{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0001","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]],\"steps\":1}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 1, [[1/5, 4/5], [1/4, 3/4]], [19/80, 61/80])","explanation_zh":"收束波核對=([1/4, 3/4], 1, [[1/5, 4/5], [1/4, 3/4]], [19/80, 61/80])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256001,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[1,5],[4,5]],[[1,4],[3,4]]],"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,4],[3,4]],\"steps\":1,\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0002","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[2,5],[3,5]],[[1,4],[3,4]]],\"steps\":2}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 2, [[31/100, 69/100], [23/80, 57/80]], [469/1600, 1131/1600])","explanation_zh":"收束波核對=([1/4, 3/4], 2, [[31/100, 69/100], [23/80, 57/80]], [469/1600, 1131/1600])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256002,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[2,5],[3,5]],[[1,4],[3,4]]],"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,4],[3,4]],\"steps\":2,\"transition\":[[[2,5],[3,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0003","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[3,5],[2,5]],[[1,4],[3,4]]],\"steps\":3}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 3, [[411/1000, 589/1000], [589/1600, 1011/1600]], [12123/32000, 19877/32000])","explanation_zh":"收束波核對=([1/4, 3/4], 3, [[411/1000, 589/1000], [589/1600, 1011/1600]], [12123/32000, 19877/32000])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256003,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[3,5],[2,5]],[[1,4],[3,4]]],"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,4],[3,4]],\"steps\":3,\"transition\":[[[3,5],[2,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0004","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[4,5],[1,5]],[[1,4],[3,4]]],\"steps\":1}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 1, [[4/5, 1/5], [1/4, 3/4]], [31/80, 49/80])","explanation_zh":"收束波核對=([1/4, 3/4], 1, [[4/5, 1/5], [1/4, 3/4]], [31/80, 49/80])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256004,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[4,5],[1,5]],[[1,4],[3,4]]],"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,4],[3,4]],\"steps\":1,\"transition\":[[[4,5],[1,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0005","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]],\"steps\":2}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 2, [[11/25, 14/25], [7/20, 13/20]], [149/400, 251/400])","explanation_zh":"收束波核對=([1/4, 3/4], 2, [[11/25, 14/25], [7/20, 13/20]], [149/400, 251/400])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256005,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[1,5],[4,5]],[[1,2],[1,2]]],"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,4],[3,4]],\"steps\":2,\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0006","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[2,5],[3,5]],[[1,2],[1,2]]],\"steps\":3}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 3, [[227/500, 273/500], [91/200, 109/200]], [1819/4000, 2181/4000])","explanation_zh":"收束波核對=([1/4, 3/4], 3, [[227/500, 273/500], [91/200, 109/200]], [1819/4000, 2181/4000])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256006,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[2,5],[3,5]],[[1,2],[1,2]]],"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,4],[3,4]],\"steps\":3,\"transition\":[[[2,5],[3,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0007","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[3,5],[2,5]],[[1,2],[1,2]]],\"steps\":1}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 1, [[3/5, 2/5], [1/2, 1/2]], [21/40, 19/40])","explanation_zh":"收束波核對=([1/4, 3/4], 1, [[3/5, 2/5], [1/2, 1/2]], [21/40, 19/40])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256007,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[3,5],[2,5]],[[1,2],[1,2]]],"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,4],[3,4]],\"steps\":1,\"transition\":[[[3,5],[2,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0008","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[4,5],[1,5]],[[1,2],[1,2]]],\"steps\":2}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 2, [[37/50, 13/50], [13/20, 7/20]], [269/400, 131/400])","explanation_zh":"收束波核對=([1/4, 3/4], 2, [[37/50, 13/50], [13/20, 7/20]], [269/400, 131/400])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256008,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[4,5],[1,5]],[[1,2],[1,2]]],"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,4],[3,4]],\"steps\":2,\"transition\":[[[4,5],[1,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0009","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[1,5],[4,5]],[[3,4],[1,4]]],\"steps\":3}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 3, [[199/500, 301/500], [903/1600, 697/1600]], [16729/32000, 15271/32000])","explanation_zh":"收束波核對=([1/4, 3/4], 3, [[199/500, 301/500], [903/1600, 697/1600]], [16729/32000, 15271/32000])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256009,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[1,5],[4,5]],[[3,4],[1,4]]],"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,4],[3,4]],\"steps\":3,\"transition\":[[[1,5],[4,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0010","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[2,5],[3,5]],[[3,4],[1,4]]],\"steps\":1}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 1, [[2/5, 3/5], [3/4, 1/4]], [53/80, 27/80])","explanation_zh":"收束波核對=([1/4, 3/4], 1, [[2/5, 3/5], [3/4, 1/4]], [53/80, 27/80])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256010,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[2,5],[3,5]],[[3,4],[1,4]]],"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,4],[3,4]],\"steps\":1,\"transition\":[[[2,5],[3,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0011","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[3,5],[2,5]],[[3,4],[1,4]]],\"steps\":2}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 2, [[33/50, 17/50], [51/80, 29/80]], [1029/1600, 571/1600])","explanation_zh":"收束波核對=([1/4, 3/4], 2, [[33/50, 17/50], [51/80, 29/80]], [1029/1600, 571/1600])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256011,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[3,5],[2,5]],[[3,4],[1,4]]],"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,4],[3,4]],\"steps\":2,\"transition\":[[[3,5],[2,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0012","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[4,5],[1,5]],[[3,4],[1,4]]],\"steps\":3}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 3, [[1579/2000, 421/2000], [1263/1600, 337/1600]], [25261/32000, 6739/32000])","explanation_zh":"收束波核對=([1/4, 3/4], 3, [[1579/2000, 421/2000], [1263/1600, 337/1600]], [25261/32000, 6739/32000])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256012,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[4,5],[1,5]],[[3,4],[1,4]]],"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,4],[3,4]],\"steps\":3,\"transition\":[[[4,5],[1,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0013","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]],\"steps\":1}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 