{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r05-homeomorphism-0001","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"map\":[0,1,2,3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055001,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"map":[0,1,2,3,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,1,2,3,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0002","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"map\":[0,1,2,4,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055002,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"map":[0,1,2,4,3],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,1,2,4,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0003","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2,3,4],[0,1,3],[0,2,4],[1,2,3,4],[1,3],[2,4]],\"map\":[0,1,3,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055003,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2,3,4],[0,1,3],[0,2,4],[1,2,3,4],[1,3],[2,4]],"map":[0,1,3,2,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,1,3,2,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,3],[0,2,4],[0],[1,2,3,4],[1,3],[2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0004","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2,3,4],[0,1,3],[0,2,4],[1,2,3,4],[1,3],[2,4]],\"map\":[0,1,3,4,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055004,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2,3,4],[0,1,3],[0,2,4],[1,2,3,4],[1,3],[2,4]],"map":[0,1,3,4,2],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,1,3,4,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,3],[0,2,4],[0],[1,2,3,4],[1,3],[2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0005","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2,3,4],[0,1,4],[0,2,3],[1,2,3,4],[1,4],[2,3]],\"map\":[0,1,4,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055005,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2,3,4],[0,1,4],[0,2,3],[1,2,3,4],[1,4],[2,3]],"map":[0,1,4,2,3],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,1,4,2,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,4],[0,2,3],[0],[1,2,3,4],[1,4],[2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0006","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2,3,4],[0,1,4],[0,2,3],[1,2,3,4],[1,4],[2,3]],\"map\":[0,1,4,3,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055006,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2,3,4],[0,1,4],[0,2,3],[1,2,3,4],[1,4],[2,3]],"map":[0,1,4,3,2],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,1,4,3,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,4],[0,2,3],[0],[1,2,3,4],[1,4],[2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0007","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"map\":[0,2,1,3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055007,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"map":[0,2,1,3,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,2,1,3,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0008","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"map\":[0,2,1,4,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055008,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"map":[0,2,1,4,3],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,2,1,4,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0009","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2,3,4],[0,1,4],[0,2,3],[1,2,3,4],[1,4],[2,3]],\"map\":[0,2,3,1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055009,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2,3,4],[0,1,4],[0,2,3],[1,2,3,4],[1,4],[2,3]],"map":[0,2,3,1,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,2,3,1,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,4],[0,2,3],[0],[1,2,3,4],[1,4],[2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0010","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2,3,4],[0,1,4],[0,2,3],[1,2,3,4],[1,4],[2,3]],\"map\":[0,2,3,4,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055010,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2,3,4],[0,1,4],[0,2,3],[1,2,3,4],[1,4],[2,3]],"map":[0,2,3,4,1],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,2,3,4,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,4],[0,2,3],[0],[1,2,3,4],[1,4],[2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0011","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2,3,4],[0,1,3],[0,2,4],[1,2,3,4],[1,3],[2,4]],\"map\":[0,2,4,1,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055011,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2,3,4],[0,1,3],[0,2,4],[1,2,3,4],[1,3],[2,4]],"map":[0,2,4,1,3],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,2,4,1,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,3],[0,2,4],[0],[1,2,3,4],[1,3],[2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0012","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2,3,4],[0,1,3],[0,2,4],[1,2,3,4],[1,3],[2,4]],\"map\":[0,2,4,3,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055012,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2,3,4],[0,1,3],[0,2,4],[1,2,3,4],[1,3],[2,4]],"map":[0,2,4,3,1],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,2,4,3,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,3],[0,2,4],[0],[1,2,3,4],[1,3],[2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0013","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2,3,4],[0,1,3],[0,2,4],[1,2,3,4],[1,3],[2,4]],\"map\":[0,3,1,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055013,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2,3,4],[0,1,3],[0,2,4],[1,2,3,4],[1,3],[2,4]],"map":[0,3,1,2,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,3,1,2,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,3],[0,2,4],[0],[1,2,3,4],[1,3],[2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0014","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2,3,4],[0,1,3],[0,2,4],[1,2,3,4],[1,3],[2,4]],\"map\":[0,3,1,4,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055014,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2,3,4],[0,1,3],[0,2,4],[1,2,3,4],[1,3],[2,4]],"map":[0,3,1,4,2],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,3,1,4,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,3],[0,2,4],[0],[1,2,3,4],[1,3],[2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0015","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2,3,4],[0,1,4],[0,2,3],[1,2,3,4],[1,4],[2,3]],\"map\":[0,3,2,1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055015,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2,3,4],[0,1,4],[0,2,3],[1,2,3,4],[1,4],[2,3]],"map":[0,3,2,1,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,3,2,1,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,4],[0,2,3],[0],[1,2,3,4],[1,4],[2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0016","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2,3,4],[0,1,