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{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0027","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[1,0,0,0],\"left_event\":[1],\"right_event\":[2,3]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 0, 0], [1], [2, 3], [1, 2, 3], [0], [], [0], [0], 是)","explanation_zh":"收束波核對=([1, 0, 0, 0], [1], [2, 3], [1, 2, 3], [0], [], [0], [0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255027,"parameters":{"source_n":4,"target_n":4,"mapping":[1,0,0,0],"left_event":[1],"right_event":[2,3],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[1],\"mapping\":[1,0,0,0],\"right_event\":[2,3],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0028","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[2,0,0,0],\"left_event\":[2],\"right_event\":[0,3]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([2, 0, 0, 0], [2], [0, 3], [0, 2, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], 是)","explanation_zh":"收束波核對=([2, 0, 0, 0], [2], [0, 3], [0, 2, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255028,"parameters":{"source_n":4,"target_n":4,"mapping":[2,0,0,0],"left_event":[2],"right_event":[0,3],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[2],\"mapping\":[2,0,0,0],\"right_event\":[0,3],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0029","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[3,0,0,0],\"left_event\":[3],\"right_event\":[0,1]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([3, 0, 0, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], 是)","explanation_zh":"收束波核對=([3, 0, 0, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255029,"parameters":{"source_n":4,"target_n":4,"mapping":[3,0,0,0],"left_event":[3],"right_event":[0,1],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[3],\"mapping\":[3,0,0,0],\"right_event\":[0,1],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0030","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[0,1,0,0],\"left_event\":[0],\"right_event\":[1,2]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 0], [0], [1, 2], [0, 1, 2], [0, 2, 3], [1], [0, 1, 2, 3], [0, 1, 2, 3], 是)","explanation_zh":"收束波核對=([0, 1, 0, 0], [0], [1, 2], [0, 1, 2], [0, 2, 3], [1], [0, 1, 2, 3], [0, 1, 2, 3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255030,"parameters":{"source_n":4,"target_n":4,"mapping":[0,1,0,0],"left_event":[0],"right_event":[1,2],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[0],\"mapping\":[0,1,0,0],\"right_event\":[1,2],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0031","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[1,1,0,0],\"left_event\":[1],\"right_event\":[2,3]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 0, 0], [1], [2, 3], [1, 2, 3], [0, 1], [], [0, 1], [0, 1], 是)","explanation_zh":"收束波核對=([1, 1, 0, 0], [1], [2, 3], [1, 2, 3], [0, 1], [], [0, 1], [0, 1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255031,"parameters":{"source_n":4,"target_n":4,"mapping":[1,1,0,0],"left_event":[1],"right_event":[2,3],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[1],\"mapping\":[1,1,0,0],\"right_event\":[2,3],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0032","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[2,1,0,0],\"left_event\":[2],\"right_event\":[0,3]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0], [2], [0, 3], [0, 2, 3], [0], [2, 3], [0, 2, 3], [0, 2, 3], 是)","explanation_zh":"收束波核對=([2, 1, 0, 0], [2], [0, 3], [0, 2, 3], [0], [2, 3], [0, 2, 3], [0, 2, 3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255032,"parameters":{"source_n":4,"target_n":4,"mapping":[2,1,0,0],"left_event":[2],"right_event":[0,3],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[2],\"mapping\":[2,1,0,0],\"right_event\":[0,3],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0034","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[0,2,0,0],\"left_event\":[0],\"right_event\":[1,2]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 0, 0], [0], [1, 2], [0, 1, 2], [0, 2, 3], [1], [0, 1, 2, 3], [0, 1, 2, 3], 是)","explanation_zh":"收束波核對=([0, 2, 0, 0], [0], [1, 2], [0, 1, 2], [0, 2, 3], [1], [0, 1, 2, 3], [0, 1, 2, 3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255034,"parameters":{"source_n":4,"target_n":4,"mapping":[0,2,0,0],"left_event":[0],"right_event":[1,2],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[0],\"mapping\":[0,2,0,0],\"right_event\":[1,2],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0035","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[1,2,0,0],\"left_event\":[1],\"right_event\":[2,3]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 0, 0], [1], [2, 3], [1, 2, 3], [0], [1], [0, 1], [0, 1], 是)","explanation_zh":"收束波核對=([1, 2, 0, 0], [1], [2, 3], [1, 2, 3], [0], [1], [0, 1], [0, 1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255035,"parameters":{"source_n":4,"target_n":4,"mapping":[1,2,0,0],"left_event":[1],"right_event":[2,3],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[1],\"mapping\":[1,2,0,0],\"right_event\":[2,3],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0036","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[2,2,0,0],\"left_event\":[2],\"right_event\":[0,3]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 0, 0], [2], [0, 3], [0, 2, 3], [0, 1], [2, 3], [0, 1, 2, 3], [0, 1, 2, 3], 是)","explanation_zh":"收束波核對=([2, 2, 0, 0], [2], [0, 3], [0, 2, 3], [0, 1], [2, 3], [0, 1, 2, 3], [0, 1, 2, 3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255036,"parameters":{"source_n":4,"target_n":4,"mapping":[2,2,0,0],"left_event":[2],"right_event":[0,3],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[2],\"mapping\":[2,2,0,0],\"right_event\":[0,3],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0038","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[0,3,0,0],\"left_event\":[0],\"right_event\":[1,2]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([0, 3, 0, 0], [0], [1, 2], [0, 1, 2], [0, 2, 3], [], [0, 2, 3], [0, 2, 3], 是)","explanation_zh":"收束波核對=([0, 3, 0, 0], [0], [1, 2], [0, 1, 2], [0, 2, 3], [], [0, 2, 3], [0, 2, 3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255038,"parameters":{"source_n":4,"target_n":4,"mapping":[0,3,0,0],"left_event":[0],"right_event":[1,2],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[0],\"mapping\":[0,3,0,0],\"right_event\":[1,2],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0039","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[1,3,0,0],\"left_event\":[1],\"right_event\":[2,3]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 0, 0], [1], [2, 3], [1, 2, 3], [0], [1], [0, 1], [0, 1], 是)","explanation_zh":"收束波核對=([1, 3, 0, 0], [1], [2, 3], [1, 2, 3], [0], [1], [0, 1], [0, 1], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255039,"parameters":{"source_n":4,"target_n":4,"mapping":[1,3,0,0],"left_event":[1],"right_event":[2,3],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[1],\"mapping\":[1,3,0,0],\"right_event\":[2,3],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0041","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[3,3,0,0],\"left_event\":[3],\"right_event\":[0,1]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([3, 3, 0, 0], [3], [0, 1], [0, 1, 3], [0, 1], [2, 3], [0, 1, 2, 3], [0, 1, 2, 3], 是)","explanation_zh":"收束波核對=([3, 3, 0, 0], [3], [0, 1], [0, 1, 3], [0, 1], [2, 3], [0, 1, 2, 3], [0, 1, 2, 3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255041,"parameters":{"source_n":4,"target_n":4,"mapping":[3,3,0,0],"left_event":[3],"right_event":[0,1],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[3],\"mapping\":[3,3,0,0],\"right_event\":[0,1],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0042","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[0,0,1,0],\"left_event\":[0],\"right_event\":[1,2]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 1, 0], [0], [1, 2], [0, 1, 2], [0, 1, 3], [2], [0, 1, 2, 3], [0, 1, 2, 3], 是)","explanation_zh":"收束波核對=([0, 0, 1, 0], [0], [1, 2], [0, 1, 2], [0, 1, 3], [2], [0, 1, 2, 3], [0, 1, 2, 3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255042,"parameters":{"source_n":4,"target_n":4,"mapping":[0,0,1,0],"left_event":[0],"right_event":[1,2],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[0],\"mapping\":[0,0,1,0],\"right_event\":[1,2],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0043","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[1,0,1,0],\"left_event\":[1],\"right_event\":[2,3]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1, 0], [1], [2, 3], [1, 2, 3], [0, 2], [], [0, 2], [0, 2], 是)","explanation_zh":"收束波核對=([1, 0, 1, 0], [1], [2, 3], [1, 2, 3], [0, 2], [], [0, 2], [0, 2], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255043,"parameters":{"source_n":4,"target_n":4,"mapping":[1,0,1,0],"left_event":[1],"right_event":[2,3],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[1],\"mapping\":[1,0,1,0],\"right_event\":[2,3],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0045","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[3,0,1,0],\"left_event\":[3],\"right_event\":[0,1]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([3, 