1, [[1/5, 4/5], [1/4, 3/4]], [9/40, 31/40])","explanation_zh":"收束波核對=([1/2, 1/2], 1, [[1/5, 4/5], [1/4, 3/4]], [9/40, 31/40])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256013,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[1,5],[4,5]],[[1,4],[3,4]]],"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,2],[1,2]],\"steps\":1,\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0014","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[2,5],[3,5]],[[1,4],[3,4]]],\"steps\":2}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 2, [[31/100, 69/100], [23/80, 57/80]], [239/800, 561/800])","explanation_zh":"收束波核對=([1/2, 1/2], 2, [[31/100, 69/100], [23/80, 57/80]], [239/800, 561/800])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256014,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[2,5],[3,5]],[[1,4],[3,4]]],"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,2],[1,2]],\"steps\":2,\"transition\":[[[2,5],[3,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0015","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[3,5],[2,5]],[[1,4],[3,4]]],\"steps\":3}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 3, [[411/1000, 589/1000], [589/1600, 1011/1600]], [6233/16000, 9767/16000])","explanation_zh":"收束波核對=([1/2, 1/2], 3, [[411/1000, 589/1000], [589/1600, 1011/1600]], [6233/16000, 9767/16000])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256015,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[3,5],[2,5]],[[1,4],[3,4]]],"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,2],[1,2]],\"steps\":3,\"transition\":[[[3,5],[2,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0016","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[4,5],[1,5]],[[1,4],[3,4]]],\"steps\":1}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 1, [[4/5, 1/5], [1/4, 3/4]], [21/40, 19/40])","explanation_zh":"收束波核對=([1/2, 1/2], 1, [[4/5, 1/5], [1/4, 3/4]], [21/40, 19/40])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256016,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[4,5],[1,5]],[[1,4],[3,4]]],"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,2],[1,2]],\"steps\":1,\"transition\":[[[4,5],[1,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0017","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]],\"steps\":2}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 2, [[11/25, 14/25], [7/20, 13/20]], [79/200, 121/200])","explanation_zh":"收束波核對=([1/2, 1/2], 2, [[11/25, 14/25], [7/20, 13/20]], [79/200, 121/200])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256017,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[1,5],[4,5]],[[1,2],[1,2]]],"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,2],[1,2]],\"steps\":2,\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0018","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[2,5],[3,5]],[[1,2],[1,2]]],\"steps\":3}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 3, [[227/500, 273/500], [91/200, 109/200]], [909/2000, 1091/2000])","explanation_zh":"收束波核對=([1/2, 1/2], 3, [[227/500, 273/500], [91/200, 109/200]], [909/2000, 1091/2000])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256018,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[2,5],[3,5]],[[1,2],[1,2]]],"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,2],[1,2]],\"steps\":3,\"transition\":[[[2,5],[3,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0019","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[3,5],[2,5]],[[1,2],[1,2]]],\"steps\":1}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 1, [[3/5, 2/5], [1/2, 1/2]], [11/20, 9/20])","explanation_zh":"收束波核對=([1/2, 1/2], 1, [[3/5, 2/5], [1/2, 1/2]], [11/20, 9/20])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256019,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[3,5],[2,5]],[[1,2],[1,2]]],"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,2],[1,2]],\"steps\":1,\"transition\":[[[3,5],[2,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0020","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[4,5],[1,5]],[[1,2],[1,2]]],\"steps\":2}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 2, [[37/50, 13/50], [13/20, 7/20]], [139/200, 61/200])","explanation_zh":"收束波核對=([1/2, 1/2], 2, [[37/50, 13/50], [13/20, 7/20]], [139/200, 61/200])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256020,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[4,5],[1,5]],[[1,2],[1,2]]],"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,2],[1,2]],\"steps\":2,\"transition\":[[[4,5],[1,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0021","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[1,5],[4,5]],[[3,4],[1,4]]],\"steps\":3}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 3, [[199/500, 301/500], [903/1600, 697/1600]], [7699/16000, 8301/16000])","explanation_zh":"收束波核對=([1/2, 1/2], 3, [[199/500, 301/500], [903/1600, 697/1600]], [7699/16000, 8301/16000])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256021,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[1,5],[4,5]],[[3,4],[1,4]]],"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,2],[1,2]],\"steps\":3,\"transition\":[[[1,5],[4,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0022","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[2,5],[3,5]],[[3,4],[1,4]]],\"steps\":1}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 1, [[2/5, 3/5], [3/4, 1/4]], [23/40, 17/40])","explanation_zh":"收束波核對=([1/2, 1/2], 1, [[2/5, 3/5], [3/4, 1/4]], [23/40, 17/40])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256022,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[2,5],[3,5]],[[3,4],[1,4]]],"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,2],[1,2]],\"steps\":1,\"transition\":[[[2,5],[3,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0023","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[3,5],[2,5]],[[3,4],[1,4]]],\"steps\":2}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 2, [[33/50, 17/50], [51/80, 29/80]], [519/800, 281/800])","explanation_zh":"收束波核對=([1/2, 1/2], 2, [[33/50, 17/50], [51/80, 29/80]], [519/800, 281/800])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256023,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[3,5],[2,5]],[[3,4],[1,4]]],"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,2],[1,2]],\"steps\":2,\"transition\":[[[3,5],[2,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0024","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[4,5],[1,5]],[[3,4],[1,4]]],\"steps\":3}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 3, [[1579/2000, 421/2000], [1263/1600, 337/1600]], [12631/16000, 3369/16000])","explanation_zh":"收束波核對=([1/2, 1/2], 3, [[1579/2000, 421/2000], [1263/1600, 337/1600]], [12631/16000, 3369/16000])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256024,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[4,5],[1,5]],[[3,4],[1,4]]],"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[1,2],[1,2]],\"steps\":3,\"transition\":[[[4,5],[1,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0025","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-law；資料={\"initial\":[[3,4],[1,4]],\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]],\"steps\":1}。