4],[0,2,3],[1,2,3,4],[1,4],[2,3]],\"map\":[0,3,2,4,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055016,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2,3,4],[0,1,4],[0,2,3],[1,2,3,4],[1,4],[2,3]],"map":[0,3,2,4,1],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,3,2,4,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,4],[0,2,3],[0],[1,2,3,4],[1,4],[2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0017","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"map\":[0,3,4,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055017,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"map":[0,3,4,1,2],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,3,4,1,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0018","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"map\":[0,3,4,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055018,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"map":[0,3,4,2,1],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,3,4,2,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0019","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2,3,4],[0,1,4],[0,2,3],[1,2,3,4],[1,4],[2,3]],\"map\":[0,4,1,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055019,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2,3,4],[0,1,4],[0,2,3],[1,2,3,4],[1,4],[2,3]],"map":[0,4,1,2,3],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,4,1,2,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,4],[0,2,3],[0],[1,2,3,4],[1,4],[2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0020","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2,3,4],[0,1,4],[0,2,3],[1,2,3,4],[1,4],[2,3]],\"map\":[0,4,1,3,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055020,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2,3,4],[0,1,4],[0,2,3],[1,2,3,4],[1,4],[2,3]],"map":[0,4,1,3,2],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,4,1,3,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,4],[0,2,3],[0],[1,2,3,4],[1,4],[2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0021","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2,3,4],[0,1,3],[0,2,4],[1,2,3,4],[1,3],[2,4]],\"map\":[0,4,2,1,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055021,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2,3,4],[0,1,3],[0,2,4],[1,2,3,4],[1,3],[2,4]],"map":[0,4,2,1,3],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,4,2,1,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,3],[0,2,4],[0],[1,2,3,4],[1,3],[2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0022","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2,3,4],[0,1,3],[0,2,4],[1,2,3,4],[1,3],[2,4]],\"map\":[0,4,2,3,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055022,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2,3,4],[0,1,3],[0,2,4],[1,2,3,4],[1,3],[2,4]],"map":[0,4,2,3,1],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,4,2,3,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,3],[0,2,4],[0],[1,2,3,4],[1,3],[2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0023","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"map\":[0,4,3,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055023,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"map":[0,4,3,1,2],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,4,3,1,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0024","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"map\":[0,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055024,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"map":[0,4,3,2,1],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,4,3,2,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0025","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-topology-audit；資料={\"n\":5,\"source_topology\":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],\"target_topology\":[[],[0,1,2],[0,1,2,3,4],[0,2],[0,2,3,4],[1],[1,3,4],[3,4]],\"map\":[1,0,2,3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 是, 是, complete_open_set_preimage_audit)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055025,"parameters":{"n":5,"source_topology":[[],[0],[0,1,2],[0,1,2,3,4],[0,3,4],[1,2],[1,2,3,4],[3,4]],"target_topology":[[],[0,1,2],[0,1,2,3,4],[0,2],[0,2,3,4],[1],[1,3,4],[3,4]],"map":[1,0,2,3,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-topology-audit","semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[1,0,2,3,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,2,3,4],[0,2],[1,3,4],[1],[3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0026","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,1,2,3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055026,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,1,2,3,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,1,2,3,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0027","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,1,2,4,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055027,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,1,2,4,3],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,1,2,4,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0028","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,1,3,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055028,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,1,3,2,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,1,3,2,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0029","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,1,3,4,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055029,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,1,3,4,2],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,1,3,4,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0030","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,1,4,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055030,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,1,4,2,3],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,1,4,2,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0031","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,1,4,3,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055031,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,1,4,3,2],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,1,4,3,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0032","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,2,1,3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055032,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,2,1,3,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,2,1,3,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0033","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,2,1,4,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055033,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,2,1,4,3],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