{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261023-r06-discrete-stochastic-process-0001","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[0,0,1],[0,1,1],[1,1,0]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[0, 0, 1], [0, 1, 1], [1, 1, 0]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[0, 0, 1], [0, 1, 1], [1, 1, 0]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236001,"parameters":{"paths":[[0,0,1],[0,1,1],[1,1,0]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[0,0,1],[0,1,1],[1,1,0]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0002","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[1,1,2],[1,2,2],[2,2,1]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[1, 1, 2], [1, 2, 2], [2, 2, 1]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[1, 1, 2], [1, 2, 2], [2, 2, 1]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236002,"parameters":{"paths":[[1,1,2],[1,2,2],[2,2,1]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[1,1,2],[1,2,2],[2,2,1]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0003","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[2,2,3],[2,3,3],[3,3,2]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[2, 2, 3], [2, 3, 3], [3, 3, 2]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[2, 2, 3], [2, 3, 3], [3, 3, 2]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236003,"parameters":{"paths":[[2,2,3],[2,3,3],[3,3,2]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[2,2,3],[2,3,3],[3,3,2]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0004","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[3,3,4],[3,4,4],[4,4,3]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[3, 3, 4], [3, 4, 4], [4, 4, 3]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[3, 3, 4], [3, 4, 4], [4, 4, 3]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236004,"parameters":{"paths":[[3,3,4],[3,4,4],[4,4,3]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[3,3,4],[3,4,4],[4,4,3]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0005","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[4,4,5],[4,5,5],[5,5,4]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[4, 4, 5], [4, 5, 5], [5, 5, 4]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[4, 4, 5], [4, 5, 5], [5, 5, 4]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236005,"parameters":{"paths":[[4,4,5],[4,5,5],[5,5,4]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[4,4,5],[4,5,5],[5,5,4]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0006","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[5,5,6],[5,6,6],[6,6,5]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[5, 5, 6], [5, 6, 6], [6, 6, 5]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[5, 5, 6], [5, 6, 6], [6, 6, 5]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236006,"parameters":{"paths":[[5,5,6],[5,6,6],[6,6,5]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[5,5,6],[5,6,6],[6,6,5]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0007","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[6,6,7],[6,7,7],[7,7,6]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[6, 6, 7], [6, 7, 7], [7, 7, 6]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[6, 6, 7], [6, 7, 7], [7, 7, 6]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236007,"parameters":{"paths":[[6,6,7],[6,7,7],[7,7,6]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[6,6,7],[6,7,7],[7,7,6]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0008","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[7,7,8],[7,8,8],[8,8,7]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[7, 7, 8], [7, 8, 8], [8, 8, 7]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[7, 7, 8], [7, 8, 8], [8, 8, 7]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236008,"parameters":{"paths":[[7,7,8],[7,8,8],[8,8,7]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[7,7,8],[7,8,8],[8,8,7]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0009","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[8,8,9],[8,9,9],[9,9,8]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[8, 8, 9], [8, 9, 9], [9, 9, 8]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[8, 8, 9], [8, 9, 9], [9, 9, 8]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236009,"parameters":{"paths":[[8,8,9],[8,9,9],[9,9,8]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[8,8,9],[8,9,9],[9,9,8]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0010","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[9,9,10],[9,10,10],[10,10,9]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[9, 9, 10], [9, 10, 10], [10, 10, 9]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[9, 9, 10], [9, 10, 10], [10, 10, 9]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236010,"parameters":{"paths":[[9,9,10],[9,10,10],[10,10,9]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[10,10,9],[9,10,10],[9,9,10]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0011","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[10,10,11],[10,11,11],[11,11,10]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[10, 10, 11], [10, 11, 11], [11, 11, 10]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[10, 10, 11], [10, 11, 11], [11, 11, 10]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236011,"parameters":{"paths":[[10,10,11],[10,11,11],[11,11,10]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[10,10,11],[10,11,11],[11,11,10]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0012","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[11,11,12],[11,12,12],[12,12,11]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[11, 11, 12], [11, 12, 12], [12, 12, 11]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[11, 11, 12], [11, 12, 12], [12, 12, 11]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236012,"parameters":{"paths":[[11,11,12],[11,12,12],[12,12,11]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[11,11,12],[11,12,12],[12,12,11]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0013","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[12,12,13],[12,13,13],[13,13,12]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[12, 12, 13], [12, 13, 13], [13, 13, 12]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[12, 12, 13], [12, 13, 13], [13, 13, 12]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236013,"parameters":{"paths":[[12,12,13],[12,13,13],[13,13,12]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[12,12,13],[12,13,13],[13,13,12]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0014","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[13,13,14],[13,14,14],[14,14,13]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[13, 13, 14], [13, 14, 14], [14, 14, 13]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[13, 13, 14], [13, 14, 14], [14, 14, 13]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236014,"parameters":{"paths":[[13,13,14],[13,14,14],[14,14,13]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[13,13,14],[13,14,14],[14,14,13]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0015","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[14,14,15],[14,15,15],[15,15,14]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[14, 14, 15], [14, 15, 15], [15, 15, 14]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[14, 14, 15], [14, 15, 15], [15, 15, 