請依 [initial,steps,power_matrix,law] 欄序回答。","answer":{"kind":"expression","value":"([3/4, 1/4], 1, [[1/5, 4/5], [1/4, 3/4]], [17/80, 63/80])","explanation_zh":"收束波核對=([3/4, 1/4], 1, [[1/5, 4/5], [1/4, 3/4]], [17/80, 63/80])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-law","seed":202610256025,"parameters":{"initial":[[3,4],[1,4]],"transition":[[[1,5],[4,5]],[[1,4],[3,4]]],"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-law","semantic_template":"qf-mod25-markovdeep-n-step-law/{\"initial\":[[3,4],[1,4]],\"steps\":1,\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0026","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]],\"from\":0,\"to\":1,\"steps\":1}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 1, [[1/5, 4/5], [1/4, 3/4]], 4/5)","explanation_zh":"收束波核對=(0, 1, 1, [[1/5, 4/5], [1/4, 3/4]], 4/5)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256026,"parameters":{"transition":[[[1,5],[4,5]],[[1,4],[3,4]]],"from":0,"to":1,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":0,\"steps\":1,\"to\":1,\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0027","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[2,5],[3,5]],[[1,4],[3,4]]],\"from\":1,\"to\":0,\"steps\":1}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 1, [[2/5, 3/5], [1/4, 3/4]], 1/4)","explanation_zh":"收束波核對=(1, 0, 1, [[2/5, 3/5], [1/4, 3/4]], 1/4)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256027,"parameters":{"transition":[[[2,5],[3,5]],[[1,4],[3,4]]],"from":1,"to":0,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":1,\"steps\":1,\"to\":0,\"transition\":[[[2,5],[3,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0028","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[3,5],[2,5]],[[1,4],[3,4]]],\"from\":0,\"to\":1,\"steps\":1}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 1, [[3/5, 2/5], [1/4, 3/4]], 2/5)","explanation_zh":"收束波核對=(0, 1, 1, [[3/5, 2/5], [1/4, 3/4]], 2/5)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256028,"parameters":{"transition":[[[3,5],[2,5]],[[1,4],[3,4]]],"from":0,"to":1,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":0,\"steps\":1,\"to\":1,\"transition\":[[[3,5],[2,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0029","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[4,5],[1,5]],[[1,4],[3,4]]],\"from\":1,\"to\":0,\"steps\":1}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 1, [[4/5, 1/5], [1/4, 3/4]], 1/4)","explanation_zh":"收束波核對=(1, 0, 1, [[4/5, 1/5], [1/4, 3/4]], 1/4)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256029,"parameters":{"transition":[[[4,5],[1,5]],[[1,4],[3,4]]],"from":1,"to":0,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":1,\"steps\":1,\"to\":0,\"transition\":[[[4,5],[1,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0030","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]],\"from\":0,\"to\":1,\"steps\":1}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 1, [[1/5, 4/5], [1/2, 1/2]], 4/5)","explanation_zh":"收束波核對=(0, 1, 1, [[1/5, 4/5], [1/2, 1/2]], 4/5)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256030,"parameters":{"transition":[[[1,5],[4,5]],[[1,2],[1,2]]],"from":0,"to":1,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":0,\"steps\":1,\"to\":1,\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0031","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[2,5],[3,5]],[[1,2],[1,2]]],\"from\":1,\"to\":0,\"steps\":1}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 1, [[2/5, 3/5], [1/2, 1/2]], 1/2)","explanation_zh":"收束波核對=(1, 0, 1, [[2/5, 3/5], [1/2, 1/2]], 1/2)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256031,"parameters":{"transition":[[[2,5],[3,5]],[[1,2],[1,2]]],"from":1,"to":0,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":1,\"steps\":1,\"to\":0,\"transition\":[[[2,5],[3,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0032","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[3,5],[2,5]],[[1,2],[1,2]]],\"from\":0,\"to\":1,\"steps\":1}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 1, [[3/5, 2/5], [1/2, 1/2]], 2/5)","explanation_zh":"收束波核對=(0, 1, 1, [[3/5, 2/5], [1/2, 1/2]], 2/5)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256032,"parameters":{"transition":[[[3,5],[2,5]],[[1,2],[1,2]]],"from":0,"to":1,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":0,\"steps\":1,\"to\":1,\"transition\":[[[3,5],[2,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0033","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[4,5],[1,5]],[[1,2],[1,2]]],\"from\":1,\"to\":0,\"steps\":1}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 1, [[4/5, 1/5], [1/2, 1/2]], 1/2)","explanation_zh":"收束波核對=(1, 0, 1, [[4/5, 1/5], [1/2, 1/2]], 1/2)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256033,"parameters":{"transition":[[[4,5],[1,5]],[[1,2],[1,2]]],"from":1,"to":0,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":1,\"steps\":1,\"to\":0,\"transition\":[[[4,5],[1,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0034","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[1,5],[4,5]],[[3,4],[1,4]]],\"from\":0,\"to\":1,\"steps\":1}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 1, [[1/5, 4/5], [3/4, 1/4]], 4/5)","explanation_zh":"收束波核對=(0, 1, 1, [[1/5, 4/5], [3/4, 1/4]], 4/5)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256034,"parameters":{"transition":[[[1,5],[4,5]],[[3,4],[1,4]]],"from":0,"to":1,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":0,\"steps\":1,\"to\":1,\"transition\":[[[1,5],[4,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0035","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[2,5],[3,5]],[[3,4],[1,4]]],\"from\":1,\"to\":0,\"steps\":1}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 1, [[2/5, 3/5], [3/4, 1/4]], 3/4)","explanation_zh":"收束波核對=(1, 0, 1, [[2/5, 3/5], [3/4, 1/4]], 3/4)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256035,"parameters":{"transition":[[[2,5],[3,5]],[[3,4],[1,4]]],"from":1,"to":0,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":1,\"steps\":1,\"to\":0,\"transition\":[[[2,5],[3,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0036","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[3,5],[2,5]],[[3,4],[1,4]]],\"from\":0,\"to\":1,\"steps\":1}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 1, [[3/5, 2/5], [3/4, 1/4]], 2/5)","explanation_zh":"收束波核對=(0, 1, 1, [[3/5, 2/5], [3/4, 1/4]], 2/5)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256036,"parameters":{"transition":[[[3,5],[2,5]],[[3,4],[1,4]]],"from":0,"to":1,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":0,\"steps\":1,\"to\":1,\"transition\":[[[3,5],[2,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0037","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[4,5],[1,5]],[[3,4],[1,4]]],\"from\":1,\"to\":0,\"steps\":1}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 1, [[4/5, 1/5], [3/4, 1/4]], 3/4)","explanation_zh":"收束波核對=(1, 0, 1, [[4/5, 1/5], [3/4, 1/4]], 3/4)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256037,"parameters":{"transition":[[[4,5],[1,5]],[[3,4],[1,4]]],"from":1,"to":0,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":1,\"steps\":1,\"to\":0,\"transition\":[[[4,5],[1,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0038","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]],\"from\":0,\"to\":1,\"steps\":2}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 2, [[6/25, 19/25], [19/80, 61/80]], 19/25)","explanation_zh":"收束波核對=(0, 