{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r03-finite-markov-chain-0001","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[1,3],[2,3]],[[1,2],[1,2]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[1/3, 2/3], [1/2, 1/2]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[1/3, 2/3], [1/2, 1/2]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243001,"parameters":{"transition":[[[1,3],[2,3]],[[1,2],[1,2]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[1,3],[2,3]],[[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、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0002","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[1,2],[1,2]],[[3,5],[2,5]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[1/2, 1/2], [3/5, 2/5]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[1/2, 1/2], [3/5, 2/5]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243002,"parameters":{"transition":[[[1,2],[1,2]],[[3,5],[2,5]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[1,2],[1,2]],[[3,5],[2,5]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0003","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[3,5],[2,5]],[[2,3],[1,3]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[3/5, 2/5], [2/3, 1/3]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[3/5, 2/5], [2/3, 1/3]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243003,"parameters":{"transition":[[[3,5],[2,5]],[[2,3],[1,3]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[3,5],[2,5]],[[2,3],[1,3]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0004","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[2,3],[1,3]],[[5,7],[2,7]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[2/3, 1/3], [5/7, 2/7]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[2/3, 1/3], [5/7, 2/7]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243004,"parameters":{"transition":[[[2,3],[1,3]],[[5,7],[2,7]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[2,3],[1,3]],[[5,7],[2,7]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0005","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[5,7],[2,7]],[[3,4],[1,4]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[5/7, 2/7], [3/4, 1/4]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[5/7, 2/7], [3/4, 1/4]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243005,"parameters":{"transition":[[[5,7],[2,7]],[[3,4],[1,4]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[5,7],[2,7]],[[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、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0006","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[3,4],[1,4]],[[7,9],[2,9]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[3/4, 1/4], [7/9, 2/9]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[3/4, 1/4], [7/9, 2/9]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243006,"parameters":{"transition":[[[3,4],[1,4]],[[7,9],[2,9]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[3,4],[1,4]],[[7,9],[2,9]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0007","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[7,9],[2,9]],[[4,5],[1,5]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[7/9, 2/9], [4/5, 1/5]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[7/9, 2/9], [4/5, 1/5]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243007,"parameters":{"transition":[[[7,9],[2,9]],[[4,5],[1,5]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[7,9],[2,9]],[[4,5],[1,5]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0008","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[4,5],[1,5]],[[9,11],[2,11]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[4/5, 1/5], [9/11, 2/11]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[4/5, 1/5], [9/11, 2/11]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243008,"parameters":{"transition":[[[4,5],[1,5]],[[9,11],[2,11]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[4,5],[1,5]],[[9,11],[2,11]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0009","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[9,11],[2,11]],[[5,6],[1,6]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[9/11, 2/11], [5/6, 1/6]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[9/11, 2/11], [5/6, 1/6]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243009,"parameters":{"transition":[[[9,11],[2,11]],[[5,6],[1,6]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[9,11],[2,11]],[[5,6],[1,6]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0010","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[5,6],[1,6]],[[11,13],[2,13]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[5/6, 1/6], [11/13, 2/13]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[5/6, 1/6], [11/13, 2/13]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243010,"parameters":{"transition":[[[5,6],[1,6]],[[11,13],[2,13]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[5,6],[1,6]],[[11,13],[2,13]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0011","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[11,13],[2,13]],[[6,7],[1,7]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[11/13, 2/13], [6/7, 1/7]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[11/13, 2/13], [6/7, 1/7]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243011,"parameters":{"transition":[[[11,13],[2,13]],[[6,7],[1,7]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[11,13],[2,13]],[[6,7],[1,7]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0012","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[6,7],[1,7]],[[13,15],[2,15]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[6/7, 1/7], [13/15, 2/15]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[6/7, 1/7], [13/15, 2/15]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243012,"parameters":{"transition":[[[6,7],[1,7]],[[13,15],[2,15]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[6,7],[1,7]],[[13,15],[2,15]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0013","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[13,15],[2,15]],[[7,8],[1,8]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[13/15, 2/15], [7/8, 1/8]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[13/15, 2/15], [7/8, 1/8]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243013,"parameters":{"transition":[[[13,15],[2,15]],[[7,8],[1,8]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[13,15],[2,15]],[[7,8],[1,8]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0014","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[7,8],[1,8]],[[15,17],[2,17]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[7/8, 1/8], [15/17, 2/17]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[7/8, 1/8], [15/17, 2/17]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243014,"parameters":{"transition":[[[7,8],[1,8]],[[15,17],[2,17]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[7,8],[1,8]],[[15,17],[2,17]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0015","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[15,17],[2,17]],[[8,9],[1,9]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[15/17, 2/17], [8/9, 1/9]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[15/17, 2/17], [8/9, 