0, 1, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], 是)","explanation_zh":"收束波核對=([3, 0, 1, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255045,"parameters":{"source_n":4,"target_n":4,"mapping":[3,0,1,0],"left_event":[3],"right_event":[0,1],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[3],\"mapping\":[3,0,1,0],\"right_event\":[0,1],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0047","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[1,1,1,0],\"left_event\":[1],\"right_event\":[2,3]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1, 0], [1], [2, 3], [1, 2, 3], [0, 1, 2], [], [0, 1, 2], [0, 1, 2], 是)","explanation_zh":"收束波核對=([1, 1, 1, 0], [1], [2, 3], [1, 2, 3], [0, 1, 2], [], [0, 1, 2], [0, 1, 2], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255047,"parameters":{"source_n":4,"target_n":4,"mapping":[1,1,1,0],"left_event":[1],"right_event":[2,3],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[1],\"mapping\":[1,1,1,0],\"right_event\":[2,3],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0048","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[2,1,1,0],\"left_event\":[2],\"right_event\":[0,3]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 1, 0], [2], [0, 3], [0, 2, 3], [0], [3], [0, 3], [0, 3], 是)","explanation_zh":"收束波核對=([2, 1, 1, 0], [2], [0, 3], [0, 2, 3], [0], [3], [0, 3], [0, 3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255048,"parameters":{"source_n":4,"target_n":4,"mapping":[2,1,1,0],"left_event":[2],"right_event":[0,3],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[2],\"mapping\":[2,1,1,0],\"right_event\":[0,3],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0049","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[3,1,1,0],\"left_event\":[3],\"right_event\":[0,1]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([3, 1, 1, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], 是)","explanation_zh":"收束波核對=([3, 1, 1, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255049,"parameters":{"source_n":4,"target_n":4,"mapping":[3,1,1,0],"left_event":[3],"right_event":[0,1],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[3],\"mapping\":[3,1,1,0],\"right_event\":[0,1],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0050","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-preimage-union-law；資料={\"source_n\":4,\"target_n\":4,\"mapping\":[0,2,1,0],\"left_event\":[0],\"right_event\":[1,2]}。請依 [mapping,left,right,union,left_preimage,right_preimage,union_preimage,preimage_union,equal] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 1, 0], [0], [1, 2], [0, 1, 2], [0, 3], [1, 2], [0, 1, 2, 3], [0, 1, 2, 3], 是)","explanation_zh":"收束波核對=([0, 2, 1, 0], [0], [1, 2], [0, 1, 2], [0, 3], [1, 2], [0, 1, 2, 3], [0, 1, 2, 3], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255050,"parameters":{"source_n":4,"target_n":4,"mapping":[0,2,1,0],"left_event":[0],"right_event":[1,2],"domain":"Finite_preimage_law_density","operation":"measdeep-preimage-union-law","semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[0],\"mapping\":[0,2,1,0],\"right_event\":[1,2],\"source_n\":4,\"target_n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0051","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":0}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 0, [0, 0, 0, 0], [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 0, [0, 0, 0, 0], [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255051,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":0,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0],[1,2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0052","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[1],[0,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":1}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 1, [1, 1, 1, 1], [[[], [], 是], [[0], [], 是], [[1], [0, 1, 2, 3], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 1, [1, 1, 1, 1], [[[], [], 是], [[0], [], 是], [[1], [0, 1, 2, 3], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255052,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[1],[0,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":1,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0053","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,1],[2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":2}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 2, [2, 2, 2, 2], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [0, 1, 2, 3], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [], 是], [[0, 1, 3], [], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 2, [2, 2, 2, 2], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [0, 1, 2, 3], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [], 是], [[0, 1, 3], [], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255053,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,1],[2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":2,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0054","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[2],[0,1,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":3}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 3, [3, 3, 3, 3], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [], 是], [[0, 1, 2], [], 是], [[3], [0, 1, 2, 3], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 3, [3, 3, 3, 3], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [], 是], [[0, 1, 2], [], 是], [[3], [0, 1, 2, 3], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255054,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[2],[0,1,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":3,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,3],[2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0055","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,2],[1,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":0}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 0, [0, 0, 0, 0], [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 0, [0, 0, 0, 0], [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255055,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,2],[1,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":0,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2],[1,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0056","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[1,2],[0,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":1}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 1, [1, 1, 1, 1], [[[], [], 是], [[0], [], 是], [[1], [0, 1, 2, 3], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 1, [1, 1, 1, 1], [[[], [], 是], [[0], [], 是], [[1], [0, 1, 2, 3], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255056,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[1,2],[0,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":1,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,3],[1,2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0057","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,1,2],[3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":2}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 2, [2, 2, 2, 2], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [0, 1, 2, 3], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [], 是], [[0, 1, 3], [], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 2, [2, 2, 2, 2], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [0, 1, 2, 3], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [], 是], [[0, 1, 3], [], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255057,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,1,2],[3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":2,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,2],[3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0058","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":3}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 3, [3, 3, 3, 3], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [], 是], [[0, 1, 2], [], 是], [[3], [0, 1, 2, 3], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 3, [3, 3, 3, 3], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [], 是], [[0, 1, 2], [], 是], [[3], [0, 1, 2, 3], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255058,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":3,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0],[1,2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0059","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[1],[0,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":0}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 0, [0, 0, 0, 0], [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 0, [0, 0, 0, 0], [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255059,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[1],[0,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":0,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0060","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,1],[2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":1}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 