1, 2, [[6/25, 19/25], [19/80, 61/80]], 19/25)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256038,"parameters":{"transition":[[[1,5],[4,5]],[[1,4],[3,4]]],"from":0,"to":1,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":0,\"steps\":2,\"to\":1,\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0039","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[2,5],[3,5]],[[1,4],[3,4]]],\"from\":1,\"to\":0,\"steps\":2}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 2, [[31/100, 69/100], [23/80, 57/80]], 23/80)","explanation_zh":"收束波核對=(1, 0, 2, [[31/100, 69/100], [23/80, 57/80]], 23/80)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256039,"parameters":{"transition":[[[2,5],[3,5]],[[1,4],[3,4]]],"from":1,"to":0,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":1,\"steps\":2,\"to\":0,\"transition\":[[[2,5],[3,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0040","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[3,5],[2,5]],[[1,4],[3,4]]],\"from\":0,\"to\":1,\"steps\":2}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 2, [[23/50, 27/50], [27/80, 53/80]], 27/50)","explanation_zh":"收束波核對=(0, 1, 2, [[23/50, 27/50], [27/80, 53/80]], 27/50)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256040,"parameters":{"transition":[[[3,5],[2,5]],[[1,4],[3,4]]],"from":0,"to":1,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":0,\"steps\":2,\"to\":1,\"transition\":[[[3,5],[2,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0041","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[4,5],[1,5]],[[1,4],[3,4]]],\"from\":1,\"to\":0,\"steps\":2}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 2, [[69/100, 31/100], [31/80, 49/80]], 31/80)","explanation_zh":"收束波核對=(1, 0, 2, [[69/100, 31/100], [31/80, 49/80]], 31/80)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256041,"parameters":{"transition":[[[4,5],[1,5]],[[1,4],[3,4]]],"from":1,"to":0,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":1,\"steps\":2,\"to\":0,\"transition\":[[[4,5],[1,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0042","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]],\"from\":0,\"to\":1,\"steps\":2}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 2, [[11/25, 14/25], [7/20, 13/20]], 14/25)","explanation_zh":"收束波核對=(0, 1, 2, [[11/25, 14/25], [7/20, 13/20]], 14/25)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256042,"parameters":{"transition":[[[1,5],[4,5]],[[1,2],[1,2]]],"from":0,"to":1,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":0,\"steps\":2,\"to\":1,\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0043","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[2,5],[3,5]],[[1,2],[1,2]]],\"from\":1,\"to\":0,\"steps\":2}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 2, [[23/50, 27/50], [9/20, 11/20]], 9/20)","explanation_zh":"收束波核對=(1, 0, 2, [[23/50, 27/50], [9/20, 11/20]], 9/20)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256043,"parameters":{"transition":[[[2,5],[3,5]],[[1,2],[1,2]]],"from":1,"to":0,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":1,\"steps\":2,\"to\":0,\"transition\":[[[2,5],[3,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0044","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[3,5],[2,5]],[[1,2],[1,2]]],\"from\":0,\"to\":1,\"steps\":2}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 2, [[14/25, 11/25], [11/20, 9/20]], 11/25)","explanation_zh":"收束波核對=(0, 1, 2, [[14/25, 11/25], [11/20, 9/20]], 11/25)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256044,"parameters":{"transition":[[[3,5],[2,5]],[[1,2],[1,2]]],"from":0,"to":1,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":0,\"steps\":2,\"to\":1,\"transition\":[[[3,5],[2,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0045","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[4,5],[1,5]],[[1,2],[1,2]]],\"from\":1,\"to\":0,\"steps\":2}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 2, [[37/50, 13/50], [13/20, 7/20]], 13/20)","explanation_zh":"收束波核對=(1, 0, 2, [[37/50, 13/50], [13/20, 7/20]], 13/20)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256045,"parameters":{"transition":[[[4,5],[1,5]],[[1,2],[1,2]]],"from":1,"to":0,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":1,\"steps\":2,\"to\":0,\"transition\":[[[4,5],[1,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0046","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[1,5],[4,5]],[[3,4],[1,4]]],\"from\":0,\"to\":1,\"steps\":2}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 2, [[16/25, 9/25], [27/80, 53/80]], 9/25)","explanation_zh":"收束波核對=(0, 1, 2, [[16/25, 9/25], [27/80, 53/80]], 9/25)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256046,"parameters":{"transition":[[[1,5],[4,5]],[[3,4],[1,4]]],"from":0,"to":1,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":0,\"steps\":2,\"to\":1,\"transition\":[[[1,5],[4,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0047","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[2,5],[3,5]],[[3,4],[1,4]]],\"from\":1,\"to\":0,\"steps\":2}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 2, [[61/100, 39/100], [39/80, 41/80]], 39/80)","explanation_zh":"收束波核對=(1, 0, 2, [[61/100, 39/100], [39/80, 41/80]], 39/80)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256047,"parameters":{"transition":[[[2,5],[3,5]],[[3,4],[1,4]]],"from":1,"to":0,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":1,\"steps\":2,\"to\":0,\"transition\":[[[2,5],[3,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0048","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[3,5],[2,5]],[[3,4],[1,4]]],\"from\":0,\"to\":1,\"steps\":2}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 2, [[33/50, 17/50], [51/80, 29/80]], 17/50)","explanation_zh":"收束波核對=(0, 1, 2, [[33/50, 17/50], [51/80, 29/80]], 17/50)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256048,"parameters":{"transition":[[[3,5],[2,5]],[[3,4],[1,4]]],"from":0,"to":1,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":0,\"steps\":2,\"to\":1,\"transition\":[[[3,5],[2,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0049","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[4,5],[1,5]],[[3,4],[1,4]]],\"from\":1,\"to\":0,\"steps\":2}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 2, [[79/100, 21/100], [63/80, 17/80]], 63/80)","explanation_zh":"收束波核對=(1, 0, 2, [[79/100, 21/100], [63/80, 17/80]], 63/80)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256049,"parameters":{"transition":[[[4,5],[1,5]],[[3,4],[1,4]]],"from":1,"to":0,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":1,\"steps\":2,\"to\":0,\"transition\":[[[4,5],[1,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0050","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-n-step-entry；資料={\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]],\"from\":0,\"to\":1,\"steps\":3}。