1/9]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243015,"parameters":{"transition":[[[15,17],[2,17]],[[8,9],[1,9]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[15,17],[2,17]],[[8,9],[1,9]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0016","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[8,9],[1,9]],[[17,19],[2,19]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[8/9, 1/9], [17/19, 2/19]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[8/9, 1/9], [17/19, 2/19]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243016,"parameters":{"transition":[[[8,9],[1,9]],[[17,19],[2,19]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[8,9],[1,9]],[[17,19],[2,19]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0017","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[17,19],[2,19]],[[9,10],[1,10]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[17/19, 2/19], [9/10, 1/10]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[17/19, 2/19], [9/10, 1/10]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243017,"parameters":{"transition":[[[17,19],[2,19]],[[9,10],[1,10]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[17,19],[2,19]],[[9,10],[1,10]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0018","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[9,10],[1,10]],[[19,21],[2,21]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[9/10, 1/10], [19/21, 2/21]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[9/10, 1/10], [19/21, 2/21]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243018,"parameters":{"transition":[[[9,10],[1,10]],[[19,21],[2,21]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[9,10],[1,10]],[[19,21],[2,21]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0019","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[19,21],[2,21]],[[10,11],[1,11]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[19/21, 2/21], [10/11, 1/11]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[19/21, 2/21], [10/11, 1/11]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243019,"parameters":{"transition":[[[19,21],[2,21]],[[10,11],[1,11]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[19,21],[2,21]],[[10,11],[1,11]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0020","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[10,11],[1,11]],[[21,23],[2,23]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[10/11, 1/11], [21/23, 2/23]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[10/11, 1/11], [21/23, 2/23]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243020,"parameters":{"transition":[[[10,11],[1,11]],[[21,23],[2,23]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[10,11],[1,11]],[[21,23],[2,23]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0021","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[21,23],[2,23]],[[11,12],[1,12]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[21/23, 2/23], [11/12, 1/12]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[21/23, 2/23], [11/12, 1/12]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243021,"parameters":{"transition":[[[21,23],[2,23]],[[11,12],[1,12]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[21,23],[2,23]],[[11,12],[1,12]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0022","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[11,12],[1,12]],[[23,25],[2,25]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[11/12, 1/12], [23/25, 2/25]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[11/12, 1/12], [23/25, 2/25]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243022,"parameters":{"transition":[[[11,12],[1,12]],[[23,25],[2,25]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[11,12],[1,12]],[[23,25],[2,25]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0023","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[23,25],[2,25]],[[12,13],[1,13]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[23/25, 2/25], [12/13, 1/13]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[23/25, 2/25], [12/13, 1/13]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243023,"parameters":{"transition":[[[23,25],[2,25]],[[12,13],[1,13]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[23,25],[2,25]],[[12,13],[1,13]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0024","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[12,13],[1,13]],[[25,27],[2,27]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[12/13, 1/13], [25/27, 2/27]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[12/13, 1/13], [25/27, 2/27]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243024,"parameters":{"transition":[[[12,13],[1,13]],[[25,27],[2,27]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[12,13],[1,13]],[[25,27],[2,27]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0025","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-kernel-audit；資料={\"transition\":[[[25,27],[2,27]],[[13,14],[1,14]]]}。請依 [n,transition,nonnegative,row_sums,row_stochastic,row_convention] 欄序回答。","answer":{"kind":"expression","value":"(2, [[25/27, 2/27], [13/14, 1/14]], 是, [1, 1], 是, rows=current state, columns=next state)","explanation_zh":"依賴結構核對=(2, [[25/27, 2/27], [13/14, 1/14]], 是, [1, 1], 是, rows=current state, columns=next 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-kernel-audit","seed":202610243025,"parameters":{"transition":[[[25,27],[2,27]],[[13,14],[1,14]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-kernel-audit","semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[25,27],[2,27]],[[13,14],[1,14]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0026","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[1,3],[2,3]],[[1,2],[1,2]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/2], [[1/3, 2/3], [1/2, 1/2]], [5/12, 7/12])","explanation_zh":"依賴結構核對=([1/2, 1/2], [[1/3, 2/3], [1/2, 1/2]], [5/12, 7/12])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243026,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[1,3],[2,3]],[[1,2],[1,2]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[1,2],[1,2]],\"transition\":[[[1,3],[2,3]],[[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、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0027","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[2,3],[1,3]],\"transition\":[[[1,2],[1,2]],[[3,5],[2,5]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([2/3, 1/3], [[1/2, 1/2], [3/5, 2/5]], [8/15, 7/15])","explanation_zh":"依賴結構核對=([2/3, 1/3], [[1/2, 1/2], [3/5, 2/5]], [8/15, 7/15])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243027,"parameters":{"initial":[[2,3],[1,3]],"transition":[[[1,2],[1,2]],[[3,5],[2,5]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[2,3],[1,3]],\"transition\":[[[1,2],[1,2]],[[3,5],[2,5]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0028","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[3,4],[1,4]],\"transition\":[[[3,5],[2,5]],[[2,3],[1,3]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([3/4, 1/4], [[3/5, 2/5], [2/3, 1/3]], [37/60, 23/60])","explanation_zh":"依賴結構核對=([3/4, 1/4], [[3/5, 2/5], [2/3, 1/3]], [37/60, 23/60])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243028,"parameters":{"initial":[[3,4],[1,4]],"transition":[[[3,5],[2,5]],[[2,3],[1,3]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[3,4],[1,4]],\"transition\":[[[3,5],[2,5]],[[2,3],[1,3]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0029","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[4,5],[1,5]],\"transition\":[[[2,3],[1,3]],[[5,7],[2,7]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([4/5, 