1, [1, 1, 1, 1], [[[], [], 是], [[0], [], 是], [[1], [0, 1, 2, 3], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 1, [1, 1, 1, 1], [[[], [], 是], [[0], [], 是], [[1], [0, 1, 2, 3], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255060,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,1],[2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":1,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0061","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[2],[0,1,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":2}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 2, [2, 2, 2, 2], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [0, 1, 2, 3], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [], 是], [[0, 1, 3], [], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 2, [2, 2, 2, 2], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [0, 1, 2, 3], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [], 是], [[0, 1, 3], [], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255061,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[2],[0,1,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":2,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,3],[2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0062","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,2],[1,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":3}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 3, [3, 3, 3, 3], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [], 是], [[0, 1, 2], [], 是], [[3], [0, 1, 2, 3], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 3, [3, 3, 3, 3], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [], 是], [[0, 1, 2], [], 是], [[3], [0, 1, 2, 3], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255062,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,2],[1,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":3,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2],[1,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0063","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[1,2],[0,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":0}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 0, [0, 0, 0, 0], [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 0, [0, 0, 0, 0], [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255063,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[1,2],[0,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":0,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,3],[1,2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0064","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,1,2],[3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":1}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 1, [1, 1, 1, 1], [[[], [], 是], [[0], [], 是], [[1], [0, 1, 2, 3], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 1, [1, 1, 1, 1], [[[], [], 是], [[0], [], 是], [[1], [0, 1, 2, 3], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255064,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,1,2],[3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":1,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,2],[3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0065","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":2}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 2, [2, 2, 2, 2], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [0, 1, 2, 3], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [], 是], [[0, 1, 3], [], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 2, [2, 2, 2, 2], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [0, 1, 2, 3], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [], 是], [[0, 1, 3], [], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255065,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":2,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0],[1,2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0066","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[1],[0,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":3}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 3, [3, 3, 3, 3], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [], 是], [[0, 1, 2], [], 是], [[3], [0, 1, 2, 3], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 3, [3, 3, 3, 3], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [], 是], [[0, 1, 2], [], 是], [[3], [0, 1, 2, 3], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255066,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[1],[0,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":3,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0067","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,1],[2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":0}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 0, [0, 0, 0, 0], [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 0, [0, 0, 0, 0], [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255067,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,1],[2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":0,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0068","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[2],[0,1,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":1}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 1, [1, 1, 1, 1], [[[], [], 是], [[0], [], 是], [[1], [0, 1, 2, 3], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 1, [1, 1, 1, 1], [[[], [], 是], [[0], [], 是], [[1], [0, 1, 2, 3], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255068,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[2],[0,1,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":1,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,3],[2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0069","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,2],[1,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":2}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 2, [2, 2, 2, 2], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [0, 1, 2, 3], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [], 是], [[0, 1, 3], [], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 2, [2, 2, 2, 2], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [0, 1, 2, 3], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [], 是], [[0, 1, 3], [], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255069,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,2],[1,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":2,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2],[1,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0070","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[1,2],[0,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":3}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 3, [3, 3, 3, 3], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [], 是], [[0, 1, 2], [], 是], [[3], [0, 1, 2, 3], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 3, [3, 3, 3, 3], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [], 是], [[0, 1, 2], [], 是], [[3], [0, 1, 2, 3], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255070,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[1,2],[0,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":3,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,3],[1,2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0071","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,1,2],[3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":0}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 0, [0, 0, 0, 0], [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 0, [0, 0, 0, 0], [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255071,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,1,2],[3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":0,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,2],[3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0072","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":1}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 1, [1, 1, 1, 1], [[[], [], 是], [[0], [], 是], [[1], [0, 1, 2, 3], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 1, [1, 1, 1, 1], [[[], [], 是], [[0], [], 是], [[1], [0, 1, 2, 3], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255072,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":1,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0],[1,2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0073","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[1],[0,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":2}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 2, [2, 2, 2, 2], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [0, 1, 2, 3], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [], 是], [[0, 1, 3], [], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 2, [2, 2, 2, 2], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [0, 