14]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236015,"parameters":{"paths":[[14,14,15],[14,15,15],[15,15,14]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[14,14,15],[14,15,15],[15,15,14]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0016","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[15,15,16],[15,16,16],[16,16,15]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[15, 15, 16], [15, 16, 16], [16, 16, 15]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[15, 15, 16], [15, 16, 16], [16, 16, 15]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236016,"parameters":{"paths":[[15,15,16],[15,16,16],[16,16,15]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[15,15,16],[15,16,16],[16,16,15]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0017","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[16,16,17],[16,17,17],[17,17,16]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[16, 16, 17], [16, 17, 17], [17, 17, 16]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[16, 16, 17], [16, 17, 17], [17, 17, 16]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236017,"parameters":{"paths":[[16,16,17],[16,17,17],[17,17,16]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[16,16,17],[16,17,17],[17,17,16]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0018","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[17,17,18],[17,18,18],[18,18,17]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[17, 17, 18], [17, 18, 18], [18, 18, 17]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[17, 17, 18], [17, 18, 18], [18, 18, 17]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236018,"parameters":{"paths":[[17,17,18],[17,18,18],[18,18,17]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[17,17,18],[17,18,18],[18,18,17]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0019","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[18,18,19],[18,19,19],[19,19,18]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[18, 18, 19], [18, 19, 19], [19, 19, 18]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[18, 18, 19], [18, 19, 19], [19, 19, 18]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236019,"parameters":{"paths":[[18,18,19],[18,19,19],[19,19,18]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[18,18,19],[18,19,19],[19,19,18]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0020","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[19,19,20],[19,20,20],[20,20,19]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[19, 19, 20], [19, 20, 20], [20, 20, 19]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[19, 19, 20], [19, 20, 20], [20, 20, 19]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236020,"parameters":{"paths":[[19,19,20],[19,20,20],[20,20,19]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[19,19,20],[19,20,20],[20,20,19]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0021","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[20,20,21],[20,21,21],[21,21,20]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[20, 20, 21], [20, 21, 21], [21, 21, 20]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[20, 20, 21], [20, 21, 21], [21, 21, 20]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236021,"parameters":{"paths":[[20,20,21],[20,21,21],[21,21,20]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[20,20,21],[20,21,21],[21,21,20]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0022","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[21,21,22],[21,22,22],[22,22,21]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[21, 21, 22], [21, 22, 22], [22, 22, 21]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[21, 21, 22], [21, 22, 22], [22, 22, 21]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236022,"parameters":{"paths":[[21,21,22],[21,22,22],[22,22,21]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[21,21,22],[21,22,22],[22,22,21]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0023","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[22,22,23],[22,23,23],[23,23,22]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[22, 22, 23], [22, 23, 23], [23, 23, 22]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[22, 22, 23], [22, 23, 23], [23, 23, 22]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236023,"parameters":{"paths":[[22,22,23],[22,23,23],[23,23,22]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[22,22,23],[22,23,23],[23,23,22]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0024","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[23,23,24],[23,24,24],[24,24,23]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[23, 23, 24], [23, 24, 24], [24, 24, 23]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[23, 23, 24], [23, 24, 24], [24, 24, 23]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236024,"parameters":{"paths":[[23,23,24],[23,24,24],[24,24,23]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[23,23,24],[23,24,24],[24,24,23]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0025","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-path-law-qualification；資料={\"paths\":[[24,24,25],[24,25,25],[25,25,24]],\"masses\":[[1,4],[1,4],[1,2]]}。請依 [paths,masses,horizon,nonnegative,sum,qualified] 欄序回答。","answer":{"kind":"expression","value":"([[24, 24, 25], [24, 25, 25], [25, 25, 24]], [1/4, 1/4, 1/2], 3, 是, 1, 是)","explanation_zh":"現代橋梁核對=([[24, 24, 25], [24, 25, 25], [25, 25, 24]], [1/4, 1/4, 1/2], 3, 是, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-path-law-qualification","seed":202610236025,"parameters":{"paths":[[24,24,25],[24,25,25],[25,25,24]],"masses":[[1,4],[1,4],[1,2]],"domain":"Finite_horizon_joint_path_law","operation":"process-path-law-qualification","semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[24,24,25],[24,25,25],[25,25,24]]}"}},"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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0026","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[0,0,1],[0,1,1],[1,1,0]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 1/2], [1, 1/2]])","explanation_zh":"現代橋梁核對=(0, [[0, 1/2], [1, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236026,"parameters":{"paths":[[0,0,1],[0,1,1],[1,1,0]],"masses":[[1,4],[1,4],[1,2]],"time":0,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[0,0,1],[0,1,1],[1,1,0]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0027","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[1,1,2],[1,2,2],[2,2,1]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":1}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(1, [[1, 1/4], [2, 3/4]])","explanation_zh":"現代橋梁核對=(1, [[1, 1/4], [2, 3/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236027,"parameters":{"paths":[[1,1,2],[1,2,2],[2,2,1]],"masses":[[1,4],[1,4],[1,2]],"time":1,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[1,1,2],[1,2,2],[2,2,1]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0028","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[2,2,3],[2,3,3],[3,3,2]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":2}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(2, [[2, 1/2], [3, 1/2]])","explanation_zh":"現代橋梁核對=(2, [[2, 1/2], [3, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236028,"parameters":{"paths":[[2,2,3],[2,3,3],[3,3,2]],"masses":[[1,4],[1,4],[1,2]],"time":2,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[2,2,3],[2,3,3],[3,3,2]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0029","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[3,3,4],[3,4,4],[4,4,3]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, [[3, 