,2,1,4,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0034","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,2,3,1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055034,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,2,3,1,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,2,3,1,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0035","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,2,3,4,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055035,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,2,3,4,1],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,2,3,4,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0036","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,2,4,1,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055036,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,2,4,1,3],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,2,4,1,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0037","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,2,4,3,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055037,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,2,4,3,1],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,2,4,3,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0038","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,3,1,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055038,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,3,1,2,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,3,1,2,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0039","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,3,1,4,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055039,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,3,1,4,2],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,3,1,4,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0040","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,3,2,1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055040,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,3,2,1,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,3,2,1,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0041","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,3,2,4,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055041,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,3,2,4,1],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,3,2,4,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0042","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,3,4,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055042,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,3,4,1,2],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,3,4,1,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0043","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,3,4,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055043,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,3,4,2,1],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,3,4,2,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0044","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,4,1,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055044,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,4,1,2,3],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,4,1,2,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0045","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,4,1,3,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055045,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,4,1,3,2],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,4,1,3,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0046","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,4,2,1,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055046,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,4,2,1,3],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,4,2,1,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0047","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,4,2,3,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055047,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,4,2,3,1],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,4,2,3,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0048","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,4,3,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055048,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,4,3,1,2],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,4,3,1,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0049","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[0,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055049,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[0,4,3,2,1],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,4,3,2,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0050","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-continuous-bijection-boundary；資料={\"n\":5,\"source_topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],\"target_topology\":[[],[0,1,2,3,4]],\"map\":[1,0,2,3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)","explanation_zh":"Wave5第五層核對=(是, 是, 是, 是, 否, 否, continuous_bijection_inverse_continuity_required)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055050,"parameters":{"n":5,"source_topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4],[0,1,2,4],[0,1,3],[0,1,3,4],[0,1,4],[0,2],[0,2,3],[0,2,3,4],[0,2,4],[0,3],[0,3,4],[0,4],[1],[1,2],[1,2,3],[1,2,3,4],[1,2,4],[1,3],[1,3,4],[1,4],[2],[2,3],[2,3,4],[2,4],[3],[3,4],[4]],"target_topology":[[],[0,1,2,3,4]],"map":[1,0,2,3,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-continuous-bijection-boundary","semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[1,0,2,3,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0051","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[0,1,2,3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 1, 2, 3, 4], 是, 是, 是, finite_discrete_scope_only)","explanation_zh":"Wave5第五層核對=(5, [0, 1, 2, 3, 4], 是, 是, 是, finite_discrete_scope_only)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055051,"parameters":{"n":5,"map":[0,1,2,3,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-discrete-bijection","semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,1,2,3,4],\"n\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0052","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[0,1,2,4,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 1, 2, 4, 3], 是, 是, 是, finite_discrete_scope_only)","explanation_zh":"Wave5第五層核對=(5, [0, 1, 2, 4, 3], 是, 是, 是, finite_discrete_scope_only)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055052,"parameters":{"n":5,"map":[0,1,2,4,3],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-discrete-bijection","semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,1,2,4,3],\"n\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0053","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[0,1,3,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 