請依 [from,to,steps,power_matrix,entry] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 3, [[119/500, 381/500], [381/1600, 1219/1600]], 381/500)","explanation_zh":"收束波核對=(0, 1, 3, [[119/500, 381/500], [381/1600, 1219/1600]], 381/500)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-n-step-entry","seed":202610256050,"parameters":{"transition":[[[1,5],[4,5]],[[1,4],[3,4]]],"from":0,"to":1,"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-n-step-entry","semantic_template":"qf-mod25-markovdeep-n-step-entry/{\"from\":0,\"steps\":3,\"to\":1,\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0051","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]],\"from\":0,\"to\":1,\"steps\":1}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 0, 1, 1, 1/4, 4/5, 1/5)","explanation_zh":"收束波核對=([1/4, 3/4], 0, 1, 1, 1/4, 4/5, 1/5)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256051,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[1,5],[4,5]],[[1,4],[3,4]]],"from":0,"to":1,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":0,\"initial\":[[1,4],[3,4]],\"steps\":1,\"to\":1,\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0052","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[2,5],[3,5]],[[1,4],[3,4]]],\"from\":1,\"to\":0,\"steps\":2}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 1, 0, 2, 3/4, 23/80, 69/320)","explanation_zh":"收束波核對=([1/4, 3/4], 1, 0, 2, 3/4, 23/80, 69/320)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256052,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[2,5],[3,5]],[[1,4],[3,4]]],"from":1,"to":0,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":1,\"initial\":[[1,4],[3,4]],\"steps\":2,\"to\":0,\"transition\":[[[2,5],[3,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0053","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[3,5],[2,5]],[[1,4],[3,4]]],\"from\":0,\"to\":1,\"steps\":3}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 0, 1, 3, 1/4, 589/1000, 589/4000)","explanation_zh":"收束波核對=([1/4, 3/4], 0, 1, 3, 1/4, 589/1000, 589/4000)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256053,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[3,5],[2,5]],[[1,4],[3,4]]],"from":0,"to":1,"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":0,\"initial\":[[1,4],[3,4]],\"steps\":3,\"to\":1,\"transition\":[[[3,5],[2,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0054","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[4,5],[1,5]],[[1,4],[3,4]]],\"from\":1,\"to\":0,\"steps\":1}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 1, 0, 1, 3/4, 1/4, 3/16)","explanation_zh":"收束波核對=([1/4, 3/4], 1, 0, 1, 3/4, 1/4, 3/16)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256054,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[4,5],[1,5]],[[1,4],[3,4]]],"from":1,"to":0,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":1,\"initial\":[[1,4],[3,4]],\"steps\":1,\"to\":0,\"transition\":[[[4,5],[1,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0055","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]],\"from\":0,\"to\":1,\"steps\":2}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 0, 1, 2, 1/4, 14/25, 7/50)","explanation_zh":"收束波核對=([1/4, 3/4], 0, 1, 2, 1/4, 14/25, 7/50)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256055,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[1,5],[4,5]],[[1,2],[1,2]]],"from":0,"to":1,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":0,\"initial\":[[1,4],[3,4]],\"steps\":2,\"to\":1,\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0056","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[2,5],[3,5]],[[1,2],[1,2]]],\"from\":1,\"to\":0,\"steps\":3}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 1, 0, 3, 3/4, 91/200, 273/800)","explanation_zh":"收束波核對=([1/4, 3/4], 1, 0, 3, 3/4, 91/200, 273/800)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256056,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[2,5],[3,5]],[[1,2],[1,2]]],"from":1,"to":0,"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":1,\"initial\":[[1,4],[3,4]],\"steps\":3,\"to\":0,\"transition\":[[[2,5],[3,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0057","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[3,5],[2,5]],[[1,2],[1,2]]],\"from\":0,\"to\":1,\"steps\":1}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 0, 1, 1, 1/4, 2/5, 1/10)","explanation_zh":"收束波核對=([1/4, 3/4], 0, 1, 1, 1/4, 2/5, 1/10)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256057,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[3,5],[2,5]],[[1,2],[1,2]]],"from":0,"to":1,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":0,\"initial\":[[1,4],[3,4]],\"steps\":1,\"to\":1,\"transition\":[[[3,5],[2,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0058","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[4,5],[1,5]],[[1,2],[1,2]]],\"from\":1,\"to\":0,\"steps\":2}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 1, 0, 2, 3/4, 13/20, 39/80)","explanation_zh":"收束波核對=([1/4, 3/4], 1, 0, 2, 3/4, 13/20, 39/80)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256058,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[4,5],[1,5]],[[1,2],[1,2]]],"from":1,"to":0,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":1,\"initial\":[[1,4],[3,4]],\"steps\":2,\"to\":0,\"transition\":[[[4,5],[1,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0059","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[1,5],[4,5]],[[3,4],[1,4]]],\"from\":0,\"to\":1,\"steps\":3}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 0, 1, 3, 1/4, 301/500, 301/2000)","explanation_zh":"收束波核對=([1/4, 3/4], 0, 1, 3, 1/4, 301/500, 301/2000)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256059,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[1,5],[4,5]],[[3,4],[1,4]]],"from":0,"to":1,"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":0,\"initial\":[[1,4],[3,4]],\"steps\":3,\"to\":1,\"transition\":[[[1,5],[4,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0060","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[2,5],[3,5]],[[3,4],[1,4]]],\"from\":1,\"to\":0,\"steps\":1}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 1, 0, 1, 3/4, 3/4, 9/16)","explanation_zh":"收束波核對=([1/4, 3/4], 1, 0, 1, 3/4, 3/4, 9/16)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256060,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[2,5],[3,5]],[[3,4],[1,4]]],"from":1,"to":0,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":1,\"initial\":[[1,4],[3,4]],\"steps\":1,\"to\":0,\"transition\":[[[2,5],[3,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0061","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[3,5],[2,5]],[[3,4],[1,4]]],\"from\":0,\"to\":1,\"steps\":2}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 0, 1, 2, 1/4, 17/50, 17/200)","explanation_zh":"收束波核對=([1/4, 3/4], 0, 1, 2, 1/4, 17/50, 17/200)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256061,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[3,5],[2,5]],[[3,4],[1,4]]],"from":0,"to":1,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":0,\"initial\":[[1,4],[3,4]],\"steps\":2,\"to\":1,\"transition\":[[[3,5],[2,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0062","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,4],[3,4]],\"transition\":[[[4,5],[1,5]],[[3,4],[1,4]]],\"from\":1,\"to\":0,\"steps\":3}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], 1, 0, 3, 3/4, 1263/1600, 3789/6400)","explanation_zh":"收束波核對=([1/4, 3/4], 1, 0, 3, 3/4, 1263/1600, 3789/6400)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256062,"parameters":{"initial":[[1,4],[3,4]],"transition":[[[4,5],[1,5]],[[3,4],[1,4]]],"from":1,"to":0,"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":1,\"initial\":[[1,4],[3,4]],\"steps\":3,\"to\":0,\"transition\":[[[4,5],[1,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0063","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]],\"from\":0,\"to\":1,\"steps\":1}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 