1/5], [[2/3, 1/3], [5/7, 2/7]], [71/105, 34/105])","explanation_zh":"依賴結構核對=([4/5, 1/5], [[2/3, 1/3], [5/7, 2/7]], [71/105, 34/105])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243029,"parameters":{"initial":[[4,5],[1,5]],"transition":[[[2,3],[1,3]],[[5,7],[2,7]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[4,5],[1,5]],\"transition\":[[[2,3],[1,3]],[[5,7],[2,7]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0030","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[5,6],[1,6]],\"transition\":[[[5,7],[2,7]],[[3,4],[1,4]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([5/6, 1/6], [[5/7, 2/7], [3/4, 1/4]], [121/168, 47/168])","explanation_zh":"依賴結構核對=([5/6, 1/6], [[5/7, 2/7], [3/4, 1/4]], [121/168, 47/168])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243030,"parameters":{"initial":[[5,6],[1,6]],"transition":[[[5,7],[2,7]],[[3,4],[1,4]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[5,6],[1,6]],\"transition\":[[[5,7],[2,7]],[[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、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0031","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[6,7],[1,7]],\"transition\":[[[3,4],[1,4]],[[7,9],[2,9]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([6/7, 1/7], [[3/4, 1/4], [7/9, 2/9]], [95/126, 31/126])","explanation_zh":"依賴結構核對=([6/7, 1/7], [[3/4, 1/4], [7/9, 2/9]], [95/126, 31/126])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243031,"parameters":{"initial":[[6,7],[1,7]],"transition":[[[3,4],[1,4]],[[7,9],[2,9]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[6,7],[1,7]],\"transition\":[[[3,4],[1,4]],[[7,9],[2,9]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0032","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[7,8],[1,8]],\"transition\":[[[7,9],[2,9]],[[4,5],[1,5]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([7/8, 1/8], [[7/9, 2/9], [4/5, 1/5]], [281/360, 79/360])","explanation_zh":"依賴結構核對=([7/8, 1/8], [[7/9, 2/9], [4/5, 1/5]], [281/360, 79/360])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243032,"parameters":{"initial":[[7,8],[1,8]],"transition":[[[7,9],[2,9]],[[4,5],[1,5]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[7,8],[1,8]],\"transition\":[[[7,9],[2,9]],[[4,5],[1,5]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0033","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[8,9],[1,9]],\"transition\":[[[4,5],[1,5]],[[9,11],[2,11]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([8/9, 1/9], [[4/5, 1/5], [9/11, 2/11]], [397/495, 98/495])","explanation_zh":"依賴結構核對=([8/9, 1/9], [[4/5, 1/5], [9/11, 2/11]], [397/495, 98/495])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243033,"parameters":{"initial":[[8,9],[1,9]],"transition":[[[4,5],[1,5]],[[9,11],[2,11]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[8,9],[1,9]],\"transition\":[[[4,5],[1,5]],[[9,11],[2,11]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0034","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[9,10],[1,10]],\"transition\":[[[9,11],[2,11]],[[5,6],[1,6]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([9/10, 1/10], [[9/11, 2/11], [5/6, 1/6]], [541/660, 119/660])","explanation_zh":"依賴結構核對=([9/10, 1/10], [[9/11, 2/11], [5/6, 1/6]], [541/660, 119/660])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243034,"parameters":{"initial":[[9,10],[1,10]],"transition":[[[9,11],[2,11]],[[5,6],[1,6]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[9,10],[1,10]],\"transition\":[[[9,11],[2,11]],[[5,6],[1,6]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0035","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[10,11],[1,11]],\"transition\":[[[5,6],[1,6]],[[11,13],[2,13]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([10/11, 1/11], [[5/6, 1/6], [11/13, 2/13]], [358/429, 71/429])","explanation_zh":"依賴結構核對=([10/11, 1/11], [[5/6, 1/6], [11/13, 2/13]], [358/429, 71/429])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243035,"parameters":{"initial":[[10,11],[1,11]],"transition":[[[5,6],[1,6]],[[11,13],[2,13]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[10,11],[1,11]],\"transition\":[[[5,6],[1,6]],[[11,13],[2,13]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0036","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[11,12],[1,12]],\"transition\":[[[11,13],[2,13]],[[6,7],[1,7]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([11/12, 1/12], [[11/13, 2/13], [6/7, 1/7]], [925/1092, 167/1092])","explanation_zh":"依賴結構核對=([11/12, 1/12], [[11/13, 2/13], [6/7, 1/7]], [925/1092, 167/1092])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243036,"parameters":{"initial":[[11,12],[1,12]],"transition":[[[11,13],[2,13]],[[6,7],[1,7]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[11,12],[1,12]],\"transition\":[[[11,13],[2,13]],[[6,7],[1,7]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0037","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[12,13],[1,13]],\"transition\":[[[6,7],[1,7]],[[13,15],[2,15]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([12/13, 1/13], [[6/7, 1/7], [13/15, 2/15]], [1171/1365, 194/1365])","explanation_zh":"依賴結構核對=([12/13, 1/13], [[6/7, 1/7], [13/15, 2/15]], [1171/1365, 194/1365])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243037,"parameters":{"initial":[[12,13],[1,13]],"transition":[[[6,7],[1,7]],[[13,15],[2,15]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[12,13],[1,13]],\"transition\":[[[6,7],[1,7]],[[13,15],[2,15]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0038","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[13,14],[1,14]],\"transition\":[[[13,15],[2,15]],[[7,8],[1,8]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([13/14, 1/14], [[13/15, 2/15], [7/8, 1/8]], [1457/1680, 223/1680])","explanation_zh":"依賴結構核對=([13/14, 1/14], [[13/15, 2/15], [7/8, 1/8]], [1457/1680, 223/1680])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243038,"parameters":{"initial":[[13,14],[1,14]],"transition":[[[13,15],[2,15]],[[7,8],[1,8]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[13,14],[1,14]],\"transition\":[[[13,15],[2,15]],[[7,8],[1,8]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0039","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[14,15],[1,15]],\"transition\":[[[7,8],[1,8]],[[15,17],[2,17]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([14/15, 1/15], [[7/8, 1/8], [15/17, 2/17]], [893/1020, 127/1020])","explanation_zh":"依賴結構核對=([14/15, 1/15], [[7/8, 1/8], [15/17, 2/17]], [893/1020, 127/1020])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243039,"parameters":{"initial":[[14,15],[1,15]],"transition":[[[7,8],[1,8]],[[15,17],[2,17]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[14,15],[1,15]],\"transition\":[[[7,8],[1,8]],[[15,17],[2,17]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0040","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[15,16],[1,16]],\"transition\":[[[15,17],[2,17]],[[8,9],[1,9]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([15/16, 1/16], [[15/17, 2/17], [8/9, 1/9]], [2161/2448, 287/2448])","explanation_zh":"依賴結構核對=([15/16, 1/16], [[15/17, 2/17], [8/9, 1/9]], [2161/2448, 