1, 3, 2, 4], 是, 是, 是, finite_discrete_scope_only)","explanation_zh":"Wave5第五層核對=(5, [0, 1, 3, 2, 4], 是, 是, 是, finite_discrete_scope_only)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055053,"parameters":{"n":5,"map":[0,1,3,2,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-discrete-bijection","semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,1,3,2,4],\"n\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0054","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[0,1,3,4,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 1, 3, 4, 2], 是, 是, 是, finite_discrete_scope_only)","explanation_zh":"Wave5第五層核對=(5, [0, 1, 3, 4, 2], 是, 是, 是, finite_discrete_scope_only)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055054,"parameters":{"n":5,"map":[0,1,3,4,2],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-discrete-bijection","semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,1,3,4,2],\"n\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0055","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[0,1,4,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 1, 4, 2, 3], 是, 是, 是, finite_discrete_scope_only)","explanation_zh":"Wave5第五層核對=(5, [0, 1, 4, 2, 3], 是, 是, 是, finite_discrete_scope_only)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055055,"parameters":{"n":5,"map":[0,1,4,2,3],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-discrete-bijection","semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,1,4,2,3],\"n\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0056","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[0,1,4,3,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 1, 4, 3, 2], 是, 是, 是, finite_discrete_scope_only)","explanation_zh":"Wave5第五層核對=(5, [0, 1, 4, 3, 2], 是, 是, 是, finite_discrete_scope_only)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055056,"parameters":{"n":5,"map":[0,1,4,3,2],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-discrete-bijection","semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,1,4,3,2],\"n\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r05-homeomorphism-0063","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[0,3,1,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 3, 1, 2, 4], 是, 是, 是, finite_discrete_scope_only)","explanation_zh":"Wave5第五層核對=(5, [0, 3, 1, 2, 4], 是, 是, 是, finite_discrete_scope_only)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055063,"parameters":{"n":5,"map":[0,3,1,2,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-discrete-bijection","semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,3,1,2,4],\"n\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r05-homeomorphism-0065","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[0,3,2,1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 3, 2, 1, 4], 是, 是, 是, finite_discrete_scope_only)","explanation_zh":"Wave5第五層核對=(5, [0, 3, 2, 1, 4], 是, 是, 是, finite_discrete_scope_only)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055065,"parameters":{"n":5,"map":[0,3,2,1,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-discrete-bijection","semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,3,2,1,4],\"n\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0066","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[0,3,2,4,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 3, 2, 4, 1], 是, 是, 是, finite_discrete_scope_only)","explanation_zh":"Wave5第五層核對=(5, [0, 3, 2, 4, 1], 是, 是, 是, finite_discrete_scope_only)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055066,"parameters":{"n":5,"map":[0,3,2,4,1],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-discrete-bijection","semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,3,2,4,1],\"n\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r05-homeomorphism-0068","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[0,3,4,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 3, 4, 2, 1], 是, 是, 是, finite_discrete_scope_only)","explanation_zh":"Wave5第五層核對=(5, [0, 3, 4, 2, 1], 是, 是, 是, finite_discrete_scope_only)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055068,"parameters":{"n":5,"map":[0,3,4,2,1],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-discrete-bijection","semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,3,4,2,1],\"n\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0069","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[0,4,1,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 4, 1, 2, 3], 是, 是, 是, finite_discrete_scope_only)","explanation_zh":"Wave5第五層核對=(5, [0, 4, 1, 2, 3], 是, 是, 是, finite_discrete_scope_only)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055069,"parameters":{"n":5,"map":[0,4,1,2,3],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-discrete-bijection","semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,4,1,2,3],\"n\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r05-homeomorphism-0072","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[0,4,2,3,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 4, 2, 3, 1], 是, 是, 是, finite_discrete_scope_only)","explanation_zh":"Wave5第五層核對=(5, [0, 4, 2, 3, 1], 是, 是, 是, finite_discrete_scope_only)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055072,"parameters":{"n":5,"map":[0,4,2,3,1],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-discrete-bijection","semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,4,2,3,1],\"n\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0073","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[0,4,3,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 4, 3, 1, 2], 是, 是, 是, finite_discrete_scope_only)","explanation_zh":"Wave5第五層核對=(5, [0, 4, 3, 1, 2], 是, 是, 是, finite_discrete_scope_only)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055073,"parameters":{"n":5,"map":[0,4,3,1,2],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-discrete-bijection","semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,4,3,1,2],\"n\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0074","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[0,4,3,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [0, 4, 3, 2, 1], 是, 是, 是, finite_discrete_scope_only)","explanation_zh":"Wave5第五層核對=(5, [0, 4, 3, 2, 1], 是, 是, 是, finite_discrete_scope_only)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055074,"parameters":{"n":5,"map":[0,4,3,2,1],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-discrete-bijection","semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,4,3,2,1],\"n\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0075","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-finite-discrete-bijection；資料={\"n\":5,\"map\":[1,0,2,3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [1, 