0, 1, 1, 1/2, 4/5, 2/5)","explanation_zh":"收束波核對=([1/2, 1/2], 0, 1, 1, 1/2, 4/5, 2/5)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256063,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[1,5],[4,5]],[[1,4],[3,4]]],"from":0,"to":1,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":0,\"initial\":[[1,2],[1,2]],\"steps\":1,\"to\":1,\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0064","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[2,5],[3,5]],[[1,4],[3,4]]],\"from\":1,\"to\":0,\"steps\":2}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 1, 0, 2, 1/2, 23/80, 23/160)","explanation_zh":"收束波核對=([1/2, 1/2], 1, 0, 2, 1/2, 23/80, 23/160)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256064,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[2,5],[3,5]],[[1,4],[3,4]]],"from":1,"to":0,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":1,\"initial\":[[1,2],[1,2]],\"steps\":2,\"to\":0,\"transition\":[[[2,5],[3,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0065","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[3,5],[2,5]],[[1,4],[3,4]]],\"from\":0,\"to\":1,\"steps\":3}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 0, 1, 3, 1/2, 589/1000, 589/2000)","explanation_zh":"收束波核對=([1/2, 1/2], 0, 1, 3, 1/2, 589/1000, 589/2000)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256065,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[3,5],[2,5]],[[1,4],[3,4]]],"from":0,"to":1,"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":0,\"initial\":[[1,2],[1,2]],\"steps\":3,\"to\":1,\"transition\":[[[3,5],[2,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0066","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[4,5],[1,5]],[[1,4],[3,4]]],\"from\":1,\"to\":0,\"steps\":1}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 1, 0, 1, 1/2, 1/4, 1/8)","explanation_zh":"收束波核對=([1/2, 1/2], 1, 0, 1, 1/2, 1/4, 1/8)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256066,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[4,5],[1,5]],[[1,4],[3,4]]],"from":1,"to":0,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":1,\"initial\":[[1,2],[1,2]],\"steps\":1,\"to\":0,\"transition\":[[[4,5],[1,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0067","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]],\"from\":0,\"to\":1,\"steps\":2}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 0, 1, 2, 1/2, 14/25, 7/25)","explanation_zh":"收束波核對=([1/2, 1/2], 0, 1, 2, 1/2, 14/25, 7/25)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256067,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[1,5],[4,5]],[[1,2],[1,2]]],"from":0,"to":1,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":0,\"initial\":[[1,2],[1,2]],\"steps\":2,\"to\":1,\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0068","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[2,5],[3,5]],[[1,2],[1,2]]],\"from\":1,\"to\":0,\"steps\":3}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 1, 0, 3, 1/2, 91/200, 91/400)","explanation_zh":"收束波核對=([1/2, 1/2], 1, 0, 3, 1/2, 91/200, 91/400)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256068,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[2,5],[3,5]],[[1,2],[1,2]]],"from":1,"to":0,"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":1,\"initial\":[[1,2],[1,2]],\"steps\":3,\"to\":0,\"transition\":[[[2,5],[3,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0069","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[3,5],[2,5]],[[1,2],[1,2]]],\"from\":0,\"to\":1,\"steps\":1}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 0, 1, 1, 1/2, 2/5, 1/5)","explanation_zh":"收束波核對=([1/2, 1/2], 0, 1, 1, 1/2, 2/5, 1/5)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256069,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[3,5],[2,5]],[[1,2],[1,2]]],"from":0,"to":1,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":0,\"initial\":[[1,2],[1,2]],\"steps\":1,\"to\":1,\"transition\":[[[3,5],[2,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0070","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[4,5],[1,5]],[[1,2],[1,2]]],\"from\":1,\"to\":0,\"steps\":2}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 1, 0, 2, 1/2, 13/20, 13/40)","explanation_zh":"收束波核對=([1/2, 1/2], 1, 0, 2, 1/2, 13/20, 13/40)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256070,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[4,5],[1,5]],[[1,2],[1,2]]],"from":1,"to":0,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":1,\"initial\":[[1,2],[1,2]],\"steps\":2,\"to\":0,\"transition\":[[[4,5],[1,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0071","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[1,5],[4,5]],[[3,4],[1,4]]],\"from\":0,\"to\":1,\"steps\":3}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 0, 1, 3, 1/2, 301/500, 301/1000)","explanation_zh":"收束波核對=([1/2, 1/2], 0, 1, 3, 1/2, 301/500, 301/1000)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256071,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[1,5],[4,5]],[[3,4],[1,4]]],"from":0,"to":1,"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":0,\"initial\":[[1,2],[1,2]],\"steps\":3,\"to\":1,\"transition\":[[[1,5],[4,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0072","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[2,5],[3,5]],[[3,4],[1,4]]],\"from\":1,\"to\":0,\"steps\":1}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 1, 0, 1, 1/2, 3/4, 3/8)","explanation_zh":"收束波核對=([1/2, 1/2], 1, 0, 1, 1/2, 3/4, 3/8)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256072,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[2,5],[3,5]],[[3,4],[1,4]]],"from":1,"to":0,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":1,\"initial\":[[1,2],[1,2]],\"steps\":1,\"to\":0,\"transition\":[[[2,5],[3,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0073","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[3,5],[2,5]],[[3,4],[1,4]]],\"from\":0,\"to\":1,\"steps\":2}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 0, 1, 2, 1/2, 17/50, 17/100)","explanation_zh":"收束波核對=([1/2, 1/2], 0, 1, 2, 1/2, 17/50, 17/100)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256073,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[3,5],[2,5]],[[3,4],[1,4]]],"from":0,"to":1,"steps":2,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":0,\"initial\":[[1,2],[1,2]],\"steps\":2,\"to\":1,\"transition\":[[[3,5],[2,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0074","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[4,5],[1,5]],[[3,4],[1,4]]],\"from\":1,\"to\":0,\"steps\":3}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], 1, 0, 3, 1/2, 1263/1600, 1263/3200)","explanation_zh":"收束波核對=([1/2, 1/2], 1, 0, 3, 1/2, 1263/1600, 1263/3200)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256074,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[4,5],[1,5]],[[3,4],[1,4]]],"from":1,"to":0,"steps":3,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":1,\"initial\":[[1,2],[1,2]],\"steps\":3,\"to\":0,\"transition\":[[[4,5],[1,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0075","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={\"initial\":[[3,4],[1,4]],\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]],\"from\":0,\"to\":1,\"steps\":1}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。","answer":{"kind":"expression","value":"([3/4, 