1/2], [4, 1/2]])","explanation_zh":"現代橋梁核對=(0, [[3, 1/2], [4, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236029,"parameters":{"paths":[[3,3,4],[3,4,4],[4,4,3]],"masses":[[1,4],[1,4],[1,2]],"time":0,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[3,3,4],[3,4,4],[4,4,3]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0030","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[4,4,5],[4,5,5],[5,5,4]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":1}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(1, [[4, 1/4], [5, 3/4]])","explanation_zh":"現代橋梁核對=(1, [[4, 1/4], [5, 3/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236030,"parameters":{"paths":[[4,4,5],[4,5,5],[5,5,4]],"masses":[[1,4],[1,4],[1,2]],"time":1,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[4,4,5],[4,5,5],[5,5,4]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0031","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[5,5,6],[5,6,6],[6,6,5]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":2}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(2, [[5, 1/2], [6, 1/2]])","explanation_zh":"現代橋梁核對=(2, [[5, 1/2], [6, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236031,"parameters":{"paths":[[5,5,6],[5,6,6],[6,6,5]],"masses":[[1,4],[1,4],[1,2]],"time":2,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[5,5,6],[5,6,6],[6,6,5]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0032","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[6,6,7],[6,7,7],[7,7,6]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, [[6, 1/2], [7, 1/2]])","explanation_zh":"現代橋梁核對=(0, [[6, 1/2], [7, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236032,"parameters":{"paths":[[6,6,7],[6,7,7],[7,7,6]],"masses":[[1,4],[1,4],[1,2]],"time":0,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[6,6,7],[6,7,7],[7,7,6]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0033","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[7,7,8],[7,8,8],[8,8,7]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":1}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(1, [[7, 1/4], [8, 3/4]])","explanation_zh":"現代橋梁核對=(1, [[7, 1/4], [8, 3/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236033,"parameters":{"paths":[[7,7,8],[7,8,8],[8,8,7]],"masses":[[1,4],[1,4],[1,2]],"time":1,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[7,7,8],[7,8,8],[8,8,7]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0034","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[8,8,9],[8,9,9],[9,9,8]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":2}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(2, [[8, 1/2], [9, 1/2]])","explanation_zh":"現代橋梁核對=(2, [[8, 1/2], [9, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236034,"parameters":{"paths":[[8,8,9],[8,9,9],[9,9,8]],"masses":[[1,4],[1,4],[1,2]],"time":2,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[8,8,9],[8,9,9],[9,9,8]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0035","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[9,9,10],[9,10,10],[10,10,9]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, [[9, 1/2], [10, 1/2]])","explanation_zh":"現代橋梁核對=(0, [[9, 1/2], [10, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236035,"parameters":{"paths":[[9,9,10],[9,10,10],[10,10,9]],"masses":[[1,4],[1,4],[1,2]],"time":0,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[10,10,9],[9,10,10],[9,9,10]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0036","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[10,10,11],[10,11,11],[11,11,10]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":1}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(1, [[10, 1/4], [11, 3/4]])","explanation_zh":"現代橋梁核對=(1, [[10, 1/4], [11, 3/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236036,"parameters":{"paths":[[10,10,11],[10,11,11],[11,11,10]],"masses":[[1,4],[1,4],[1,2]],"time":1,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[10,10,11],[10,11,11],[11,11,10]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0037","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[11,11,12],[11,12,12],[12,12,11]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":2}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(2, [[11, 1/2], [12, 1/2]])","explanation_zh":"現代橋梁核對=(2, [[11, 1/2], [12, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236037,"parameters":{"paths":[[11,11,12],[11,12,12],[12,12,11]],"masses":[[1,4],[1,4],[1,2]],"time":2,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[11,11,12],[11,12,12],[12,12,11]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0038","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[12,12,13],[12,13,13],[13,13,12]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, [[12, 1/2], [13, 1/2]])","explanation_zh":"現代橋梁核對=(0, [[12, 1/2], [13, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236038,"parameters":{"paths":[[12,12,13],[12,13,13],[13,13,12]],"masses":[[1,4],[1,4],[1,2]],"time":0,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[12,12,13],[12,13,13],[13,13,12]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0039","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[13,13,14],[13,14,14],[14,14,13]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":1}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(1, [[13, 1/4], [14, 3/4]])","explanation_zh":"現代橋梁核對=(1, [[13, 1/4], [14, 3/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236039,"parameters":{"paths":[[13,13,14],[13,14,14],[14,14,13]],"masses":[[1,4],[1,4],[1,2]],"time":1,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[13,13,14],[13,14,14],[14,14,13]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0040","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[14,14,15],[14,15,15],[15,15,14]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":2}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(2, [[14, 1/2], [15, 1/2]])","explanation_zh":"現代橋梁核對=(2, [[14, 1/2], [15, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236040,"parameters":{"paths":[[14,14,15],[14,15,15],[15,15,14]],"masses":[[1,4],[1,4],[1,2]],"time":2,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[14,14,15],[14,15,15],[15,15,14]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0041","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[15,15,16],[15,16,16],[16,16,15]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, [[15, 1/2], [16, 1/2]])","explanation_zh":"現代橋梁核對=(0, [[15, 1/2], [16, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236041,"parameters":{"paths":[[15,15,16],[15,16,16],[16,16,15]],"masses":[[1,4],[1,4],[1,2]],"time":0,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[15,15,16],[15,16,16],[16,16,15]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0042","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[16,16,17],[16,17,17],[17,17,16]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":1}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(1, [[16, 1/4], [17, 3/4]])","explanation_zh":"現代橋梁核對=(1, [[16, 1/4], [17, 