0, 2, 3, 4], 是, 是, 是, finite_discrete_scope_only)","explanation_zh":"Wave5第五層核對=(5, [1, 0, 2, 3, 4], 是, 是, 是, finite_discrete_scope_only)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055075,"parameters":{"n":5,"map":[1,0,2,3,4],"domain":"Complete_finite_topological_spaces","operation":"homeo-finite-discrete-bijection","semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[1,0,2,3,4],\"n\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0076","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,1,2,3,4],\"source_scale\":1,\"target_scale\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 1, 2, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 1, 2, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055076,"parameters":{"n":5,"map":[0,1,2,3,4],"source_scale":1,"target_scale":2,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,1,2,3,4],\"n\":5,\"source_scale\":1,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0077","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,1,2,4,3],\"source_scale\":2,\"target_scale\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 2, 3, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 2, 3, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055077,"parameters":{"n":5,"map":[0,1,2,4,3],"source_scale":2,"target_scale":3,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,1,2,4,3],\"n\":5,\"source_scale\":2,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0078","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,1,3,2,4],\"source_scale\":3,\"target_scale\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 3, 4, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 3, 4, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055078,"parameters":{"n":5,"map":[0,1,3,2,4],"source_scale":3,"target_scale":4,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,1,3,2,4],\"n\":5,\"source_scale\":3,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0079","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,1,3,4,2],\"source_scale\":4,\"target_scale\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 4, 5, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 4, 5, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055079,"parameters":{"n":5,"map":[0,1,3,4,2],"source_scale":4,"target_scale":5,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,1,3,4,2],\"n\":5,\"source_scale\":4,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0080","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,1,4,2,3],\"source_scale\":5,\"target_scale\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 5, 6, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 5, 6, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055080,"parameters":{"n":5,"map":[0,1,4,2,3],"source_scale":5,"target_scale":6,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,1,4,2,3],\"n\":5,\"source_scale\":5,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0081","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,1,4,3,2],\"source_scale\":6,\"target_scale\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 6, 7, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 6, 7, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055081,"parameters":{"n":5,"map":[0,1,4,3,2],"source_scale":6,"target_scale":7,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,1,4,3,2],\"n\":5,\"source_scale\":6,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0082","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,2,1,3,4],\"source_scale\":7,\"target_scale\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 7, 8, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 7, 8, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055082,"parameters":{"n":5,"map":[0,2,1,3,4],"source_scale":7,"target_scale":8,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,2,1,3,4],\"n\":5,\"source_scale\":7,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0083","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,2,1,4,3],\"source_scale\":8,\"target_scale\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 8, 9, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 8, 9, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055083,"parameters":{"n":5,"map":[0,2,1,4,3],"source_scale":8,"target_scale":9,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,2,1,4,3],\"n\":5,\"source_scale\":8,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0084","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,2,3,1,4],\"source_scale\":9,\"target_scale\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 9, 10, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 9, 10, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055084,"parameters":{"n":5,"map":[0,2,3,1,4],"source_scale":9,"target_scale":10,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,2,3,1,4],\"n\":5,\"source_scale\":9,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0085","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,2,3,4,1],\"source_scale\":10,\"target_scale\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 10, 11, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 10, 11, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055085,"parameters":{"n":5,"map":[0,2,3,4,1],"source_scale":10,"target_scale":11,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,2,3,4,1],\"n\":5,\"source_scale\":10,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0086","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,2,4,1,3],\"source_scale\":11,\"target_scale\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 11, 12, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 11, 12, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055086,"parameters":{"n":5,"map":[0,2,4,1,3],"source_scale":11,"target_scale":12,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,2,4,1,3],\"n\":5,\"source_scale\":11,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0087","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,2,4,3,1],\"source_scale\":12,\"target_scale\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 12, 13, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 12, 13, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055087,"parameters":{"n":5,"map":[0,2,4,3,1],"source_scale":12,"target_scale":13,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,2,4,3,1],\"n\":5,\"source_scale\":12,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0088","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,3,1,2,4],\"source_scale\":13,\"target_scale\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 13, 14, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 13, 14, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055088,"parameters":{"n":5,"map":[0,3,1,2,4],"source_scale":13,"target_scale":14,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,3,1,2,4],\"n\":5,\"source_scale\":13,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0089","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,3,1,4,2],\"source_scale\":14,\"target_scale\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 14, 15, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 14, 15, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055089,"parameters":{"n":5,"map":[0,3,1,4,2],"source_scale":14,"target_scale":15,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,3,1,4,2],\"n\":5,\"source_scale\":14,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0090","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,3,2,1,4],\"source_scale\":15,\"target_scale\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 15, 16, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 15, 16, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055090,"parameters":{"n":5,"map":[0,3,2,1,4],"source_scale":15,"target_scale":16,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,3,2,1,4],\"n\":5,\"source_scale\":15,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0091","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,3,2,4,1],\"source_scale\":16,\"target_scale\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 16, 17, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 16, 17, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055091,"parameters":{"n":5,"map":[0,3,2,4,1],"source_scale":16,"target_scale":17,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,3,2,4,1],\"n\":5,\"source_scale\":16,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0092","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,3,4,1,2],\"source_scale\":17,\"target_scale\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 17, 18, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 17, 18, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055092,"parameters":{"n":5,"map":[0,3,4,1,2],"source_scale":17,"target_scale":18,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,3,4,1,2],\"n\":5,\"source_scale\":17,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0093","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,3,4,2,1],\"source_scale\":18,\"target_scale\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 18, 19, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 18, 19, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055093,"parameters":{"n":5,"map":[0,3,4,2,1],"source_scale":18,"target_scale":19,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,3,4,2,1],\"n\":5,\"source_scale\":18,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0094","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,4,1,2,3],\"source_scale\":19,\"target_scale\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 19, 20, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 19, 20, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055094,"parameters":{"n":5,"map":[0,4,1,2,3],"source_scale":19,"target_scale":20,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,4,1,2,3],\"n\":5,\"source_scale\":19,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0095","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,4,1,3,2],\"source_scale\":20,\"target_scale\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 20, 21, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 20, 21, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055095,"parameters":{"n":5,"map":[0,4,1,3,2],"source_scale":20,"target_scale":21,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,4,1,3,2],\"n\":5,\"source_scale\":20,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0096","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,4,2,1,3],\"source_scale\":21,\"target_scale\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 21, 22, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 21, 22, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055096,"parameters":{"n":5,"map":[0,4,2,1,3],"source_scale":21,"target_scale":22,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,4,2,1,3],\"n\":5,\"source_scale\":21,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0097","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,4,2,3,1],\"source_scale\":22,\"target_scale\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 22, 23, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 22, 23, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055097,"parameters":{"n":5,"map":[0,4,2,3,1],"source_scale":22,"target_scale":23,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,4,2,3,1],\"n\":5,\"source_scale\":22,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0098","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,4,3,1,2],\"source_scale\":23,\"target_scale\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 23, 24, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 23, 24, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055098,"parameters":{"n":5,"map":[0,4,3,1,2],"source_scale":23,"target_scale":24,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,4,3,1,2],\"n\":5,\"source_scale\":23,\"target_scale\":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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0099","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[0,4,3,2,1],\"source_scale\":24,\"target_scale\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 24, 25, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 24, 25, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055099,"parameters":{"n":5,"map":[0,4,3,2,1],"source_scale":24,"target_scale":25,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,4,3,2,1],\"n\":5,\"source_scale\":24,\"target_scale\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、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-20261105-r05-homeomorphism-0100","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","language":"zh-Hant","source_mko_ids":["mko-homeomorphism"],"title_zh":"同胚","stem_zh":"完整有限來源與目標拓撲、雙射及全部開集逆像。操作=homeo-metric-nonimplication；資料={\"n\":5,\"map\":[1,0,2,3,4],\"source_scale\":25,\"target_scale\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 否, 25, 26, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation_zh":"Wave5第五層核對=(5, 是, 否, 25, 26, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)；依完整明示資料獨立重算；homeomorphism同時要求f連續及逆映射連續；一般continuous bijection不足；有限離散雙射結論不得外推，也不推出等距、可微或幾何全等。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["同胚"]},"variant":{"family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055100,"parameters":{"n":5,"map":[1,0,2,3,4],"source_scale":25,"target_scale":26,"domain":"Complete_finite_topological_spaces","operation":"homeo-metric-nonimplication","semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[1,0,2,3,4],\"n\":5,\"source_scale\":25,\"target_scale\":26}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}