{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0001","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":5,\"a\":-7,\"b\":-11,\"x\":0}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(5, -7, -11, 0, 4)","explanation_zh":"應用基礎核對=(5, -7, -11, 0, 4)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201001,"parameters":{"modulus":5,"a":-7,"b":-11,"x":0,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":-7,\"b\":-11,\"modulus\":5,\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0002","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":6,\"a\":-6,\"b\":-9,\"x\":1}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(6, -6, -9, 1, 3)","explanation_zh":"應用基礎核對=(6, -6, -9, 1, 3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201002,"parameters":{"modulus":6,"a":-6,"b":-9,"x":1,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":-6,\"b\":-9,\"modulus\":6,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0003","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":7,\"a\":-5,\"b\":-7,\"x\":2}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(7, -5, -7, 2, 4)","explanation_zh":"應用基礎核對=(7, -5, -7, 2, 4)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201003,"parameters":{"modulus":7,"a":-5,"b":-7,"x":2,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":-5,\"b\":-7,\"modulus\":7,\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0004","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":8,\"a\":-4,\"b\":-5,\"x\":3}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(8, -4, -5, 3, 7)","explanation_zh":"應用基礎核對=(8, -4, -5, 3, 7)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201004,"parameters":{"modulus":8,"a":-4,"b":-5,"x":3,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":-4,\"b\":-5,\"modulus\":8,\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0005","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":9,\"a\":-3,\"b\":-3,\"x\":4}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(9, -3, -3, 4, 3)","explanation_zh":"應用基礎核對=(9, -3, -3, 4, 3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201005,"parameters":{"modulus":9,"a":-3,"b":-3,"x":4,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":-3,\"b\":-3,\"modulus\":9,\"x\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0006","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":10,\"a\":-2,\"b\":-1,\"x\":5}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(10, -2, -1, 5, 9)","explanation_zh":"應用基礎核對=(10, -2, -1, 5, 9)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201006,"parameters":{"modulus":10,"a":-2,"b":-1,"x":5,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":-2,\"b\":-1,\"modulus\":10,\"x\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0007","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":5,\"a\":-1,\"b\":1,\"x\":1}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(5, -1, 1, 1, 0)","explanation_zh":"應用基礎核對=(5, -1, 1, 1, 0)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201007,"parameters":{"modulus":5,"a":-1,"b":1,"x":1,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":-1,\"b\":1,\"modulus\":5,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0008","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":6,\"a\":0,\"b\":3,\"x\":1}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 3, 1, 3)","explanation_zh":"應用基礎核對=(6, 0, 3, 1, 3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201008,"parameters":{"modulus":6,"a":0,"b":3,"x":1,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":0,\"b\":3,\"modulus\":6,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0009","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":7,\"a\":1,\"b\":5,\"x\":1}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(7, 1, 5, 1, 6)","explanation_zh":"應用基礎核對=(7, 1, 5, 1, 6)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201009,"parameters":{"modulus":7,"a":1,"b":5,"x":1,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":1,\"b\":5,\"modulus\":7,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0010","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":8,\"a\":2,\"b\":7,\"x\":1}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(8, 2, 7, 1, 1)","explanation_zh":"應用基礎核對=(8, 2, 7, 1, 1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201010,"parameters":{"modulus":8,"a":2,"b":7,"x":1,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":2,\"b\":7,\"modulus\":8,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0011","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":9,\"a\":3,\"b\":9,\"x\":1}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(9, 3, 9, 1, 3)","explanation_zh":"應用基礎核對=(9, 3, 9, 1, 3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201011,"parameters":{"modulus":9,"a":3,"b":9,"x":1,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":3,\"b\":9,\"modulus\":9,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0012","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":10,\"a\":4,\"b\":11,\"x\":1}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(10, 