1, 2, 3], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0, 1, 2, 3], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [], 是], [[1, 3], [], 是], [[0, 1, 3], [], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255073,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[1],[0,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":2,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0074","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,1],[2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":3}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 3, [3, 3, 3, 3], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [], 是], [[0, 1, 2], [], 是], [[3], [0, 1, 2, 3], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 3, [3, 3, 3, 3], [[[], [], 是], [[0], [], 是], [[1], [], 是], [[0, 1], [], 是], [[2], [], 是], [[0, 2], [], 是], [[1, 2], [], 是], [[0, 1, 2], [], 是], [[3], [0, 1, 2, 3], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0, 1, 2, 3], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0, 1, 2, 3], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1, 2, 3], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255074,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,1],[2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":3,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0075","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-constant-map-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[2],[0,1,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"value\":0}。請依 [source_n,target_n,value,mapping,preimage_rows,measurable] 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 0, [0, 0, 0, 0], [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=(4, 4, 0, [0, 0, 0, 0], [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255075,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[2],[0,1,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"value":0,"domain":"Finite_preimage_law_density","operation":"measdeep-constant-map-audit","semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,3],[2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0076","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0],[1,2,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[0,0,0,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 0, 0], 是, [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是, [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([0, 0, 0, 0], 是, [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是, [[[], [], 是], [[0], [0, 1, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255076,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0],[1,2,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[0,0,0,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[0,0,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0],[1,2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0077","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[1],[0,2,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[1,0,0,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 0, 0], 是, [[[], [], 是], [[0], [1, 2, 3], 否], [[1], [0], 否], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [1, 2, 3], 否], [[1, 2], [0], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [1, 2, 3], 否], [[1, 3], [0], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [1, 2, 3], 否], [[1, 2, 3], [0], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [1, 2, 3], 是], [[1], [0], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [1, 2, 3], 是], [[1, 2], [0], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [1, 2, 3], 是], [[1, 3], [0], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [1, 2, 3], 是], [[1, 2, 3], [0], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([1, 0, 0, 0], 是, [[[], [], 是], [[0], [1, 2, 3], 否], [[1], [0], 否], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [1, 2, 3], 否], [[1, 2], [0], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [1, 2, 3], 否], [[1, 3], [0], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [1, 2, 3], 否], [[1, 2, 3], [0], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [1, 2, 3], 是], [[1], [0], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [1, 2, 3], 是], [[1, 2], [0], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [1, 2, 3], 是], [[1, 3], [0], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [1, 2, 3], 是], [[1, 2, 3], [0], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255077,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[1],[0,2,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[1,0,0,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[1,0,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0078","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,1],[2,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[2,0,0,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([2, 0, 0, 0], 是, [[[], [], 是], [[0], [1, 2, 3], 否], [[1], [], 是], [[0, 1], [1, 2, 3], 否], [[2], [0], 否], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [1, 2, 3], 否], [[1, 3], [], 是], [[0, 1, 3], [1, 2, 3], 否], [[2, 3], [0], 否], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [1, 2, 3], 是], [[1], [], 是], [[0, 1], [1, 2, 3], 是], [[2], [0], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([2, 0, 0, 0], 是, [[[], [], 是], [[0], [1, 2, 3], 否], [[1], [], 是], [[0, 1], [1, 2, 3], 否], [[2], [0], 否], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [1, 2, 3], 否], [[1, 3], [], 是], [[0, 1, 3], [1, 2, 3], 否], [[2, 3], [0], 否], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [1, 2, 3], 是], [[1], [], 是], [[0, 1], [1, 2, 3], 是], [[2], [0], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [1, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255078,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,1],[2,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[2,0,0,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[2,0,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0079","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[2],[0,1,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[3,0,0,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([3, 0, 0, 0], 是, [[[], [], 是], [[0], [1, 2, 3], 否], [[1], [], 是], [[0, 1], [1, 2, 3], 否], [[2], [], 是], [[0, 2], [1, 2, 3], 否], [[1, 2], [], 是], [[0, 1, 2], [1, 2, 3], 否], [[3], [0], 否], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0], 否], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [1, 2, 3], 是], [[1], [], 是], [[0, 1], [1, 2, 3], 是], [[2], [], 是], [[0, 2], [1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [1, 2, 3], 是], [[3], [0], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([3, 0, 0, 0], 是, [[[], [], 是], [[0], [1, 2, 3], 否], [[1], [], 是], [[0, 1], [1, 2, 3], 否], [[2], [], 是], [[0, 2], [1, 2, 3], 否], [[1, 2], [], 是], [[0, 1, 2], [1, 2, 3], 否], [[3], [0], 否], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0], 否], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [1, 2, 3], 是], [[1], [], 是], [[0, 1], [1, 2, 3], 是], [[2], [], 是], [[0, 2], [1, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [1, 2, 3], 是], [[3], [0], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255079,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[2],[0,1,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[3,0,0,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[3,0,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,3],[2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0080","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,2],[1,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[0,1,0,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 0], 是, [[[], [], 是], [[0], [0, 2, 3], 否], [[1], [1], 否], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 2, 3], 否], [[1, 2], [1], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 2, 3], 否], [[1, 3], [1], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 2, 3], 否], [[1, 2, 3], [1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [0, 2, 3], 是], [[1], [1], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 2, 3], 是], [[1, 2], [1], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 2, 3], 是], [[1, 3], [1], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 2, 3], 是], [[1, 2, 3], [1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([0, 1, 0, 0], 是, [[[], [], 是], [[0], [0, 2, 3], 否], [[1], [1], 否], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 2, 3], 否], [[1, 2], [1], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 2, 3], 否], [[1, 3], [1], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 2, 3], 否], [[1, 2, 3], [1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [0, 2, 3], 是], [[1], [1], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 2, 3], 是], [[1, 2], [1], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 2, 3], 是], [[1, 3], [1], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 2, 3], 是], [[1, 2, 