1/4], 0, 1, 1, 3/4, 4/5, 3/5)","explanation_zh":"收束波核對=([3/4, 1/4], 0, 1, 1, 3/4, 4/5, 3/5)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-endpoint-joint-probability","seed":202610256075,"parameters":{"initial":[[3,4],[1,4]],"transition":[[[1,5],[4,5]],[[1,4],[3,4]]],"from":0,"to":1,"steps":1,"domain":"Finite_exact_Markov_density","operation":"markovdeep-endpoint-joint-probability","semantic_template":"qf-mod25-markovdeep-endpoint-joint-probability/{\"from\":0,\"initial\":[[3,4],[1,4]],\"steps\":1,\"to\":1,\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0076","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]],\"path\":[0,1,0]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [4/5, 1/4], 1/5, conditional_on_initial_state)","explanation_zh":"收束波核對=([0, 1, 0], [4/5, 1/4], 1/5, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256076,"parameters":{"transition":[[[1,5],[4,5]],[[1,4],[3,4]]],"path":[0,1,0],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[0,1,0],\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0077","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[2,5],[3,5]],[[1,4],[3,4]]],\"path\":[1,0,1]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [1/4, 3/5], 3/20, conditional_on_initial_state)","explanation_zh":"收束波核對=([1, 0, 1], [1/4, 3/5], 3/20, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256077,"parameters":{"transition":[[[2,5],[3,5]],[[1,4],[3,4]]],"path":[1,0,1],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[1,0,1],\"transition\":[[[2,5],[3,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0078","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[3,5],[2,5]],[[1,4],[3,4]]],\"path\":[0,1,0]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [2/5, 1/4], 1/10, conditional_on_initial_state)","explanation_zh":"收束波核對=([0, 1, 0], [2/5, 1/4], 1/10, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256078,"parameters":{"transition":[[[3,5],[2,5]],[[1,4],[3,4]]],"path":[0,1,0],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[0,1,0],\"transition\":[[[3,5],[2,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0079","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[4,5],[1,5]],[[1,4],[3,4]]],\"path\":[1,0,1]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [1/4, 1/5], 1/20, conditional_on_initial_state)","explanation_zh":"收束波核對=([1, 0, 1], [1/4, 1/5], 1/20, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256079,"parameters":{"transition":[[[4,5],[1,5]],[[1,4],[3,4]]],"path":[1,0,1],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[1,0,1],\"transition\":[[[4,5],[1,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0080","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]],\"path\":[0,1,0]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [4/5, 1/2], 2/5, conditional_on_initial_state)","explanation_zh":"收束波核對=([0, 1, 0], [4/5, 1/2], 2/5, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256080,"parameters":{"transition":[[[1,5],[4,5]],[[1,2],[1,2]]],"path":[0,1,0],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[0,1,0],\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0081","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[2,5],[3,5]],[[1,2],[1,2]]],\"path\":[1,0,1]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [1/2, 3/5], 3/10, conditional_on_initial_state)","explanation_zh":"收束波核對=([1, 0, 1], [1/2, 3/5], 3/10, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256081,"parameters":{"transition":[[[2,5],[3,5]],[[1,2],[1,2]]],"path":[1,0,1],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[1,0,1],\"transition\":[[[2,5],[3,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0082","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[3,5],[2,5]],[[1,2],[1,2]]],\"path\":[0,1,0]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [2/5, 1/2], 1/5, conditional_on_initial_state)","explanation_zh":"收束波核對=([0, 1, 0], [2/5, 1/2], 1/5, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256082,"parameters":{"transition":[[[3,5],[2,5]],[[1,2],[1,2]]],"path":[0,1,0],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[0,1,0],\"transition\":[[[3,5],[2,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0083","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[4,5],[1,5]],[[1,2],[1,2]]],\"path\":[1,0,1]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [1/2, 1/5], 1/10, conditional_on_initial_state)","explanation_zh":"收束波核對=([1, 0, 1], [1/2, 1/5], 1/10, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256083,"parameters":{"transition":[[[4,5],[1,5]],[[1,2],[1,2]]],"path":[1,0,1],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[1,0,1],\"transition\":[[[4,5],[1,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0084","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[1,5],[4,5]],[[3,4],[1,4]]],\"path\":[0,1,0]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [4/5, 3/4], 3/5, conditional_on_initial_state)","explanation_zh":"收束波核對=([0, 1, 0], [4/5, 3/4], 3/5, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256084,"parameters":{"transition":[[[1,5],[4,5]],[[3,4],[1,4]]],"path":[0,1,0],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[0,1,0],\"transition\":[[[1,5],[4,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0085","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[2,5],[3,5]],[[3,4],[1,4]]],\"path\":[1,0,1]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [3/4, 3/5], 9/20, conditional_on_initial_state)","explanation_zh":"收束波核對=([1, 0, 1], [3/4, 3/5], 9/20, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256085,"parameters":{"transition":[[[2,5],[3,5]],[[3,4],[1,4]]],"path":[1,0,1],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[1,0,1],\"transition\":[[[2,5],[3,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0086","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[3,5],[2,5]],[[3,4],[1,4]]],\"path\":[0,1,0]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [2/5, 3/4], 3/10, conditional_on_initial_state)","explanation_zh":"收束波核對=([0, 1, 0], [2/5, 3/4], 3/10, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256086,"parameters":{"transition":[[[3,5],[2,5]],[[3,4],[1,4]]],"path":[0,1,0],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[0,1,0],\"transition\":[[[3,5],[2,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0087","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[4,5],[1,5]],[[3,4],[1,4]]],\"path\":[1,0,1]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [3/4, 1/5], 3/20, conditional_on_initial_state)","explanation_zh":"收束波核對=([1, 0, 1], [3/4, 1/5], 3/20, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256087,"parameters":{"transition":[[[4,5],[1,5]],[[3,4],[1,4]]],"path":[1,0,1],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[1,0,1],\"transition\":[[[4,5],[1,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0088","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]],\"path\":[0,1,0,1]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1], [4/5, 1/4, 4/5], 4/25, conditional_on_initial_state)","explanation_zh":"收束波核對=([0, 1, 0, 1], [4/5, 1/4, 4/5], 