4, 11, 1, 5)","explanation_zh":"應用基礎核對=(10, 4, 11, 1, 5)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201012,"parameters":{"modulus":10,"a":4,"b":11,"x":1,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":4,\"b\":11,\"modulus\":10,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0013","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":5,\"a\":5,\"b\":13,\"x\":2}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, 13, 2, 3)","explanation_zh":"應用基礎核對=(5, 5, 13, 2, 3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201013,"parameters":{"modulus":5,"a":5,"b":13,"x":2,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":5,\"b\":13,\"modulus\":5,\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0014","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":6,\"a\":6,\"b\":15,\"x\":1}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(6, 6, 15, 1, 3)","explanation_zh":"應用基礎核對=(6, 6, 15, 1, 3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201014,"parameters":{"modulus":6,"a":6,"b":15,"x":1,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":6,\"b\":15,\"modulus\":6,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0015","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":7,\"a\":7,\"b\":17,\"x\":0}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(7, 7, 17, 0, 3)","explanation_zh":"應用基礎核對=(7, 7, 17, 0, 3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201015,"parameters":{"modulus":7,"a":7,"b":17,"x":0,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":7,\"b\":17,\"modulus\":7,\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0016","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":8,\"a\":8,\"b\":19,\"x\":7}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(8, 8, 19, 7, 3)","explanation_zh":"應用基礎核對=(8, 8, 19, 7, 3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201016,"parameters":{"modulus":8,"a":8,"b":19,"x":7,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":8,\"b\":19,\"modulus\":8,\"x\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0017","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":9,\"a\":9,\"b\":21,\"x\":7}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(9, 9, 21, 7, 3)","explanation_zh":"應用基礎核對=(9, 9, 21, 7, 3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201017,"parameters":{"modulus":9,"a":9,"b":21,"x":7,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":9,\"b\":21,\"modulus\":9,\"x\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0018","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":10,\"a\":10,\"b\":23,\"x\":7}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(10, 10, 23, 7, 3)","explanation_zh":"應用基礎核對=(10, 10, 23, 7, 3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201018,"parameters":{"modulus":10,"a":10,"b":23,"x":7,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":10,\"b\":23,\"modulus\":10,\"x\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0019","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":5,\"a\":11,\"b\":25,\"x\":3}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(5, 11, 25, 3, 3)","explanation_zh":"應用基礎核對=(5, 11, 25, 3, 3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201019,"parameters":{"modulus":5,"a":11,"b":25,"x":3,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":11,\"b\":25,\"modulus\":5,\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0020","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":6,\"a\":12,\"b\":27,\"x\":1}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(6, 12, 27, 1, 3)","explanation_zh":"應用基礎核對=(6, 12, 27, 1, 3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201020,"parameters":{"modulus":6,"a":12,"b":27,"x":1,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":12,\"b\":27,\"modulus\":6,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0021","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":7,\"a\":13,\"b\":29,\"x\":6}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(7, 13, 29, 6, 2)","explanation_zh":"應用基礎核對=(7, 13, 29, 6, 2)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201021,"parameters":{"modulus":7,"a":13,"b":29,"x":6,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":13,\"b\":29,\"modulus\":7,\"x\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0022","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":8,\"a\":14,\"b\":31,\"x\":5}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(8, 14, 31, 5, 5)","explanation_zh":"應用基礎核對=(8, 14, 31, 5, 5)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201022,"parameters":{"modulus":8,"a":14,"b":31,"x":5,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":14,\"b\":31,\"modulus\":8,\"x\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0023","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":9,\"a\":15,\"b\":33,\"x\":4}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(9, 