287/2448])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243040,"parameters":{"initial":[[15,16],[1,16]],"transition":[[[15,17],[2,17]],[[8,9],[1,9]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[15,16],[1,16]],\"transition\":[[[15,17],[2,17]],[[8,9],[1,9]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0041","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[16,17],[1,17]],\"transition\":[[[8,9],[1,9]],[[17,19],[2,19]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([16/17, 1/17], [[8/9, 1/9], [17/19, 2/19]], [2585/2907, 322/2907])","explanation_zh":"依賴結構核對=([16/17, 1/17], [[8/9, 1/9], [17/19, 2/19]], [2585/2907, 322/2907])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243041,"parameters":{"initial":[[16,17],[1,17]],"transition":[[[8,9],[1,9]],[[17,19],[2,19]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[16,17],[1,17]],\"transition\":[[[8,9],[1,9]],[[17,19],[2,19]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0042","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[17,18],[1,18]],\"transition\":[[[17,19],[2,19]],[[9,10],[1,10]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([17/18, 1/18], [[17/19, 2/19], [9/10, 1/10]], [3061/3420, 359/3420])","explanation_zh":"依賴結構核對=([17/18, 1/18], [[17/19, 2/19], [9/10, 1/10]], [3061/3420, 359/3420])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243042,"parameters":{"initial":[[17,18],[1,18]],"transition":[[[17,19],[2,19]],[[9,10],[1,10]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[17,18],[1,18]],\"transition\":[[[17,19],[2,19]],[[9,10],[1,10]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0043","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[18,19],[1,19]],\"transition\":[[[9,10],[1,10]],[[19,21],[2,21]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([18/19, 1/19], [[9/10, 1/10], [19/21, 2/21]], [1796/1995, 199/1995])","explanation_zh":"依賴結構核對=([18/19, 1/19], [[9/10, 1/10], [19/21, 2/21]], [1796/1995, 199/1995])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243043,"parameters":{"initial":[[18,19],[1,19]],"transition":[[[9,10],[1,10]],[[19,21],[2,21]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[18,19],[1,19]],\"transition\":[[[9,10],[1,10]],[[19,21],[2,21]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0044","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[19,20],[1,20]],\"transition\":[[[19,21],[2,21]],[[10,11],[1,11]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([19/20, 1/20], [[19/21, 2/21], [10/11, 1/11]], [4181/4620, 439/4620])","explanation_zh":"依賴結構核對=([19/20, 1/20], [[19/21, 2/21], [10/11, 1/11]], [4181/4620, 439/4620])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243044,"parameters":{"initial":[[19,20],[1,20]],"transition":[[[19,21],[2,21]],[[10,11],[1,11]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[19,20],[1,20]],\"transition\":[[[19,21],[2,21]],[[10,11],[1,11]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0045","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[20,21],[1,21]],\"transition\":[[[10,11],[1,11]],[[21,23],[2,23]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([20/21, 1/21], [[10/11, 1/11], [21/23, 2/23]], [4831/5313, 482/5313])","explanation_zh":"依賴結構核對=([20/21, 1/21], [[10/11, 1/11], [21/23, 2/23]], [4831/5313, 482/5313])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243045,"parameters":{"initial":[[20,21],[1,21]],"transition":[[[10,11],[1,11]],[[21,23],[2,23]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[20,21],[1,21]],\"transition\":[[[10,11],[1,11]],[[21,23],[2,23]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0046","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[21,22],[1,22]],\"transition\":[[[21,23],[2,23]],[[11,12],[1,12]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([21/22, 1/22], [[21/23, 2/23], [11/12, 1/12]], [5545/6072, 527/6072])","explanation_zh":"依賴結構核對=([21/22, 1/22], [[21/23, 2/23], [11/12, 1/12]], [5545/6072, 527/6072])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243046,"parameters":{"initial":[[21,22],[1,22]],"transition":[[[21,23],[2,23]],[[11,12],[1,12]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[21,22],[1,22]],\"transition\":[[[21,23],[2,23]],[[11,12],[1,12]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0047","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[22,23],[1,23]],\"transition\":[[[11,12],[1,12]],[[23,25],[2,25]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([22/23, 1/23], [[11/12, 1/12], [23/25, 2/25]], [3163/3450, 287/3450])","explanation_zh":"依賴結構核對=([22/23, 1/23], [[11/12, 1/12], [23/25, 2/25]], [3163/3450, 287/3450])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243047,"parameters":{"initial":[[22,23],[1,23]],"transition":[[[11,12],[1,12]],[[23,25],[2,25]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[22,23],[1,23]],\"transition\":[[[11,12],[1,12]],[[23,25],[2,25]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0048","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[23,24],[1,24]],\"transition\":[[[23,25],[2,25]],[[12,13],[1,13]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([23/24, 1/24], [[23/25, 2/25], [12/13, 1/13]], [7177/7800, 623/7800])","explanation_zh":"依賴結構核對=([23/24, 1/24], [[23/25, 2/25], [12/13, 1/13]], [7177/7800, 623/7800])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243048,"parameters":{"initial":[[23,24],[1,24]],"transition":[[[23,25],[2,25]],[[12,13],[1,13]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[23,24],[1,24]],\"transition\":[[[23,25],[2,25]],[[12,13],[1,13]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0049","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[24,25],[1,25]],\"transition\":[[[12,13],[1,13]],[[25,27],[2,27]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([24/25, 1/25], [[12/13, 1/13], [25/27, 2/27]], [8101/8775, 674/8775])","explanation_zh":"依賴結構核對=([24/25, 1/25], [[12/13, 1/13], [25/27, 2/27]], [8101/8775, 674/8775])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243049,"parameters":{"initial":[[24,25],[1,25]],"transition":[[[12,13],[1,13]],[[25,27],[2,27]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[24,25],[1,25]],\"transition\":[[[12,13],[1,13]],[[25,27],[2,27]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0050","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-one-step-law；資料={\"initial\":[[25,26],[1,26]],\"transition\":[[[25,27],[2,27]],[[13,14],[1,14]]]}。