21,\"right\":21,\"value\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0096","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-22,\"right\":22,\"value\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-22, 22, 20, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-22, 22, 20, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103096,"parameters":{"left":-22,"right":22,"value":20,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-22,\"right\":22,\"value\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0097","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-23,\"right\":23,\"value\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-23, 23, 21, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-23, 23, 21, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103097,"parameters":{"left":-23,"right":23,"value":21,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-23,\"right\":23,\"value\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0098","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-24,\"right\":24,\"value\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-24, 24, 22, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-24, 24, 22, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103098,"parameters":{"left":-24,"right":24,"value":22,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-24,\"right\":24,\"value\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0099","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-25,\"right\":25,\"value\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-25, 25, 23, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-25, 25, 23, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103099,"parameters":{"left":-25,"right":25,"value":23,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-25,\"right\":25,\"value\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0100","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","language":"zh-Hant","source_mko_ids":["mko-continuous-image-of-compact"],"title_zh":"緊緻集的連續像","stem_zh":"compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-26,\"right\":26,\"value\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-26, 26, 24, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation_zh":"Wave6第四層核對=(-26, 26, 24, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["緊緻集的連續像"]},"variant":{"family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103100,"parameters":{"left":-26,"right":26,"value":24,"domain":"Finite_topological_compact_image","operation":"compactimg-reverse-implication-boundary","semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-26,\"right\":26,\"value\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}