3], [1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255080,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,2],[1,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[0,1,0,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[0,1,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2],[1,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0081","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[1,2],[0,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[1,1,0,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 否], [[1], [0, 1], 否], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [2, 3], 否], [[1, 2], [0, 1], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [2, 3], 否], [[1, 3], [0, 1], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [2, 3], 否], [[1, 2, 3], [0, 1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [0, 1], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [2, 3], 是], [[1, 2], [0, 1], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [2, 3], 是], [[1, 3], [0, 1], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([1, 1, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 否], [[1], [0, 1], 否], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [2, 3], 否], [[1, 2], [0, 1], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [2, 3], 否], [[1, 3], [0, 1], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [2, 3], 否], [[1, 2, 3], [0, 1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [0, 1], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [2, 3], 是], [[1, 2], [0, 1], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [2, 3], 是], [[1, 3], [0, 1], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255081,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[1,2],[0,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[1,1,0,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[1,1,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,3],[1,2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0082","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,1,2],[3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[2,1,0,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 否], [[1], [1], 否], [[0, 1], [1, 2, 3], 否], [[2], [0], 否], [[0, 2], [0, 2, 3], 否], [[1, 2], [0, 1], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [2, 3], 否], [[1, 3], [1], 否], [[0, 1, 3], [1, 2, 3], 否], [[2, 3], [0], 否], [[0, 2, 3], [0, 2, 3], 否], [[1, 2, 3], [0, 1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [1], 是], [[0, 1], [1, 2, 3], 是], [[2], [0], 是], [[0, 2], [0, 2, 3], 是], [[1, 2], [0, 1], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [2, 3], 是], [[1, 3], [1], 是], [[0, 1, 3], [1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([2, 1, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 否], [[1], [1], 否], [[0, 1], [1, 2, 3], 否], [[2], [0], 否], [[0, 2], [0, 2, 3], 否], [[1, 2], [0, 1], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [2, 3], 否], [[1, 3], [1], 否], [[0, 1, 3], [1, 2, 3], 否], [[2, 3], [0], 否], [[0, 2, 3], [0, 2, 3], 否], [[1, 2, 3], [0, 1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [1], 是], [[0, 1], [1, 2, 3], 是], [[2], [0], 是], [[0, 2], [0, 2, 3], 是], [[1, 2], [0, 1], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [2, 3], 是], [[1, 3], [1], 是], [[0, 1, 3], [1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255082,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,1,2],[3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[2,1,0,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[2,1,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,2],[3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0083","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0],[1,2,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[3,1,0,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([3, 1, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 否], [[1], [1], 否], [[0, 1], [1, 2, 3], 是], [[2], [], 是], [[0, 2], [2, 3], 否], [[1, 2], [1], 否], [[0, 1, 2], [1, 2, 3], 是], [[3], [0], 是], [[0, 3], [0, 2, 3], 否], [[1, 3], [0, 1], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 2, 3], 否], [[1, 2, 3], [0, 1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [1], 是], [[0, 1], [1, 2, 3], 是], [[2], [], 是], [[0, 2], [2, 3], 是], [[1, 2], [1], 是], [[0, 1, 2], [1, 2, 3], 是], [[3], [0], 是], [[0, 3], [0, 2, 3], 是], [[1, 3], [0, 1], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([3, 1, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 否], [[1], [1], 否], [[0, 1], [1, 2, 3], 是], [[2], [], 是], [[0, 2], [2, 3], 否], [[1, 2], [1], 否], [[0, 1, 2], [1, 2, 3], 是], [[3], [0], 是], [[0, 3], [0, 2, 3], 否], [[1, 3], [0, 1], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 2, 3], 否], [[1, 2, 3], [0, 1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [1], 是], [[0, 1], [1, 2, 3], 是], [[2], [], 是], [[0, 2], [2, 3], 是], [[1, 2], [1], 是], [[0, 1, 2], [1, 2, 3], 是], [[3], [0], 是], [[0, 3], [0, 2, 3], 是], [[1, 3], [0, 1], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255083,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0],[1,2,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[3,1,0,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[3,1,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0],[1,2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0084","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[1],[0,2,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[0,2,0,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 0, 0], 是, [[[], [], 是], [[0], [0, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 2, 3], 是], [[2], [1], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [1], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 2, 3], 是], [[2, 3], [1], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是, [[[], [], 是], [[0], [0, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 2, 3], 是], [[2], [1], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [1], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 2, 3], 是], [[2, 3], [1], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([0, 2, 0, 0], 是, [[[], [], 是], [[0], [0, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 2, 3], 是], [[2], [1], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [1], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 2, 3], 是], [[2, 3], [1], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是, [[[], [], 是], [[0], [0, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 2, 3], 是], [[2], [1], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [1], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [0, 2, 3], 是], [[2, 3], [1], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255084,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[1],[0,2,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[0,2,0,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[0,2,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0085","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,1],[2,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[1,2,0,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 是], [[1], [0], 否], [[0, 1], [0, 2, 3], 否], [[2], [1], 否], [[0, 2], [1, 2, 3], 否], [[1, 2], [0, 1], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [2, 3], 是], [[1, 3], [0], 否], [[0, 1, 3], [0, 2, 3], 否], [[2, 3], [1], 否], [[0, 2, 3], [1, 2, 3], 否], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [0], 是], [[0, 1], [0, 2, 3], 是], [[2], [1], 是], [[0, 2], [1, 2, 3], 是], [[1, 2], [0, 1], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [2, 3], 是], [[1, 3], [0], 是], [[0, 1, 3], [0, 2, 3], 是], [[2, 3], [1], 是], [[0, 2, 3], [1, 2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([1, 2, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 是], [[1], [0], 否], [[0, 1], [0, 2, 3], 否], [[2], [1], 否], [[0, 2], [1, 2, 3], 否], [[1, 2], [0, 1], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [2, 3], 是], [[1, 3], [0], 否], [[0, 1, 3], [0, 2, 3], 否], [[2, 3], [1], 否], [[0, 2, 3], [1, 2, 3], 否], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [0], 是], [[0, 1], [0, 2, 3], 是], [[2], [1], 是], [[0, 2], [1, 2, 3], 是], [[1, 2], [0, 1], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [2, 3], 是], [[1, 3], [0], 是], [[0, 1, 3], [0, 2, 3], 是], [[2, 3], [1], 是], [[0, 2, 3], [1, 2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255085,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,1],[2,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[1,2,0,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[1,2,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0086","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[2],[0,1,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[2,2,0,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 否], [[1], [], 是], [[0, 1], [2, 3], 否], [[2], [0, 1], 否], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0, 1], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [2, 3], 否], [[1, 3], [], 是], [[0, 1, 3], [2, 3], 否], [[2, 3], [0, 1], 否], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [], 是], [[0, 1], [2, 3], 是], [[2], [0, 1], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0, 1], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [2, 