請依 [initial,transition,next_law] 欄序回答。","answer":{"kind":"expression","value":"([25/26, 1/26], [[25/27, 2/27], [13/14, 1/14]], [9101/9828, 727/9828])","explanation_zh":"依賴結構核對=([25/26, 1/26], [[25/27, 2/27], [13/14, 1/14]], [9101/9828, 727/9828])；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-one-step-law","seed":202610243050,"parameters":{"initial":[[25,26],[1,26]],"transition":[[[25,27],[2,27]],[[13,14],[1,14]]],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-one-step-law","semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[25,26],[1,26]],\"transition\":[[[25,27],[2,27]],[[13,14],[1,14]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0051","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[1,3],[2,3]],[[1,2],[1,2]]],\"from\":0,\"to\":1}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [2/9, 1/3], 5/9)","explanation_zh":"依賴結構核對=(0, 1, [2/9, 1/3], 5/9)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243051,"parameters":{"transition":[[[1,3],[2,3]],[[1,2],[1,2]]],"from":0,"to":1,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[1,3],[2,3]],[[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、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0052","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[1,2],[1,2]],[[3,5],[2,5]]],\"from\":1,\"to\":0}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [3/10, 6/25], 27/50)","explanation_zh":"依賴結構核對=(1, 0, [3/10, 6/25], 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243052,"parameters":{"transition":[[[1,2],[1,2]],[[3,5],[2,5]]],"from":1,"to":0,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[1,2],[1,2]],[[3,5],[2,5]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0053","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[3,5],[2,5]],[[2,3],[1,3]]],\"from\":0,\"to\":1}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [6/25, 2/15], 28/75)","explanation_zh":"依賴結構核對=(0, 1, [6/25, 2/15], 28/75)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243053,"parameters":{"transition":[[[3,5],[2,5]],[[2,3],[1,3]]],"from":0,"to":1,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[3,5],[2,5]],[[2,3],[1,3]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0054","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[2,3],[1,3]],[[5,7],[2,7]]],\"from\":1,\"to\":0}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [10/21, 10/49], 100/147)","explanation_zh":"依賴結構核對=(1, 0, [10/21, 10/49], 100/147)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243054,"parameters":{"transition":[[[2,3],[1,3]],[[5,7],[2,7]]],"from":1,"to":0,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[2,3],[1,3]],[[5,7],[2,7]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0055","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[5,7],[2,7]],[[3,4],[1,4]]],\"from\":0,\"to\":1}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [10/49, 1/14], 27/98)","explanation_zh":"依賴結構核對=(0, 1, [10/49, 1/14], 27/98)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243055,"parameters":{"transition":[[[5,7],[2,7]],[[3,4],[1,4]]],"from":0,"to":1,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[5,7],[2,7]],[[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、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0056","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[3,4],[1,4]],[[7,9],[2,9]]],\"from\":1,\"to\":0}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [7/12, 14/81], 245/324)","explanation_zh":"依賴結構核對=(1, 0, [7/12, 14/81], 245/324)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243056,"parameters":{"transition":[[[3,4],[1,4]],[[7,9],[2,9]]],"from":1,"to":0,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[3,4],[1,4]],[[7,9],[2,9]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0057","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[7,9],[2,9]],[[4,5],[1,5]]],\"from\":0,\"to\":1}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [14/81, 2/45], 88/405)","explanation_zh":"依賴結構核對=(0, 1, [14/81, 2/45], 88/405)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243057,"parameters":{"transition":[[[7,9],[2,9]],[[4,5],[1,5]]],"from":0,"to":1,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[7,9],[2,9]],[[4,5],[1,5]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0058","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[4,5],[1,5]],[[9,11],[2,11]]],\"from\":1,\"to\":0}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [36/55, 18/121], 486/605)","explanation_zh":"依賴結構核對=(1, 0, [36/55, 18/121], 486/605)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243058,"parameters":{"transition":[[[4,5],[1,5]],[[9,11],[2,11]]],"from":1,"to":0,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[4,5],[1,5]],[[9,11],[2,11]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0059","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[9,11],[2,11]],[[5,6],[1,6]]],\"from\":0,\"to\":1}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [18/121, 1/33], 65/363)","explanation_zh":"依賴結構核對=(0, 1, [18/121, 1/33], 65/363)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243059,"parameters":{"transition":[[[9,11],[2,11]],[[5,6],[1,6]]],"from":0,"to":1,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[9,11],[2,11]],[[5,6],[1,6]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0060","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[5,6],[1,6]],[[11,13],[2,13]]],\"from\":1,\"to\":0}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [55/78, 22/169], 847/1014)","explanation_zh":"依賴結構核對=(1, 0, [55/78, 22/169], 847/1014)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243060,"parameters":{"transition":[[[5,6],[1,6]],[[11,13],[2,13]]],"from":1,"to":0,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[5,6],[1,6]],[[11,13],[2,13]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0061","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[11,13],[2,13]],[[6,7],[1,7]]],\"from\":0,\"to\":1}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [22/169, 2/91], 180/1183)","explanation_zh":"依賴結構核對=(0, 1, [22/169, 2/91], 180/1183)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243061,"parameters":{"transition":[[[11,13],[2,13]],[[6,7],[1,7]]],"from":0,"to":1,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[11,13],[2,13]],[[6,7],[1,7]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0062","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[6,7],[1,7]],[[13,15],[2,15]]],\"from\":1,\"to\":0}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [26/35, 26/225], 1352/1575)","explanation_zh":"依賴結構核對=(1, 0, [26/35, 26/225], 1352/1575)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243062,"parameters":{"transition":[[[6,7],[1,7]],[[13,15],[2,15]]],"from":1,"to":0,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[6,7],[1,7]],[[13,15],[2,15]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0063","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[13,15],[2,15]],[[7,8],[1,8]]],\"from\":0,\"to\":1}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [26/225, 1/60], 119/900)","explanation_zh":"依賴結構核對=(0, 1, [26/225, 1/60], 119/900)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243063,"parameters":{"transition":[[[13,15],[2,15]],[[7,8],[1,8]]],"from":0,"to":1,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[13,15],[2,15]],[[7,8],[1,8]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0064","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[7,8],[1,8]],[[15,17],[2,17]]],\"from\":1,\"to\":0}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [105/136, 30/289], 2025/2312)","explanation_zh":"依賴結構核對=(1, 0, [105/136, 30/289], 2025/2312)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243064,"parameters":{"transition":[[[7,8],[1,8]],[[15,17],[2,17]]],"from":1,"to":0,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[7,8],[1,8]],[[15,17],[2,17]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0065","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[15,17],[2,17]],[[8,9],[1,9]]],\"from\":0,\"to\":1}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [30/289, 2/153], 304/2601)","explanation_zh":"依賴結構核對=(0, 1, [30/289, 2/153], 