3/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236042,"parameters":{"paths":[[16,16,17],[16,17,17],[17,17,16]],"masses":[[1,4],[1,4],[1,2]],"time":1,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[16,16,17],[16,17,17],[17,17,16]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0043","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[17,17,18],[17,18,18],[18,18,17]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":2}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(2, [[17, 1/2], [18, 1/2]])","explanation_zh":"現代橋梁核對=(2, [[17, 1/2], [18, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236043,"parameters":{"paths":[[17,17,18],[17,18,18],[18,18,17]],"masses":[[1,4],[1,4],[1,2]],"time":2,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[17,17,18],[17,18,18],[18,18,17]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0044","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[18,18,19],[18,19,19],[19,19,18]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, [[18, 1/2], [19, 1/2]])","explanation_zh":"現代橋梁核對=(0, [[18, 1/2], [19, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236044,"parameters":{"paths":[[18,18,19],[18,19,19],[19,19,18]],"masses":[[1,4],[1,4],[1,2]],"time":0,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[18,18,19],[18,19,19],[19,19,18]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0045","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[19,19,20],[19,20,20],[20,20,19]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":1}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(1, [[19, 1/4], [20, 3/4]])","explanation_zh":"現代橋梁核對=(1, [[19, 1/4], [20, 3/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236045,"parameters":{"paths":[[19,19,20],[19,20,20],[20,20,19]],"masses":[[1,4],[1,4],[1,2]],"time":1,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[19,19,20],[19,20,20],[20,20,19]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0046","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[20,20,21],[20,21,21],[21,21,20]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":2}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(2, [[20, 1/2], [21, 1/2]])","explanation_zh":"現代橋梁核對=(2, [[20, 1/2], [21, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236046,"parameters":{"paths":[[20,20,21],[20,21,21],[21,21,20]],"masses":[[1,4],[1,4],[1,2]],"time":2,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[20,20,21],[20,21,21],[21,21,20]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0047","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[21,21,22],[21,22,22],[22,22,21]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, [[21, 1/2], [22, 1/2]])","explanation_zh":"現代橋梁核對=(0, [[21, 1/2], [22, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236047,"parameters":{"paths":[[21,21,22],[21,22,22],[22,22,21]],"masses":[[1,4],[1,4],[1,2]],"time":0,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[21,21,22],[21,22,22],[22,22,21]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0048","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[22,22,23],[22,23,23],[23,23,22]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":1}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(1, [[22, 1/4], [23, 3/4]])","explanation_zh":"現代橋梁核對=(1, [[22, 1/4], [23, 3/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236048,"parameters":{"paths":[[22,22,23],[22,23,23],[23,23,22]],"masses":[[1,4],[1,4],[1,2]],"time":1,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[22,22,23],[22,23,23],[23,23,22]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0049","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[23,23,24],[23,24,24],[24,24,23]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":2}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(2, [[23, 1/2], [24, 1/2]])","explanation_zh":"現代橋梁核對=(2, [[23, 1/2], [24, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236049,"parameters":{"paths":[[23,23,24],[23,24,24],[24,24,23]],"masses":[[1,4],[1,4],[1,2]],"time":2,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[23,23,24],[23,24,24],[24,24,23]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0050","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-marginal；資料={\"paths\":[[24,24,25],[24,25,25],[25,25,24]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0}。請依 [time,marginal_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, [[24, 1/2], [25, 1/2]])","explanation_zh":"現代橋梁核對=(0, [[24, 1/2], [25, 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-marginal","seed":202610236050,"parameters":{"paths":[[24,24,25],[24,25,25],[25,25,24]],"masses":[[1,4],[1,4],[1,2]],"time":0,"domain":"Finite_horizon_joint_path_law","operation":"process-time-marginal","semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[24,24,25],[24,25,25],[25,25,24]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0051","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[0,0,1],[0,1,1],[1,1,0]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0,\"event\":[1]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(0, [1], 1/2)","explanation_zh":"現代橋梁核對=(0, [1], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236051,"parameters":{"paths":[[0,0,1],[0,1,1],[1,1,0]],"masses":[[1,4],[1,4],[1,2]],"time":0,"event":[1],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[1],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[0,0,1],[0,1,1],[1,1,0]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0052","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[1,1,2],[1,2,2],[2,2,1]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":1,\"event\":[2]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(1, [2], 3/4)","explanation_zh":"現代橋梁核對=(1, [2], 3/4)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236052,"parameters":{"paths":[[1,1,2],[1,2,2],[2,2,1]],"masses":[[1,4],[1,4],[1,2]],"time":1,"event":[2],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[2],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[1,1,2],[1,2,2],[2,2,1]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0053","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[2,2,3],[2,3,3],[3,3,2]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":2,\"event\":[3]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(2, [3], 1/2)","explanation_zh":"現代橋梁核對=(2, [3], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236053,"parameters":{"paths":[[2,2,3],[2,3,3],[3,3,2]],"masses":[[1,4],[1,4],[1,2]],"time":2,"event":[3],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[3],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[2,2,3],[2,3,3],[3,3,2]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0054","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[3,3,4],[3,4,4],[4,4,3]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0,\"event\":[4]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(0, [4], 1/2)","explanation_zh":"現代橋梁核對=(0, [4], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236054,"parameters":{"paths":[[3,3,4],[3,4,4],[4,4,3]],"masses":[[1,4],[1,4],[1,2]],"time":0,"event":[4],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[4],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[3,3,4],[3,4,4],[4,4,3]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0055","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[4,4,5],[4,5,5],[5,5,4]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":1,\"event\":[5]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(1, [5], 3/4)","explanation_zh":"現代橋梁核對=(1, [5], 