3], 是], [[2, 3], [0, 1], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([2, 2, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 否], [[1], [], 是], [[0, 1], [2, 3], 否], [[2], [0, 1], 否], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0, 1], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [2, 3], 否], [[1, 3], [], 是], [[0, 1, 3], [2, 3], 否], [[2, 3], [0, 1], 否], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [], 是], [[0, 1], [2, 3], 是], [[2], [0, 1], 是], [[0, 2], [0, 1, 2, 3], 是], [[1, 2], [0, 1], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [2, 3], 是], [[1, 3], [], 是], [[0, 1, 3], [2, 3], 是], [[2, 3], [0, 1], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255086,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[2],[0,1,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[2,2,0,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[2,2,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,3],[2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0087","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,2],[1,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[3,2,0,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([3, 2, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 否], [[1], [], 是], [[0, 1], [2, 3], 否], [[2], [1], 否], [[0, 2], [1, 2, 3], 否], [[1, 2], [1], 否], [[0, 1, 2], [1, 2, 3], 否], [[3], [0], 否], [[0, 3], [0, 2, 3], 否], [[1, 3], [0], 否], [[0, 1, 3], [0, 2, 3], 否], [[2, 3], [0, 1], 否], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [], 是], [[0, 1], [2, 3], 是], [[2], [1], 是], [[0, 2], [1, 2, 3], 是], [[1, 2], [1], 是], [[0, 1, 2], [1, 2, 3], 是], [[3], [0], 是], [[0, 3], [0, 2, 3], 是], [[1, 3], [0], 是], [[0, 1, 3], [0, 2, 3], 是], [[2, 3], [0, 1], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([3, 2, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 否], [[1], [], 是], [[0, 1], [2, 3], 否], [[2], [1], 否], [[0, 2], [1, 2, 3], 否], [[1, 2], [1], 否], [[0, 1, 2], [1, 2, 3], 否], [[3], [0], 否], [[0, 3], [0, 2, 3], 否], [[1, 3], [0], 否], [[0, 1, 3], [0, 2, 3], 否], [[2, 3], [0, 1], 否], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [], 是], [[0, 1], [2, 3], 是], [[2], [1], 是], [[0, 2], [1, 2, 3], 是], [[1, 2], [1], 是], [[0, 1, 2], [1, 2, 3], 是], [[3], [0], 是], [[0, 3], [0, 2, 3], 是], [[1, 3], [0], 是], [[0, 1, 3], [0, 2, 3], 是], [[2, 3], [0, 1], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255087,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,2],[1,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[3,2,0,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[3,2,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2],[1,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0088","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[1,2],[0,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[0,3,0,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([0, 3, 0, 0], 是, [[[], [], 是], [[0], [0, 2, 3], 否], [[1], [], 是], [[0, 1], [0, 2, 3], 否], [[2], [], 是], [[0, 2], [0, 2, 3], 否], [[1, 2], [], 是], [[0, 1, 2], [0, 2, 3], 否], [[3], [1], 否], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [1], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [1], 否], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [0, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 2, 3], 是], [[3], [1], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [1], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [1], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([0, 3, 0, 0], 是, [[[], [], 是], [[0], [0, 2, 3], 否], [[1], [], 是], [[0, 1], [0, 2, 3], 否], [[2], [], 是], [[0, 2], [0, 2, 3], 否], [[1, 2], [], 是], [[0, 1, 2], [0, 2, 3], 否], [[3], [1], 否], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [1], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [1], 否], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [0, 2, 3], 是], [[1], [], 是], [[0, 1], [0, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [0, 2, 3], 是], [[3], [1], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [1], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [1], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255088,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[1,2],[0,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[0,3,0,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[0,3,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,3],[1,2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0089","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,1,2],[3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[1,3,0,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 否], [[1], [0], 否], [[0, 1], [0, 2, 3], 否], [[2], [], 是], [[0, 2], [2, 3], 否], [[1, 2], [0], 否], [[0, 1, 2], [0, 2, 3], 否], [[3], [1], 否], [[0, 3], [1, 2, 3], 否], [[1, 3], [0, 1], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [1], 否], [[0, 2, 3], [1, 2, 3], 否], [[1, 2, 3], [0, 1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [0], 是], [[0, 1], [0, 2, 3], 是], [[2], [], 是], [[0, 2], [2, 3], 是], [[1, 2], [0], 是], [[0, 1, 2], [0, 2, 3], 是], [[3], [1], 是], [[0, 3], [1, 2, 3], 是], [[1, 3], [0, 1], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [1], 是], [[0, 2, 3], [1, 2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([1, 3, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 否], [[1], [0], 否], [[0, 1], [0, 2, 3], 否], [[2], [], 是], [[0, 2], [2, 3], 否], [[1, 2], [0], 否], [[0, 1, 2], [0, 2, 3], 否], [[3], [1], 否], [[0, 3], [1, 2, 3], 否], [[1, 3], [0, 1], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [1], 否], [[0, 2, 3], [1, 2, 3], 否], [[1, 2, 3], [0, 1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [0], 是], [[0, 1], [0, 2, 3], 是], [[2], [], 是], [[0, 2], [2, 3], 是], [[1, 2], [0], 是], [[0, 1, 2], [0, 2, 3], 是], [[3], [1], 是], [[0, 3], [1, 2, 3], 是], [[1, 3], [0, 1], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [1], 是], [[0, 2, 3], [1, 2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255089,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,1,2],[3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[1,3,0,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[1,3,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,2],[3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0090","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0],[1,2,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[2,3,0,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([2, 3, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 否], [[1], [], 是], [[0, 1], [2, 3], 否], [[2], [0], 是], [[0, 2], [0, 2, 3], 否], [[1, 2], [0], 是], [[0, 1, 2], [0, 2, 3], 否], [[3], [1], 否], [[0, 3], [1, 2, 3], 是], [[1, 3], [1], 否], [[0, 1, 3], [1, 2, 3], 是], [[2, 3], [0, 1], 否], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [], 是], [[0, 1], [2, 3], 是], [[2], [0], 是], [[0, 2], [0, 2, 3], 是], [[1, 2], [0], 是], [[0, 1, 2], [0, 2, 3], 是], [[3], [1], 是], [[0, 3], [1, 2, 3], 是], [[1, 3], [1], 是], [[0, 1, 3], [1, 2, 3], 是], [[2, 3], [0, 1], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([2, 3, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 否], [[1], [], 是], [[0, 1], [2, 3], 否], [[2], [0], 是], [[0, 2], [0, 2, 3], 否], [[1, 2], [0], 是], [[0, 1, 2], [0, 2, 3], 否], [[3], [1], 否], [[0, 3], [1, 2, 3], 是], [[1, 3], [1], 否], [[0, 1, 3], [1, 2, 3], 是], [[2, 3], [0, 1], 否], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [], 是], [[0, 1], [2, 3], 是], [[2], [0], 是], [[0, 2], [0, 2, 3], 是], [[1, 2], [0], 是], [[0, 1, 2], [0, 2, 3], 是], [[3], [1], 是], [[0, 3], [1, 2, 3], 是], [[1, 3], [1], 是], [[0, 1, 3], [1, 2, 3], 是], [[2, 3], [0, 1], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255090,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0],[1,2,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[2,3,0,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[2,3,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0],[1,2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0091","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[1],[0,2,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[3,3,0,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([3, 3, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 否], [[1], [], 是], [[0, 1], [2, 3], 否], [[2], [], 是], [[0, 2], [2, 3], 否], [[1, 2], [], 是], [[0, 1, 2], [2, 3], 否], [[3], [0, 1], 否], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0, 1], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0, 1], 否], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [], 是], [[0, 1], [2, 3], 是], [[2], [], 是], [[0, 2], [2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [2, 3], 是], [[3], [0, 1], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0, 1], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0, 1], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([3, 3, 0, 0], 是, [[[], [], 是], [[0], [2, 3], 否], [[1], [], 是], [[0, 1], [2, 3], 否], [[2], [], 是], [[0, 2], [2, 3], 否], [[1, 2], [], 是], [[0, 1, 2], [2, 3], 否], [[3], [0, 1], 否], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0, 1], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0, 