304/2601)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243065,"parameters":{"transition":[[[15,17],[2,17]],[[8,9],[1,9]]],"from":0,"to":1,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[15,17],[2,17]],[[8,9],[1,9]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0066","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[8,9],[1,9]],[[17,19],[2,19]]],\"from\":1,\"to\":0}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [136/171, 34/361], 2890/3249)","explanation_zh":"依賴結構核對=(1, 0, [136/171, 34/361], 2890/3249)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243066,"parameters":{"transition":[[[8,9],[1,9]],[[17,19],[2,19]]],"from":1,"to":0,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[8,9],[1,9]],[[17,19],[2,19]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0067","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[17,19],[2,19]],[[9,10],[1,10]]],\"from\":0,\"to\":1}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [34/361, 1/95], 189/1805)","explanation_zh":"依賴結構核對=(0, 1, [34/361, 1/95], 189/1805)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243067,"parameters":{"transition":[[[17,19],[2,19]],[[9,10],[1,10]]],"from":0,"to":1,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[17,19],[2,19]],[[9,10],[1,10]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0068","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[9,10],[1,10]],[[19,21],[2,21]]],\"from\":1,\"to\":0}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [57/70, 38/441], 3971/4410)","explanation_zh":"依賴結構核對=(1, 0, [57/70, 38/441], 3971/4410)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243068,"parameters":{"transition":[[[9,10],[1,10]],[[19,21],[2,21]]],"from":1,"to":0,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[9,10],[1,10]],[[19,21],[2,21]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0069","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[19,21],[2,21]],[[10,11],[1,11]]],\"from\":0,\"to\":1}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [38/441, 2/231], 460/4851)","explanation_zh":"依賴結構核對=(0, 1, [38/441, 2/231], 460/4851)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243069,"parameters":{"transition":[[[19,21],[2,21]],[[10,11],[1,11]]],"from":0,"to":1,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[19,21],[2,21]],[[10,11],[1,11]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0070","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[10,11],[1,11]],[[21,23],[2,23]]],\"from\":1,\"to\":0}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [210/253, 42/529], 5292/5819)","explanation_zh":"依賴結構核對=(1, 0, [210/253, 42/529], 5292/5819)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243070,"parameters":{"transition":[[[10,11],[1,11]],[[21,23],[2,23]]],"from":1,"to":0,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[10,11],[1,11]],[[21,23],[2,23]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0071","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[21,23],[2,23]],[[11,12],[1,12]]],\"from\":0,\"to\":1}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [42/529, 1/138], 275/3174)","explanation_zh":"依賴結構核對=(0, 1, [42/529, 1/138], 275/3174)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243071,"parameters":{"transition":[[[21,23],[2,23]],[[11,12],[1,12]]],"from":0,"to":1,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[21,23],[2,23]],[[11,12],[1,12]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0072","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[11,12],[1,12]],[[23,25],[2,25]]],\"from\":1,\"to\":0}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [253/300, 46/625], 6877/7500)","explanation_zh":"依賴結構核對=(1, 0, [253/300, 46/625], 6877/7500)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243072,"parameters":{"transition":[[[11,12],[1,12]],[[23,25],[2,25]]],"from":1,"to":0,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[11,12],[1,12]],[[23,25],[2,25]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0073","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[23,25],[2,25]],[[12,13],[1,13]]],\"from\":0,\"to\":1}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [46/625, 2/325], 648/8125)","explanation_zh":"依賴結構核對=(0, 1, [46/625, 2/325], 648/8125)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243073,"parameters":{"transition":[[[23,25],[2,25]],[[12,13],[1,13]]],"from":0,"to":1,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[23,25],[2,25]],[[12,13],[1,13]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0074","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[12,13],[1,13]],[[25,27],[2,27]]],\"from\":1,\"to\":0}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, [100/117, 50/729], 8750/9477)","explanation_zh":"依賴結構核對=(1, 0, [100/117, 50/729], 8750/9477)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243074,"parameters":{"transition":[[[12,13],[1,13]],[[25,27],[2,27]]],"from":1,"to":0,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[12,13],[1,13]],[[25,27],[2,27]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0075","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-two-step-entry；資料={\"transition\":[[[25,27],[2,27]],[[13,14],[1,14]]],\"from\":0,\"to\":1}。請依 [from,to,middle_state_terms,two_step_probability] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, [50/729, 1/189], 377/5103)","explanation_zh":"依賴結構核對=(0, 1, [50/729, 1/189], 377/5103)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-two-step-entry","seed":202610243075,"parameters":{"transition":[[[25,27],[2,27]],[[13,14],[1,14]]],"from":0,"to":1,"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-two-step-entry","semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[25,27],[2,27]],[[13,14],[1,14]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0076","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[1,2],[1,2]],\"transition\":[[[1,3],[2,3]],[[1,2],[1,2]]],\"path\":[0,1,0]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [1/2, 2/3, 1/2], 1/6)","explanation_zh":"依賴結構核對=([0, 1, 0], [1/2, 2/3, 1/2], 1/6)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243076,"parameters":{"initial":[[1,2],[1,2]],"transition":[[[1,3],[2,3]],[[1,2],[1,2]]],"path":[0,1,0],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[1,2],[1,2]],\"path\":[0,1,0],\"transition\":[[[1,3],[2,3]],[[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、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0077","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[2,3],[1,3]],\"transition\":[[[1,2],[1,2]],[[3,5],[2,5]]],\"path\":[1,0,1]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [1/3, 3/5, 1/2], 1/10)","explanation_zh":"依賴結構核對=([1, 0, 1], [1/3, 3/5, 1/2], 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243077,"parameters":{"initial":[[2,3],[1,3]],"transition":[[[1,2],[1,2]],[[3,5],[2,5]]],"path":[1,0,1],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[2,3],[1,3]],\"path\":[1,0,1],\"transition\":[[[1,2],[1,2]],[[3,5],[2,5]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0078","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[3,4],[1,4]],\"transition\":[[[3,5],[2,5]],[[2,3],[1,3]]],\"path\":[0,1,0]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [3/4, 2/5, 2/3], 1/5)","explanation_zh":"依賴結構核對=([0, 1, 0], [3/4, 2/5, 2/3], 