3/4)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236055,"parameters":{"paths":[[4,4,5],[4,5,5],[5,5,4]],"masses":[[1,4],[1,4],[1,2]],"time":1,"event":[5],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[5],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[4,4,5],[4,5,5],[5,5,4]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0056","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[5,5,6],[5,6,6],[6,6,5]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":2,\"event\":[6]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(2, [6], 1/2)","explanation_zh":"現代橋梁核對=(2, [6], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236056,"parameters":{"paths":[[5,5,6],[5,6,6],[6,6,5]],"masses":[[1,4],[1,4],[1,2]],"time":2,"event":[6],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[6],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[5,5,6],[5,6,6],[6,6,5]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0057","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[6,6,7],[6,7,7],[7,7,6]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0,\"event\":[7]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(0, [7], 1/2)","explanation_zh":"現代橋梁核對=(0, [7], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236057,"parameters":{"paths":[[6,6,7],[6,7,7],[7,7,6]],"masses":[[1,4],[1,4],[1,2]],"time":0,"event":[7],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[7],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[6,6,7],[6,7,7],[7,7,6]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0058","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[7,7,8],[7,8,8],[8,8,7]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":1,\"event\":[8]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(1, [8], 3/4)","explanation_zh":"現代橋梁核對=(1, [8], 3/4)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236058,"parameters":{"paths":[[7,7,8],[7,8,8],[8,8,7]],"masses":[[1,4],[1,4],[1,2]],"time":1,"event":[8],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[8],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[7,7,8],[7,8,8],[8,8,7]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0059","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[8,8,9],[8,9,9],[9,9,8]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":2,\"event\":[9]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(2, [9], 1/2)","explanation_zh":"現代橋梁核對=(2, [9], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236059,"parameters":{"paths":[[8,8,9],[8,9,9],[9,9,8]],"masses":[[1,4],[1,4],[1,2]],"time":2,"event":[9],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[9],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[8,8,9],[8,9,9],[9,9,8]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0060","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[9,9,10],[9,10,10],[10,10,9]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0,\"event\":[10]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(0, [10], 1/2)","explanation_zh":"現代橋梁核對=(0, [10], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236060,"parameters":{"paths":[[9,9,10],[9,10,10],[10,10,9]],"masses":[[1,4],[1,4],[1,2]],"time":0,"event":[10],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[10],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[10,10,9],[9,10,10],[9,9,10]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0061","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[10,10,11],[10,11,11],[11,11,10]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":1,\"event\":[11]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(1, [11], 3/4)","explanation_zh":"現代橋梁核對=(1, [11], 3/4)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236061,"parameters":{"paths":[[10,10,11],[10,11,11],[11,11,10]],"masses":[[1,4],[1,4],[1,2]],"time":1,"event":[11],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[11],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[10,10,11],[10,11,11],[11,11,10]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0062","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[11,11,12],[11,12,12],[12,12,11]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":2,\"event\":[12]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(2, [12], 1/2)","explanation_zh":"現代橋梁核對=(2, [12], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236062,"parameters":{"paths":[[11,11,12],[11,12,12],[12,12,11]],"masses":[[1,4],[1,4],[1,2]],"time":2,"event":[12],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[12],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[11,11,12],[11,12,12],[12,12,11]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0063","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[12,12,13],[12,13,13],[13,13,12]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0,\"event\":[13]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(0, [13], 1/2)","explanation_zh":"現代橋梁核對=(0, [13], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236063,"parameters":{"paths":[[12,12,13],[12,13,13],[13,13,12]],"masses":[[1,4],[1,4],[1,2]],"time":0,"event":[13],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[13],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[12,12,13],[12,13,13],[13,13,12]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0064","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[13,13,14],[13,14,14],[14,14,13]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":1,\"event\":[14]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(1, [14], 3/4)","explanation_zh":"現代橋梁核對=(1, [14], 3/4)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236064,"parameters":{"paths":[[13,13,14],[13,14,14],[14,14,13]],"masses":[[1,4],[1,4],[1,2]],"time":1,"event":[14],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[14],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[13,13,14],[13,14,14],[14,14,13]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0065","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[14,14,15],[14,15,15],[15,15,14]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":2,\"event\":[15]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(2, [15], 1/2)","explanation_zh":"現代橋梁核對=(2, [15], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236065,"parameters":{"paths":[[14,14,15],[14,15,15],[15,15,14]],"masses":[[1,4],[1,4],[1,2]],"time":2,"event":[15],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[15],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[14,14,15],[14,15,15],[15,15,14]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0066","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[15,15,16],[15,16,16],[16,16,15]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0,\"event\":[16]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(0, [16], 1/2)","explanation_zh":"現代橋梁核對=(0, [16], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236066,"parameters":{"paths":[[15,15,16],[15,16,16],[16,16,15]],"masses":[[1,4],[1,4],[1,2]],"time":0,"event":[16],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[16],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[15,15,16],[15,16,16],[16,16,15]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0067","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[16,16,17],[16,17,17],[17,17,16]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":1,\"event\":[17]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(1, [17], 3/4)","explanation_zh":"現代橋梁核對=(1, [17], 