1], 否], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [2, 3], 是], [[1], [], 是], [[0, 1], [2, 3], 是], [[2], [], 是], [[0, 2], [2, 3], 是], [[1, 2], [], 是], [[0, 1, 2], [2, 3], 是], [[3], [0, 1], 是], [[0, 3], [0, 1, 2, 3], 是], [[1, 3], [0, 1], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0, 1], 是], [[0, 2, 3], [0, 1, 2, 3], 是], [[1, 2, 3], [0, 1], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255091,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[1],[0,2,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[3,3,0,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[3,3,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0092","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,1],[2,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[0,0,1,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 1, 0], 是, [[[], [], 是], [[0], [0, 1, 3], 否], [[1], [2], 否], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 3], 否], [[1, 2], [2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 3], 否], [[1, 3], [2], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 3], 否], [[1, 2, 3], [2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [0, 1, 3], 是], [[1], [2], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 3], 是], [[1, 2], [2], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 3], 是], [[1, 3], [2], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 3], 是], [[1, 2, 3], [2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([0, 0, 1, 0], 是, [[[], [], 是], [[0], [0, 1, 3], 否], [[1], [2], 否], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 3], 否], [[1, 2], [2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 3], 否], [[1, 3], [2], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 3], 否], [[1, 2, 3], [2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [0, 1, 3], 是], [[1], [2], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 1, 3], 是], [[1, 2], [2], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 1, 3], 是], [[1, 3], [2], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 1, 3], 是], [[1, 2, 3], [2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255092,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,1],[2,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[0,0,1,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[0,0,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0093","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[2],[0,1,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[1,0,1,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1, 0], 是, [[[], [], 是], [[0], [1, 3], 否], [[1], [0, 2], 否], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [1, 3], 否], [[1, 2], [0, 2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [1, 3], 否], [[1, 3], [0, 2], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [1, 3], 否], [[1, 2, 3], [0, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [1, 3], 是], [[1], [0, 2], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [1, 3], 是], [[1, 2], [0, 2], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [1, 3], 是], [[1, 3], [0, 2], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [1, 3], 是], [[1, 2, 3], [0, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([1, 0, 1, 0], 是, [[[], [], 是], [[0], [1, 3], 否], [[1], [0, 2], 否], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [1, 3], 否], [[1, 2], [0, 2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [1, 3], 否], [[1, 3], [0, 2], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [1, 3], 否], [[1, 2, 3], [0, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [1, 3], 是], [[1], [0, 2], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [1, 3], 是], [[1, 2], [0, 2], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [1, 3], 是], [[1, 3], [0, 2], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [1, 3], 是], [[1, 2, 3], [0, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255093,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[2],[0,1,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[1,0,1,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[1,0,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,3],[2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0094","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,2],[1,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[2,0,1,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([2, 0, 1, 0], 是, [[[], [], 是], [[0], [1, 3], 是], [[1], [2], 否], [[0, 1], [1, 2, 3], 否], [[2], [0], 否], [[0, 2], [0, 1, 3], 否], [[1, 2], [0, 2], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [1, 3], 是], [[1, 3], [2], 否], [[0, 1, 3], [1, 2, 3], 否], [[2, 3], [0], 否], [[0, 2, 3], [0, 1, 3], 否], [[1, 2, 3], [0, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [1, 3], 是], [[1], [2], 是], [[0, 1], [1, 2, 3], 是], [[2], [0], 是], [[0, 2], [0, 1, 3], 是], [[1, 2], [0, 2], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [1, 3], 是], [[1, 3], [2], 是], [[0, 1, 3], [1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 1, 3], 是], [[1, 2, 3], [0, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([2, 0, 1, 0], 是, [[[], [], 是], [[0], [1, 3], 是], [[1], [2], 否], [[0, 1], [1, 2, 3], 否], [[2], [0], 否], [[0, 2], [0, 1, 3], 否], [[1, 2], [0, 2], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [1, 3], 是], [[1, 3], [2], 否], [[0, 1, 3], [1, 2, 3], 否], [[2, 3], [0], 否], [[0, 2, 3], [0, 1, 3], 否], [[1, 2, 3], [0, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [1, 3], 是], [[1], [2], 是], [[0, 1], [1, 2, 3], 是], [[2], [0], 是], [[0, 2], [0, 1, 3], 是], [[1, 2], [0, 2], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [1, 3], 是], [[1, 3], [2], 是], [[0, 1, 3], [1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 1, 3], 是], [[1, 2, 3], [0, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255094,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,2],[1,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[2,0,1,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[2,0,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2],[1,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0095","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[1,2],[0,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[3,0,1,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([3, 0, 1, 0], 是, [[[], [], 是], [[0], [1, 3], 否], [[1], [2], 否], [[0, 1], [1, 2, 3], 否], [[2], [], 是], [[0, 2], [1, 3], 否], [[1, 2], [2], 否], [[0, 1, 2], [1, 2, 3], 否], [[3], [0], 否], [[0, 3], [0, 1, 3], 否], [[1, 3], [0, 2], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0], 否], [[0, 2, 3], [0, 1, 3], 否], [[1, 2, 3], [0, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [1, 3], 是], [[1], [2], 是], [[0, 1], [1, 2, 3], 是], [[2], [], 是], [[0, 2], [1, 3], 是], [[1, 2], [2], 是], [[0, 1, 2], [1, 2, 3], 是], [[3], [0], 是], [[0, 3], [0, 1, 3], 是], [[1, 3], [0, 2], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 1, 3], 是], [[1, 2, 3], [0, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([3, 0, 1, 0], 是, [[[], [], 是], [[0], [1, 3], 否], [[1], [2], 否], [[0, 1], [1, 2, 3], 否], [[2], [], 是], [[0, 2], [1, 3], 否], [[1, 2], [2], 否], [[0, 1, 2], [1, 2, 3], 否], [[3], [0], 否], [[0, 3], [0, 1, 3], 否], [[1, 3], [0, 2], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0], 否], [[0, 2, 3], [0, 1, 3], 否], [[1, 2, 3], [0, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [1, 3], 是], [[1], [2], 是], [[0, 1], [1, 2, 3], 是], [[2], [], 是], [[0, 2], [1, 3], 是], [[1, 2], [2], 是], [[0, 1, 2], [1, 2, 3], 是], [[3], [0], 是], [[0, 3], [0, 1, 3], 是], [[1, 3], [0, 2], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 1, 3], 是], [[1, 2, 3], [0, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255095,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[1,2],[0,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[3,0,1,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[3,0,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,3],[1,2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0096","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,1,2],[3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[0,1,1,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 1, 0], 是, [[[], [], 是], [[0], [0, 3], 否], [[1], [1, 2], 否], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 3], 否], [[1, 2], [1, 2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 3], 否], [[1, 3], [1, 2], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 3], 否], [[1, 2, 3], [1, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [0, 3], 是], [[1], [1, 2], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 3], 是], [[1, 2], [1, 2], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 3], 是], [[1, 3], [1, 2], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 3], 是], [[1, 2, 3], [1, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([0, 1, 1, 0], 是, [[[], [], 是], [[0], [0, 3], 否], [[1], [1, 2], 否], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 3], 否], [[1, 2], [1, 2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 3], 否], [[1, 3], [1, 2], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 3], 否], [[1, 2, 3], [1, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [0, 3], 是], [[1], [1, 2], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [0, 3], 是], [[1, 2], [1, 2], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 3], 是], [[1, 3], [1, 2], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [0, 3], 是], [[1, 2, 3], [1, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255096,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,1,2],[3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[0,1,1,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[0,1,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,2],[3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0097","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0],[1,2,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[1,1,1,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1, 0], 是, [[[], [], 是], [[0], [3], 否], [[1], [0, 1, 2], 否], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [3], 否], [[1, 2], [0, 1, 2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [3], 否], [[1, 3], [0, 1, 2], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [3], 否], [[1, 2, 3], [0, 1, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [3], 是], [[1], [0, 1, 2], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [3], 是], [[1, 2], [0, 1, 2], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [3], 是], [[1, 3], [0, 1, 2], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [3], 是], [[1, 2, 3], [0, 1, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([1, 1, 1, 0], 是, [[[], [], 是], [[0], [3], 否], [[1], [0, 1, 2], 否], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [3], 否], [[1, 2], [0, 1, 2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [3], 否], [[1, 3], [0, 1, 2], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [3], 否], [[1, 2, 3], [0, 1, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [3], 是], [[1], [0, 1, 2], 是], [[0, 1], [0, 1, 2, 3], 是], [[2], [], 是], [[0, 2], [3], 是], [[1, 2], [0, 1, 2], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [3], 是], [[1, 3], [0, 1, 2], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [], 是], [[0, 2, 3], [3], 是], [[1, 2, 3], [0, 1, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255097,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0],[1,2,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[1,1,1,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[1,1,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0],[1,2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0098","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[1],[0,2,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[2,1,1,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 1, 0], 是, [[[], [], 是], [[0], [3], 否], [[1], [1, 2], 否], [[0, 1], [1, 2, 3], 否], [[2], [0], 否], [[0, 2], [0, 3], 否], [[1, 2], [0, 1, 2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [3], 否], [[1, 3], [1, 2], 否], [[0, 1, 3], [1, 2, 3], 否], [[2, 3], [0], 否], [[0, 2, 3], [0, 3], 否], [[1, 2, 3], [0, 1, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [3], 是], [[1], [1, 2], 是], [[0, 1], [1, 2, 3], 是], [[2], [0], 是], [[0, 2], [0, 3], 是], [[1, 2], [0, 1, 2], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [3], 是], [[1, 3], [1, 2], 是], [[0, 1, 3], [1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 3], 是], [[1, 2, 3], [0, 1, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([2, 1, 1, 0], 是, [[[], [], 是], [[0], [3], 否], [[1], [1, 2], 否], [[0, 1], [1, 2, 3], 否], [[2], [0], 否], [[0, 2], [0, 3], 否], [[1, 2], [0, 1, 2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [3], 否], [[1, 3], [1, 2], 否], [[0, 1, 3], [1, 2, 3], 否], [[2, 3], [0], 否], [[0, 2, 3], [0, 3], 否], [[1, 2, 3], [0, 1, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [3], 是], [[1], [1, 2], 是], [[0, 1], [1, 2, 3], 是], [[2], [0], 是], [[0, 2], [0, 3], 是], [[1, 2], [0, 1, 2], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [3], 是], [[1, 3], [1, 2], 是], [[0, 1, 3], [1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 3], 是], [[1, 2, 3], [0, 1, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255098,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[1],[0,2,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[2,1,1,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[2,1,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0099","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[0,1],[2,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[3,1,1,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([3, 1, 1, 0], 是, [[[], [], 是], [[0], [3], 否], [[1], [1, 2], 否], [[0, 1], [1, 2, 3], 否], [[2], [], 是], [[0, 2], [3], 否], [[1, 2], [1, 2], 否], [[0, 1, 2], [1, 2, 3], 否], [[3], [0], 否], [[0, 3], [0, 3], 否], [[1, 3], [0, 1, 2], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0], 否], [[0, 2, 3], [0, 3], 否], [[1, 2, 3], [0, 1, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [3], 是], [[1], [1, 2], 是], [[0, 1], [1, 2, 3], 是], [[2], [], 是], [[0, 2], [3], 是], [[1, 2], [1, 2], 是], [[0, 1, 2], [1, 2, 3], 是], [[3], [0], 是], [[0, 3], [0, 3], 是], [[1, 3], [0, 1, 2], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 3], 是], [[1, 2, 3], [0, 1, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([3, 1, 1, 0], 是, [[[], [], 是], [[0], [3], 否], [[1], [1, 2], 否], [[0, 1], [1, 2, 3], 否], [[2], [], 是], [[0, 2], [3], 否], [[1, 2], [1, 2], 否], [[0, 1, 2], [1, 2, 3], 否], [[3], [0], 否], [[0, 3], [0, 3], 否], [[1, 3], [0, 1, 2], 否], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0], 否], [[0, 2, 3], [0, 3], 否], [[1, 2, 3], [0, 1, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [3], 是], [[1], [1, 2], 是], [[0, 1], [1, 2, 3], 是], [[2], [], 是], [[0, 2], [3], 是], [[1, 2], [1, 2], 是], [[0, 1, 2], [1, 2, 3], 是], [[3], [0], 是], [[0, 3], [0, 3], 是], [[1, 3], [0, 1, 2], 是], [[0, 1, 3], [0, 1, 2, 3], 是], [[2, 3], [0], 是], [[0, 2, 3], [0, 3], 是], [[1, 2, 3], [0, 1, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255099,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[0,1],[2,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[3,1,1,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[3,1,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r05-measurable-function-0100","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","language":"zh-Hant","source_mko_ids":["mko-measurable-function"],"title_zh":"可測函數密度","stem_zh":"完整有限sigma代數與總映射。操作=measdeep-source-sigma-refinement-audit；資料={\"source_n\":4,\"target_n\":4,\"source_sigma\":[[],[2],[0,1,3],[0,1,2,3]],\"refined_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"target_sigma\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],\"mapping\":[0,2,1,0]}。請依 [mapping,is_refinement,before_rows,before_measurable,after_rows,after_measurable] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 1, 0], 是, [[[], [], 是], [[0], [0, 3], 否], [[1], [2], 是], [[0, 1], [0, 2, 3], 否], [[2], [1], 否], [[0, 2], [0, 1, 3], 是], [[1, 2], [1, 2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 3], 否], [[1, 3], [2], 是], [[0, 1, 3], [0, 2, 3], 否], [[2, 3], [1], 否], [[0, 2, 3], [0, 1, 3], 是], [[1, 2, 3], [1, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [0, 3], 是], [[1], [2], 是], [[0, 1], [0, 2, 3], 是], [[2], [1], 是], [[0, 2], [0, 1, 3], 是], [[1, 2], [1, 2], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 3], 是], [[1, 3], [2], 是], [[0, 1, 3], [0, 2, 3], 是], [[2, 3], [1], 是], [[0, 2, 3], [0, 1, 3], 是], [[1, 2, 3], [1, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)","explanation_zh":"收束波核對=([0, 2, 1, 0], 是, [[[], [], 是], [[0], [0, 3], 否], [[1], [2], 是], [[0, 1], [0, 2, 3], 否], [[2], [1], 否], [[0, 2], [0, 1, 3], 是], [[1, 2], [1, 2], 否], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 3], 否], [[1, 3], [2], 是], [[0, 1, 3], [0, 2, 3], 否], [[2, 3], [1], 否], [[0, 2, 3], [0, 1, 3], 是], [[1, 2, 3], [1, 2], 否], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 否, [[[], [], 是], [[0], [0, 3], 是], [[1], [2], 是], [[0, 1], [0, 2, 3], 是], [[2], [1], 是], [[0, 2], [0, 1, 3], 是], [[1, 2], [1, 2], 是], [[0, 1, 2], [0, 1, 2, 3], 是], [[3], [], 是], [[0, 3], [0, 3], 是], [[1, 3], [2], 是], [[0, 1, 3], [0, 2, 3], 是], [[2, 3], [1], 是], [[0, 2, 3], [0, 1, 3], 是], [[1, 2, 3], [1, 2], 是], [[0, 1, 2, 3], [0, 1, 2, 3], 是]], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；集合律一律由逆像核對；不推出連續可積保測或無限空間結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["可測函數密度"]},"variant":{"family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255100,"parameters":{"source_n":4,"target_n":4,"source_sigma":[[],[2],[0,1,3],[0,1,2,3]],"refined_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"target_sigma":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2],[3],[0,3],[1,3],[0,1,3],[2,3],[0,2,3],[1,2,3],[0,1,2,3]],"mapping":[0,2,1,0],"domain":"Finite_preimage_law_density","operation":"measdeep-source-sigma-refinement-audit","semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[0,2,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,3],[2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}