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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243078,"parameters":{"initial":[[3,4],[1,4]],"transition":[[[3,5],[2,5]],[[2,3],[1,3]]],"path":[0,1,0],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[3,4],[1,4]],\"path\":[0,1,0],\"transition\":[[[3,5],[2,5]],[[2,3],[1,3]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0079","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[4,5],[1,5]],\"transition\":[[[2,3],[1,3]],[[5,7],[2,7]]],\"path\":[1,0,1]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [1/5, 5/7, 1/3], 1/21)","explanation_zh":"依賴結構核對=([1, 0, 1], [1/5, 5/7, 1/3], 1/21)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243079,"parameters":{"initial":[[4,5],[1,5]],"transition":[[[2,3],[1,3]],[[5,7],[2,7]]],"path":[1,0,1],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[4,5],[1,5]],\"path\":[1,0,1],\"transition\":[[[2,3],[1,3]],[[5,7],[2,7]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0080","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[5,6],[1,6]],\"transition\":[[[5,7],[2,7]],[[3,4],[1,4]]],\"path\":[0,1,0]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [5/6, 2/7, 3/4], 5/28)","explanation_zh":"依賴結構核對=([0, 1, 0], [5/6, 2/7, 3/4], 5/28)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243080,"parameters":{"initial":[[5,6],[1,6]],"transition":[[[5,7],[2,7]],[[3,4],[1,4]]],"path":[0,1,0],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[5,6],[1,6]],\"path\":[0,1,0],\"transition\":[[[5,7],[2,7]],[[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、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0081","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[6,7],[1,7]],\"transition\":[[[3,4],[1,4]],[[7,9],[2,9]]],\"path\":[1,0,1]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [1/7, 7/9, 1/4], 1/36)","explanation_zh":"依賴結構核對=([1, 0, 1], [1/7, 7/9, 1/4], 1/36)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243081,"parameters":{"initial":[[6,7],[1,7]],"transition":[[[3,4],[1,4]],[[7,9],[2,9]]],"path":[1,0,1],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[6,7],[1,7]],\"path\":[1,0,1],\"transition\":[[[3,4],[1,4]],[[7,9],[2,9]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0082","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[7,8],[1,8]],\"transition\":[[[7,9],[2,9]],[[4,5],[1,5]]],\"path\":[0,1,0]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [7/8, 2/9, 4/5], 7/45)","explanation_zh":"依賴結構核對=([0, 1, 0], [7/8, 2/9, 4/5], 7/45)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243082,"parameters":{"initial":[[7,8],[1,8]],"transition":[[[7,9],[2,9]],[[4,5],[1,5]]],"path":[0,1,0],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[7,8],[1,8]],\"path\":[0,1,0],\"transition\":[[[7,9],[2,9]],[[4,5],[1,5]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0083","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[8,9],[1,9]],\"transition\":[[[4,5],[1,5]],[[9,11],[2,11]]],\"path\":[1,0,1]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [1/9, 9/11, 1/5], 1/55)","explanation_zh":"依賴結構核對=([1, 0, 1], [1/9, 9/11, 1/5], 1/55)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243083,"parameters":{"initial":[[8,9],[1,9]],"transition":[[[4,5],[1,5]],[[9,11],[2,11]]],"path":[1,0,1],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[8,9],[1,9]],\"path\":[1,0,1],\"transition\":[[[4,5],[1,5]],[[9,11],[2,11]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0084","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[9,10],[1,10]],\"transition\":[[[9,11],[2,11]],[[5,6],[1,6]]],\"path\":[0,1,0]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [9/10, 2/11, 5/6], 3/22)","explanation_zh":"依賴結構核對=([0, 1, 0], [9/10, 2/11, 5/6], 3/22)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243084,"parameters":{"initial":[[9,10],[1,10]],"transition":[[[9,11],[2,11]],[[5,6],[1,6]]],"path":[0,1,0],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[9,10],[1,10]],\"path\":[0,1,0],\"transition\":[[[9,11],[2,11]],[[5,6],[1,6]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0085","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[10,11],[1,11]],\"transition\":[[[5,6],[1,6]],[[11,13],[2,13]]],\"path\":[1,0,1]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [1/11, 11/13, 1/6], 1/78)","explanation_zh":"依賴結構核對=([1, 0, 1], [1/11, 11/13, 1/6], 1/78)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243085,"parameters":{"initial":[[10,11],[1,11]],"transition":[[[5,6],[1,6]],[[11,13],[2,13]]],"path":[1,0,1],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[10,11],[1,11]],\"path\":[1,0,1],\"transition\":[[[5,6],[1,6]],[[11,13],[2,13]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0086","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[11,12],[1,12]],\"transition\":[[[11,13],[2,13]],[[6,7],[1,7]]],\"path\":[0,1,0]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [11/12, 2/13, 6/7], 11/91)","explanation_zh":"依賴結構核對=([0, 1, 0], [11/12, 2/13, 6/7], 11/91)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243086,"parameters":{"initial":[[11,12],[1,12]],"transition":[[[11,13],[2,13]],[[6,7],[1,7]]],"path":[0,1,0],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[11,12],[1,12]],\"path\":[0,1,0],\"transition\":[[[11,13],[2,13]],[[6,7],[1,7]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0087","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[12,13],[1,13]],\"transition\":[[[6,7],[1,7]],[[13,15],[2,15]]],\"path\":[1,0,1]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [1/13, 13/15, 1/7], 1/105)","explanation_zh":"依賴結構核對=([1, 0, 1], [1/13, 13/15, 1/7], 1/105)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243087,"parameters":{"initial":[[12,13],[1,13]],"transition":[[[6,7],[1,7]],[[13,15],[2,15]]],"path":[1,0,1],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[12,13],[1,13]],\"path\":[1,0,1],\"transition\":[[[6,7],[1,7]],[[13,15],[2,15]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0088","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[13,14],[1,14]],\"transition\":[[[13,15],[2,15]],[[7,8],[1,8]]],\"path\":[0,1,0]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [13/14, 2/15, 7/8], 13/120)","explanation_zh":"依賴結構核對=([0, 1, 0], [13/14, 2/15, 7/8], 13/120)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243088,"parameters":{"initial":[[13,14],[1,14]],"transition":[[[13,15],[2,15]],[[7,8],[1,8]]],"path":[0,1,0],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[13,14],[1,14]],\"path\":[0,1,0],\"transition\":[[[13,15],[2,15]],[[7,8],[1,8]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0089","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[14,15],[1,15]],\"transition\":[[[7,8],[1,8]],[[15,17],[2,17]]],\"path\":[1,0,1]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [1/15, 15/17, 1/8], 1/136)","explanation_zh":"依賴結構核對=([1, 0, 1], [1/15, 15/17, 1/8], 1/136)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243089,"parameters":{"initial":[[14,15],[1,15]],"transition":[[[7,8],[1,8]],[[15,17],[2,17]]],"path":[1,0,1],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[14,15],[1,15]],\"path\":[1,0,1],\"transition\":[[[7,8],[1,8]],[[15,17],[2,17]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0090","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[15,16],[1,16]],\"transition\":[[[15,17],[2,17]],[[8,9],[1,9]]],\"path\":[0,1,0]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [15/16, 2/17, 8/9], 5/51)","explanation_zh":"依賴結構核對=([0, 1, 0], [15/16, 2/17, 8/9], 