3/4)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236067,"parameters":{"paths":[[16,16,17],[16,17,17],[17,17,16]],"masses":[[1,4],[1,4],[1,2]],"time":1,"event":[17],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[17],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[16,16,17],[16,17,17],[17,17,16]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0068","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[17,17,18],[17,18,18],[18,18,17]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":2,\"event\":[18]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(2, [18], 1/2)","explanation_zh":"現代橋梁核對=(2, [18], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236068,"parameters":{"paths":[[17,17,18],[17,18,18],[18,18,17]],"masses":[[1,4],[1,4],[1,2]],"time":2,"event":[18],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[18],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[17,17,18],[17,18,18],[18,18,17]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0069","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[18,18,19],[18,19,19],[19,19,18]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0,\"event\":[19]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(0, [19], 1/2)","explanation_zh":"現代橋梁核對=(0, [19], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236069,"parameters":{"paths":[[18,18,19],[18,19,19],[19,19,18]],"masses":[[1,4],[1,4],[1,2]],"time":0,"event":[19],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[19],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[18,18,19],[18,19,19],[19,19,18]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0070","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[19,19,20],[19,20,20],[20,20,19]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":1,\"event\":[20]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(1, [20], 3/4)","explanation_zh":"現代橋梁核對=(1, [20], 3/4)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236070,"parameters":{"paths":[[19,19,20],[19,20,20],[20,20,19]],"masses":[[1,4],[1,4],[1,2]],"time":1,"event":[20],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[20],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[19,19,20],[19,20,20],[20,20,19]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0071","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[20,20,21],[20,21,21],[21,21,20]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":2,\"event\":[21]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(2, [21], 1/2)","explanation_zh":"現代橋梁核對=(2, [21], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236071,"parameters":{"paths":[[20,20,21],[20,21,21],[21,21,20]],"masses":[[1,4],[1,4],[1,2]],"time":2,"event":[21],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[21],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[20,20,21],[20,21,21],[21,21,20]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0072","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[21,21,22],[21,22,22],[22,22,21]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0,\"event\":[22]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(0, [22], 1/2)","explanation_zh":"現代橋梁核對=(0, [22], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236072,"parameters":{"paths":[[21,21,22],[21,22,22],[22,22,21]],"masses":[[1,4],[1,4],[1,2]],"time":0,"event":[22],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[22],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[21,21,22],[21,22,22],[22,22,21]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0073","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[22,22,23],[22,23,23],[23,23,22]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":1,\"event\":[23]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(1, [23], 3/4)","explanation_zh":"現代橋梁核對=(1, [23], 3/4)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236073,"parameters":{"paths":[[22,22,23],[22,23,23],[23,23,22]],"masses":[[1,4],[1,4],[1,2]],"time":1,"event":[23],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[23],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[22,22,23],[22,23,23],[23,23,22]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0074","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[23,23,24],[23,24,24],[24,24,23]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":2,\"event\":[24]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(2, [24], 1/2)","explanation_zh":"現代橋梁核對=(2, [24], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236074,"parameters":{"paths":[[23,23,24],[23,24,24],[24,24,23]],"masses":[[1,4],[1,4],[1,2]],"time":2,"event":[24],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[24],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[23,23,24],[23,24,24],[24,24,23]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0075","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-time-event-probability；資料={\"paths\":[[24,24,25],[24,25,25],[25,25,24]],\"masses\":[[1,4],[1,4],[1,2]],\"time\":0,\"event\":[25]}。請依 [time,event,probability] 欄序回答。","answer":{"kind":"expression","value":"(0, [25], 1/2)","explanation_zh":"現代橋梁核對=(0, [25], 1/2)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-time-event-probability","seed":202610236075,"parameters":{"paths":[[24,24,25],[24,25,25],[25,25,24]],"masses":[[1,4],[1,4],[1,2]],"time":0,"event":[25],"domain":"Finite_horizon_joint_path_law","operation":"process-time-event-probability","semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[25],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[24,24,25],[24,25,25],[25,25,24]],\"time\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0076","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[0,0,1],[0,1,1],[1,1,0]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":0,\"time_b\":1}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [[[0, 0], 1/4], [[0, 1], 1/4], [[1, 1], 1/2]])","explanation_zh":"現代橋梁核對=(0, 1, [[[0, 0], 1/4], [[0, 1], 1/4], [[1, 1], 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236076,"parameters":{"paths":[[0,0,1],[0,1,1],[1,1,0]],"masses":[[1,4],[1,4],[1,2]],"time_a":0,"time_b":1,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[0,0,1],[0,1,1],[1,1,0]],\"time_a\":0,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0077","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[1,1,2],[1,2,2],[2,2,1]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":1,\"time_b\":2}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, [[[1, 2], 1/4], [[2, 1], 1/2], [[2, 2], 1/4]])","explanation_zh":"現代橋梁核對=(1, 2, [[[1, 2], 1/4], [[2, 1], 1/2], [[2, 2], 1/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236077,"parameters":{"paths":[[1,1,2],[1,2,2],[2,2,1]],"masses":[[1,4],[1,4],[1,2]],"time_a":1,"time_b":2,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[1,1,2],[1,2,2],[2,2,1]],\"time_a\":1,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0078","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[2,2,3],[2,3,3],[3,3,2]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":2,\"time_b\":0}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [[[2, 3], 1/2], [[3, 2], 1/2]])","explanation_zh":"現代橋梁核對=(2, 0, [[[2, 3], 1/2], [[3, 2], 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236078,"parameters":{"paths":[[2,2,3],[2,3,3],[3,3,2]],"masses":[[1,4],[1,4],[1,2]],"time_a":2,"time_b":0,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[2,2,3],[2,3,3],[3,3,2]],\"time_a\":2,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0079","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[3,3,4],[3,4,4],[4,4,3]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":0,\"time_b\":1}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [[[3, 