4/25, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256088,"parameters":{"transition":[[[1,5],[4,5]],[[1,4],[3,4]]],"path":[0,1,0,1],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[0,1,0,1],\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0089","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[2,5],[3,5]],[[1,4],[3,4]]],\"path\":[1,0,1,0]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1, 0], [1/4, 3/5, 1/4], 3/80, conditional_on_initial_state)","explanation_zh":"收束波核對=([1, 0, 1, 0], [1/4, 3/5, 1/4], 3/80, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256089,"parameters":{"transition":[[[2,5],[3,5]],[[1,4],[3,4]]],"path":[1,0,1,0],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[1,0,1,0],\"transition\":[[[2,5],[3,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0090","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[3,5],[2,5]],[[1,4],[3,4]]],\"path\":[0,1,0,1]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1], [2/5, 1/4, 2/5], 1/25, conditional_on_initial_state)","explanation_zh":"收束波核對=([0, 1, 0, 1], [2/5, 1/4, 2/5], 1/25, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256090,"parameters":{"transition":[[[3,5],[2,5]],[[1,4],[3,4]]],"path":[0,1,0,1],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[0,1,0,1],\"transition\":[[[3,5],[2,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0091","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[4,5],[1,5]],[[1,4],[3,4]]],\"path\":[1,0,1,0]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1, 0], [1/4, 1/5, 1/4], 1/80, conditional_on_initial_state)","explanation_zh":"收束波核對=([1, 0, 1, 0], [1/4, 1/5, 1/4], 1/80, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256091,"parameters":{"transition":[[[4,5],[1,5]],[[1,4],[3,4]]],"path":[1,0,1,0],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[1,0,1,0],\"transition\":[[[4,5],[1,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0092","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]],\"path\":[0,1,0,1]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1], [4/5, 1/2, 4/5], 8/25, conditional_on_initial_state)","explanation_zh":"收束波核對=([0, 1, 0, 1], [4/5, 1/2, 4/5], 8/25, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256092,"parameters":{"transition":[[[1,5],[4,5]],[[1,2],[1,2]]],"path":[0,1,0,1],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[0,1,0,1],\"transition\":[[[1,5],[4,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0093","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[2,5],[3,5]],[[1,2],[1,2]]],\"path\":[1,0,1,0]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1, 0], [1/2, 3/5, 1/2], 3/20, conditional_on_initial_state)","explanation_zh":"收束波核對=([1, 0, 1, 0], [1/2, 3/5, 1/2], 3/20, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256093,"parameters":{"transition":[[[2,5],[3,5]],[[1,2],[1,2]]],"path":[1,0,1,0],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[1,0,1,0],\"transition\":[[[2,5],[3,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0094","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[3,5],[2,5]],[[1,2],[1,2]]],\"path\":[0,1,0,1]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1], [2/5, 1/2, 2/5], 2/25, conditional_on_initial_state)","explanation_zh":"收束波核對=([0, 1, 0, 1], [2/5, 1/2, 2/5], 2/25, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256094,"parameters":{"transition":[[[3,5],[2,5]],[[1,2],[1,2]]],"path":[0,1,0,1],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[0,1,0,1],\"transition\":[[[3,5],[2,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0095","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[4,5],[1,5]],[[1,2],[1,2]]],\"path\":[1,0,1,0]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1, 0], [1/2, 1/5, 1/2], 1/20, conditional_on_initial_state)","explanation_zh":"收束波核對=([1, 0, 1, 0], [1/2, 1/5, 1/2], 1/20, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256095,"parameters":{"transition":[[[4,5],[1,5]],[[1,2],[1,2]]],"path":[1,0,1,0],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[1,0,1,0],\"transition\":[[[4,5],[1,5]],[[1,2],[1,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0096","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[1,5],[4,5]],[[3,4],[1,4]]],\"path\":[0,1,0,1]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1], [4/5, 3/4, 4/5], 12/25, conditional_on_initial_state)","explanation_zh":"收束波核對=([0, 1, 0, 1], [4/5, 3/4, 4/5], 12/25, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256096,"parameters":{"transition":[[[1,5],[4,5]],[[3,4],[1,4]]],"path":[0,1,0,1],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[0,1,0,1],\"transition\":[[[1,5],[4,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0097","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[2,5],[3,5]],[[3,4],[1,4]]],\"path\":[1,0,1,0]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1, 0], [3/4, 3/5, 3/4], 27/80, conditional_on_initial_state)","explanation_zh":"收束波核對=([1, 0, 1, 0], [3/4, 3/5, 3/4], 27/80, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256097,"parameters":{"transition":[[[2,5],[3,5]],[[3,4],[1,4]]],"path":[1,0,1,0],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[1,0,1,0],\"transition\":[[[2,5],[3,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0098","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[3,5],[2,5]],[[3,4],[1,4]]],\"path\":[0,1,0,1]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1], [2/5, 3/4, 2/5], 3/25, conditional_on_initial_state)","explanation_zh":"收束波核對=([0, 1, 0, 1], [2/5, 3/4, 2/5], 3/25, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256098,"parameters":{"transition":[[[3,5],[2,5]],[[3,4],[1,4]]],"path":[0,1,0,1],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[0,1,0,1],\"transition\":[[[3,5],[2,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0099","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[4,5],[1,5]],[[3,4],[1,4]]],\"path\":[1,0,1,0]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1, 0], [3/4, 1/5, 3/4], 9/80, conditional_on_initial_state)","explanation_zh":"收束波核對=([1, 0, 1, 0], [3/4, 1/5, 3/4], 9/80, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256099,"parameters":{"transition":[[[4,5],[1,5]],[[3,4],[1,4]]],"path":[1,0,1,0],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[1,0,1,0],\"transition\":[[[4,5],[1,5]],[[3,4],[1,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r06-finite-markov-chain-0100","batch_id":"batch-20261013-advance-20261025-r06-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限Markov密度","stem_zh":"精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]],\"path\":[0,1,0,1,0]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0], [4/5, 1/4, 4/5, 1/4], 1/25, conditional_on_initial_state)","explanation_zh":"收束波核對=([0, 1, 0, 1, 0], [4/5, 1/4, 4/5, 1/4], 1/25, conditional_on_initial_state)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；只計有限步律或條件路徑；不推出平穩不可約遍歷或極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限Markov密度"]},"variant":{"family_id":"qf-mod25-markovdeep-conditional-path-probability","seed":202610256100,"parameters":{"transition":[[[1,5],[4,5]],[[1,4],[3,4]]],"path":[0,1,0,1,0],"domain":"Finite_exact_Markov_density","operation":"markovdeep-conditional-path-probability","semantic_template":"qf-mod25-markovdeep-conditional-path-probability/{\"path\":[0,1,0,1,0],\"transition\":[[[1,5],[4,5]],[[1,4],[3,4]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}