5/51)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243090,"parameters":{"initial":[[15,16],[1,16]],"transition":[[[15,17],[2,17]],[[8,9],[1,9]]],"path":[0,1,0],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[15,16],[1,16]],\"path\":[0,1,0],\"transition\":[[[15,17],[2,17]],[[8,9],[1,9]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0091","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[16,17],[1,17]],\"transition\":[[[8,9],[1,9]],[[17,19],[2,19]]],\"path\":[1,0,1]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [1/17, 17/19, 1/9], 1/171)","explanation_zh":"依賴結構核對=([1, 0, 1], [1/17, 17/19, 1/9], 1/171)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243091,"parameters":{"initial":[[16,17],[1,17]],"transition":[[[8,9],[1,9]],[[17,19],[2,19]]],"path":[1,0,1],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[16,17],[1,17]],\"path\":[1,0,1],\"transition\":[[[8,9],[1,9]],[[17,19],[2,19]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0092","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[17,18],[1,18]],\"transition\":[[[17,19],[2,19]],[[9,10],[1,10]]],\"path\":[0,1,0]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [17/18, 2/19, 9/10], 17/190)","explanation_zh":"依賴結構核對=([0, 1, 0], [17/18, 2/19, 9/10], 17/190)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243092,"parameters":{"initial":[[17,18],[1,18]],"transition":[[[17,19],[2,19]],[[9,10],[1,10]]],"path":[0,1,0],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[17,18],[1,18]],\"path\":[0,1,0],\"transition\":[[[17,19],[2,19]],[[9,10],[1,10]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0093","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[18,19],[1,19]],\"transition\":[[[9,10],[1,10]],[[19,21],[2,21]]],\"path\":[1,0,1]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [1/19, 19/21, 1/10], 1/210)","explanation_zh":"依賴結構核對=([1, 0, 1], [1/19, 19/21, 1/10], 1/210)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243093,"parameters":{"initial":[[18,19],[1,19]],"transition":[[[9,10],[1,10]],[[19,21],[2,21]]],"path":[1,0,1],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[18,19],[1,19]],\"path\":[1,0,1],\"transition\":[[[9,10],[1,10]],[[19,21],[2,21]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0094","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[19,20],[1,20]],\"transition\":[[[19,21],[2,21]],[[10,11],[1,11]]],\"path\":[0,1,0]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [19/20, 2/21, 10/11], 19/231)","explanation_zh":"依賴結構核對=([0, 1, 0], [19/20, 2/21, 10/11], 19/231)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243094,"parameters":{"initial":[[19,20],[1,20]],"transition":[[[19,21],[2,21]],[[10,11],[1,11]]],"path":[0,1,0],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[19,20],[1,20]],\"path\":[0,1,0],\"transition\":[[[19,21],[2,21]],[[10,11],[1,11]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0095","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[20,21],[1,21]],\"transition\":[[[10,11],[1,11]],[[21,23],[2,23]]],\"path\":[1,0,1]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [1/21, 21/23, 1/11], 1/253)","explanation_zh":"依賴結構核對=([1, 0, 1], [1/21, 21/23, 1/11], 1/253)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243095,"parameters":{"initial":[[20,21],[1,21]],"transition":[[[10,11],[1,11]],[[21,23],[2,23]]],"path":[1,0,1],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[20,21],[1,21]],\"path\":[1,0,1],\"transition\":[[[10,11],[1,11]],[[21,23],[2,23]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0096","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[21,22],[1,22]],\"transition\":[[[21,23],[2,23]],[[11,12],[1,12]]],\"path\":[0,1,0]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [21/22, 2/23, 11/12], 7/92)","explanation_zh":"依賴結構核對=([0, 1, 0], [21/22, 2/23, 11/12], 7/92)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243096,"parameters":{"initial":[[21,22],[1,22]],"transition":[[[21,23],[2,23]],[[11,12],[1,12]]],"path":[0,1,0],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[21,22],[1,22]],\"path\":[0,1,0],\"transition\":[[[21,23],[2,23]],[[11,12],[1,12]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0097","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[22,23],[1,23]],\"transition\":[[[11,12],[1,12]],[[23,25],[2,25]]],\"path\":[1,0,1]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [1/23, 23/25, 1/12], 1/300)","explanation_zh":"依賴結構核對=([1, 0, 1], [1/23, 23/25, 1/12], 1/300)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243097,"parameters":{"initial":[[22,23],[1,23]],"transition":[[[11,12],[1,12]],[[23,25],[2,25]]],"path":[1,0,1],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[22,23],[1,23]],\"path\":[1,0,1],\"transition\":[[[11,12],[1,12]],[[23,25],[2,25]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0098","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[23,24],[1,24]],\"transition\":[[[23,25],[2,25]],[[12,13],[1,13]]],\"path\":[0,1,0]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [23/24, 2/25, 12/13], 23/325)","explanation_zh":"依賴結構核對=([0, 1, 0], [23/24, 2/25, 12/13], 23/325)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243098,"parameters":{"initial":[[23,24],[1,24]],"transition":[[[23,25],[2,25]],[[12,13],[1,13]]],"path":[0,1,0],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[23,24],[1,24]],\"path\":[0,1,0],\"transition\":[[[23,25],[2,25]],[[12,13],[1,13]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0099","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[24,25],[1,25]],\"transition\":[[[12,13],[1,13]],[[25,27],[2,27]]],\"path\":[1,0,1]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], [1/25, 25/27, 1/13], 1/351)","explanation_zh":"依賴結構核對=([1, 0, 1], [1/25, 25/27, 1/13], 1/351)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243099,"parameters":{"initial":[[24,25],[1,25]],"transition":[[[12,13],[1,13]],[[25,27],[2,27]]],"path":[1,0,1],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[24,25],[1,25]],\"path\":[1,0,1],\"transition\":[[[12,13],[1,13]],[[25,27],[2,27]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r03-finite-markov-chain-0100","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","language":"zh-Hant","source_mko_ids":["mko-finite-markov-chain"],"title_zh":"有限狀態Markov鏈","stem_zh":"rows=current state且columns=next state。操作=markov-path-probability；資料={\"initial\":[[25,26],[1,26]],\"transition\":[[[25,27],[2,27]],[[13,14],[1,14]]],\"path\":[0,1,0]}。請依 [path,factors,path_probability] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0], [25/26, 2/27, 13/14], 25/378)","explanation_zh":"依賴結構核對=([0, 1, 0], [25/26, 2/27, 13/14], 25/378)；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；轉移列非負且精確和為1；不推出平穩遍歷或長期收斂。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限狀態Markov鏈"]},"variant":{"family_id":"qf-mod24-markov-path-probability","seed":202610243100,"parameters":{"initial":[[25,26],[1,26]],"transition":[[[25,27],[2,27]],[[13,14],[1,14]]],"path":[0,1,0],"domain":"Exact_finite_time_homogeneous_Markov_kernel","operation":"markov-path-probability","semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[25,26],[1,26]],\"path\":[0,1,0],\"transition\":[[[25,27],[2,27]],[[13,14],[1,14]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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"}}