3], 1/4], [[3, 4], 1/4], [[4, 4], 1/2]])","explanation_zh":"現代橋梁核對=(0, 1, [[[3, 3], 1/4], [[3, 4], 1/4], [[4, 4], 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236079,"parameters":{"paths":[[3,3,4],[3,4,4],[4,4,3]],"masses":[[1,4],[1,4],[1,2]],"time_a":0,"time_b":1,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[3,3,4],[3,4,4],[4,4,3]],\"time_a\":0,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0080","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[4,4,5],[4,5,5],[5,5,4]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":1,\"time_b\":2}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, [[[4, 5], 1/4], [[5, 4], 1/2], [[5, 5], 1/4]])","explanation_zh":"現代橋梁核對=(1, 2, [[[4, 5], 1/4], [[5, 4], 1/2], [[5, 5], 1/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236080,"parameters":{"paths":[[4,4,5],[4,5,5],[5,5,4]],"masses":[[1,4],[1,4],[1,2]],"time_a":1,"time_b":2,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[4,4,5],[4,5,5],[5,5,4]],\"time_a\":1,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0081","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[5,5,6],[5,6,6],[6,6,5]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":2,\"time_b\":0}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [[[5, 6], 1/2], [[6, 5], 1/2]])","explanation_zh":"現代橋梁核對=(2, 0, [[[5, 6], 1/2], [[6, 5], 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236081,"parameters":{"paths":[[5,5,6],[5,6,6],[6,6,5]],"masses":[[1,4],[1,4],[1,2]],"time_a":2,"time_b":0,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[5,5,6],[5,6,6],[6,6,5]],\"time_a\":2,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0082","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[6,6,7],[6,7,7],[7,7,6]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":0,\"time_b\":1}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [[[6, 6], 1/4], [[6, 7], 1/4], [[7, 7], 1/2]])","explanation_zh":"現代橋梁核對=(0, 1, [[[6, 6], 1/4], [[6, 7], 1/4], [[7, 7], 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236082,"parameters":{"paths":[[6,6,7],[6,7,7],[7,7,6]],"masses":[[1,4],[1,4],[1,2]],"time_a":0,"time_b":1,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[6,6,7],[6,7,7],[7,7,6]],\"time_a\":0,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0083","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[7,7,8],[7,8,8],[8,8,7]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":1,\"time_b\":2}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, [[[7, 8], 1/4], [[8, 7], 1/2], [[8, 8], 1/4]])","explanation_zh":"現代橋梁核對=(1, 2, [[[7, 8], 1/4], [[8, 7], 1/2], [[8, 8], 1/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236083,"parameters":{"paths":[[7,7,8],[7,8,8],[8,8,7]],"masses":[[1,4],[1,4],[1,2]],"time_a":1,"time_b":2,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[7,7,8],[7,8,8],[8,8,7]],\"time_a\":1,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0084","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[8,8,9],[8,9,9],[9,9,8]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":2,\"time_b\":0}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [[[8, 9], 1/2], [[9, 8], 1/2]])","explanation_zh":"現代橋梁核對=(2, 0, [[[8, 9], 1/2], [[9, 8], 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236084,"parameters":{"paths":[[8,8,9],[8,9,9],[9,9,8]],"masses":[[1,4],[1,4],[1,2]],"time_a":2,"time_b":0,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[8,8,9],[8,9,9],[9,9,8]],\"time_a\":2,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0085","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[9,9,10],[9,10,10],[10,10,9]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":0,\"time_b\":1}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [[[10, 10], 1/2], [[9, 10], 1/4], [[9, 9], 1/4]])","explanation_zh":"現代橋梁核對=(0, 1, [[[10, 10], 1/2], [[9, 10], 1/4], [[9, 9], 1/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236085,"parameters":{"paths":[[9,9,10],[9,10,10],[10,10,9]],"masses":[[1,4],[1,4],[1,2]],"time_a":0,"time_b":1,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[10,10,9],[9,10,10],[9,9,10]],\"time_a\":0,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0086","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[10,10,11],[10,11,11],[11,11,10]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":1,\"time_b\":2}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, [[[10, 11], 1/4], [[11, 10], 1/2], [[11, 11], 1/4]])","explanation_zh":"現代橋梁核對=(1, 2, [[[10, 11], 1/4], [[11, 10], 1/2], [[11, 11], 1/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236086,"parameters":{"paths":[[10,10,11],[10,11,11],[11,11,10]],"masses":[[1,4],[1,4],[1,2]],"time_a":1,"time_b":2,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[10,10,11],[10,11,11],[11,11,10]],\"time_a\":1,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0087","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[11,11,12],[11,12,12],[12,12,11]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":2,\"time_b\":0}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [[[11, 12], 1/2], [[12, 11], 1/2]])","explanation_zh":"現代橋梁核對=(2, 0, [[[11, 12], 1/2], [[12, 11], 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236087,"parameters":{"paths":[[11,11,12],[11,12,12],[12,12,11]],"masses":[[1,4],[1,4],[1,2]],"time_a":2,"time_b":0,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[11,11,12],[11,12,12],[12,12,11]],\"time_a\":2,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0088","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[12,12,13],[12,13,13],[13,13,12]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":0,\"time_b\":1}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [[[12, 12], 1/4], [[12, 13], 1/4], [[13, 13], 1/2]])","explanation_zh":"現代橋梁核對=(0, 1, [[[12, 12], 1/4], [[12, 13], 1/4], [[13, 13], 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236088,"parameters":{"paths":[[12,12,13],[12,13,13],[13,13,12]],"masses":[[1,4],[1,4],[1,2]],"time_a":0,"time_b":1,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[12,12,13],[12,13,13],[13,13,12]],\"time_a\":0,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0089","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[13,13,14],[13,14,14],[14,14,13]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":1,\"time_b\":2}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, [[[13, 14], 1/4], [[14, 13], 1/2], [[14, 14], 1/4]])","explanation_zh":"現代橋梁核對=(1, 2, [[[13, 14], 1/4], [[14, 13], 1/2], [[14, 14], 1/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236089,"parameters":{"paths":[[13,13,14],[13,14,14],[14,14,13]],"masses":[[1,4],[1,4],[1,2]],"time_a":1,"time_b":2,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[13,13,14],[13,14,14],[14,14,13]],\"time_a\":1,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0090","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[14,14,15],[14,15,15],[15,15,14]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":2,\"time_b\":0}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [[[14, 15], 1/2], [[15, 14], 1/2]])","explanation_zh":"現代橋梁核對=(2, 0, [[[14, 15], 1/2], [[15, 14], 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236090,"parameters":{"paths":[[14,14,15],[14,15,15],[15,15,14]],"masses":[[1,4],[1,4],[1,2]],"time_a":2,"time_b":0,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[14,14,15],[14,15,15],[15,15,14]],\"time_a\":2,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0091","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[15,15,16],[15,16,16],[16,16,15]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":0,\"time_b\":1}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [[[15, 15], 1/4], [[15, 16], 1/4], [[16, 16], 1/2]])","explanation_zh":"現代橋梁核對=(0, 1, [[[15, 15], 1/4], [[15, 16], 1/4], [[16, 16], 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236091,"parameters":{"paths":[[15,15,16],[15,16,16],[16,16,15]],"masses":[[1,4],[1,4],[1,2]],"time_a":0,"time_b":1,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[15,15,16],[15,16,16],[16,16,15]],\"time_a\":0,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0092","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[16,16,17],[16,17,17],[17,17,16]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":1,\"time_b\":2}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, [[[16, 17], 1/4], [[17, 16], 1/2], [[17, 17], 1/4]])","explanation_zh":"現代橋梁核對=(1, 2, [[[16, 17], 1/4], [[17, 16], 1/2], [[17, 17], 1/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236092,"parameters":{"paths":[[16,16,17],[16,17,17],[17,17,16]],"masses":[[1,4],[1,4],[1,2]],"time_a":1,"time_b":2,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[16,16,17],[16,17,17],[17,17,16]],\"time_a\":1,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0093","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[17,17,18],[17,18,18],[18,18,17]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":2,\"time_b\":0}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [[[17, 18], 1/2], [[18, 17], 1/2]])","explanation_zh":"現代橋梁核對=(2, 0, [[[17, 18], 1/2], [[18, 17], 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236093,"parameters":{"paths":[[17,17,18],[17,18,18],[18,18,17]],"masses":[[1,4],[1,4],[1,2]],"time_a":2,"time_b":0,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[17,17,18],[17,18,18],[18,18,17]],\"time_a\":2,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0094","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[18,18,19],[18,19,19],[19,19,18]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":0,\"time_b\":1}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [[[18, 18], 1/4], [[18, 19], 1/4], [[19, 19], 1/2]])","explanation_zh":"現代橋梁核對=(0, 1, [[[18, 18], 1/4], [[18, 19], 1/4], [[19, 19], 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236094,"parameters":{"paths":[[18,18,19],[18,19,19],[19,19,18]],"masses":[[1,4],[1,4],[1,2]],"time_a":0,"time_b":1,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[18,18,19],[18,19,19],[19,19,18]],\"time_a\":0,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0095","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[19,19,20],[19,20,20],[20,20,19]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":1,\"time_b\":2}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, [[[19, 20], 1/4], [[20, 19], 1/2], [[20, 20], 1/4]])","explanation_zh":"現代橋梁核對=(1, 2, [[[19, 20], 1/4], [[20, 19], 1/2], [[20, 20], 1/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236095,"parameters":{"paths":[[19,19,20],[19,20,20],[20,20,19]],"masses":[[1,4],[1,4],[1,2]],"time_a":1,"time_b":2,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[19,19,20],[19,20,20],[20,20,19]],\"time_a\":1,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0096","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[20,20,21],[20,21,21],[21,21,20]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":2,\"time_b\":0}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [[[20, 21], 1/2], [[21, 20], 1/2]])","explanation_zh":"現代橋梁核對=(2, 0, [[[20, 21], 1/2], [[21, 20], 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236096,"parameters":{"paths":[[20,20,21],[20,21,21],[21,21,20]],"masses":[[1,4],[1,4],[1,2]],"time_a":2,"time_b":0,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[20,20,21],[20,21,21],[21,21,20]],\"time_a\":2,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0097","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[21,21,22],[21,22,22],[22,22,21]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":0,\"time_b\":1}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [[[21, 21], 1/4], [[21, 22], 1/4], [[22, 22], 1/2]])","explanation_zh":"現代橋梁核對=(0, 1, [[[21, 21], 1/4], [[21, 22], 1/4], [[22, 22], 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236097,"parameters":{"paths":[[21,21,22],[21,22,22],[22,22,21]],"masses":[[1,4],[1,4],[1,2]],"time_a":0,"time_b":1,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[21,21,22],[21,22,22],[22,22,21]],\"time_a\":0,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0098","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[22,22,23],[22,23,23],[23,23,22]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":1,\"time_b\":2}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(1, 2, [[[22, 23], 1/4], [[23, 22], 1/2], [[23, 23], 1/4]])","explanation_zh":"現代橋梁核對=(1, 2, [[[22, 23], 1/4], [[23, 22], 1/2], [[23, 23], 1/4]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236098,"parameters":{"paths":[[22,22,23],[22,23,23],[23,23,22]],"masses":[[1,4],[1,4],[1,2]],"time_a":1,"time_b":2,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[22,22,23],[22,23,23],[23,23,22]],\"time_a\":1,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0099","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[23,23,24],[23,24,24],[24,24,23]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":2,\"time_b\":0}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, [[[23, 24], 1/2], [[24, 23], 1/2]])","explanation_zh":"現代橋梁核對=(2, 0, [[[23, 24], 1/2], [[24, 23], 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236099,"parameters":{"paths":[[23,23,24],[23,24,24],[24,24,23]],"masses":[[1,4],[1,4],[1,2]],"time_a":2,"time_b":0,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[23,23,24],[23,24,24],[24,24,23]],\"time_a\":2,\"time_b\":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、Markov／可微聲明、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-20261023-r06-discrete-stochastic-process-0100","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-stochastic-process"],"title_zh":"有限時域joint process","stem_zh":"共同有限機率空間上的完整聯合路徑律。操作=process-two-time-joint；資料={\"paths\":[[24,24,25],[24,25,25],[25,25,24]],\"masses\":[[1,4],[1,4],[1,2]],\"time_a\":0,\"time_b\":1}。請依 [time_a,time_b,joint_rows] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [[[24, 24], 1/4], [[24, 25], 1/4], [[25, 25], 1/2]])","explanation_zh":"現代橋梁核對=(0, 1, [[[24, 24], 1/4], [[24, 25], 1/4], [[25, 25], 1/2]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；marginal不決定joint或Markov性；有限時域不宣稱stationarity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限時域joint process"]},"variant":{"family_id":"qf-mod23-process-two-time-joint","seed":202610236100,"parameters":{"paths":[[24,24,25],[24,25,25],[25,25,24]],"masses":[[1,4],[1,4],[1,2]],"time_a":0,"time_b":1,"domain":"Finite_horizon_joint_path_law","operation":"process-two-time-joint","semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[24,24,25],[24,25,25],[25,25,24]],\"time_a\":0,\"time_b\":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、Markov／可微聲明、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"}}