15, 33, 4, 3)","explanation_zh":"應用基礎核對=(9, 15, 33, 4, 3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201023,"parameters":{"modulus":9,"a":15,"b":33,"x":4,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":15,\"b\":33,\"modulus\":9,\"x\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0024","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":10,\"a\":16,\"b\":35,\"x\":3}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(10, 16, 35, 3, 3)","explanation_zh":"應用基礎核對=(10, 16, 35, 3, 3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201024,"parameters":{"modulus":10,"a":16,"b":35,"x":3,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":16,\"b\":35,\"modulus\":10,\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0025","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":5,\"a\":17,\"b\":37,\"x\":4}。請依 [modulus,a,b,x,T(x)] 欄序回答。","answer":{"kind":"expression","value":"(5, 17, 37, 4, 0)","explanation_zh":"應用基礎核對=(5, 17, 37, 4, 0)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-state-update","seed":202610201025,"parameters":{"modulus":5,"a":17,"b":37,"x":4,"domain":"Fin_discrete_dynamics","operation":"dynamics-state-update","semantic_template":"qf-app20-dynamics-state-update/{\"a\":17,\"b\":37,\"modulus\":5,\"x\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0026","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":5,\"a\":1,\"b\":-7,\"x0\":0,\"steps\":0}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(5, 1, -7, 0, 0, [0])","explanation_zh":"應用基礎核對=(5, 1, -7, 0, 0, [0])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201026,"parameters":{"modulus":5,"a":1,"b":-7,"x0":0,"steps":0,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":1,\"b\":-7,\"modulus\":5,\"steps\":0,\"x0\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0027","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":6,\"a\":2,\"b\":-4,\"x0\":1,\"steps\":1}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(6, 2, -4, 1, 1, [1, 4])","explanation_zh":"應用基礎核對=(6, 2, -4, 1, 1, [1, 4])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201027,"parameters":{"modulus":6,"a":2,"b":-4,"x0":1,"steps":1,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":2,\"b\":-4,\"modulus\":6,\"steps\":1,\"x0\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0028","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":7,\"a\":3,\"b\":-1,\"x0\":2,\"steps\":2}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(7, 3, -1, 2, 2, [2, 5, 0])","explanation_zh":"應用基礎核對=(7, 3, -1, 2, 2, [2, 5, 0])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201028,"parameters":{"modulus":7,"a":3,"b":-1,"x0":2,"steps":2,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":3,\"b\":-1,\"modulus\":7,\"steps\":2,\"x0\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0029","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":8,\"a\":4,\"b\":2,\"x0\":3,\"steps\":3}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(8, 4, 2, 3, 3, [3, 6, 2, 2])","explanation_zh":"應用基礎核對=(8, 4, 2, 3, 3, [3, 6, 2, 2])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201029,"parameters":{"modulus":8,"a":4,"b":2,"x0":3,"steps":3,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":4,\"b\":2,\"modulus\":8,\"steps\":3,\"x0\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0030","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":9,\"a\":5,\"b\":5,\"x0\":4,\"steps\":4}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(9, 5, 5, 4, 4, [4, 7, 4, 7, 4])","explanation_zh":"應用基礎核對=(9, 5, 5, 4, 4, [4, 7, 4, 7, 4])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201030,"parameters":{"modulus":9,"a":5,"b":5,"x0":4,"steps":4,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":5,\"b\":5,\"modulus\":9,\"steps\":4,\"x0\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0031","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":5,\"a\":6,\"b\":8,\"x0\":0,\"steps\":5}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 8, 0, 5, [0, 3, 1, 4, 2, 0])","explanation_zh":"應用基礎核對=(5, 6, 8, 0, 5, [0, 3, 1, 4, 2, 0])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201031,"parameters":{"modulus":5,"a":6,"b":8,"x0":0,"steps":5,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":6,\"b\":8,\"modulus\":5,\"steps\":5,\"x0\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0032","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":6,\"a\":7,\"b\":11,\"x0\":0,\"steps\":6}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(6, 7, 11, 0, 6, [0, 5, 4, 3, 2, 1, 0])","explanation_zh":"應用基礎核對=(6, 7, 11, 0, 6, [0, 5, 4, 3, 2, 1, 0])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201032,"parameters":{"modulus":6,"a":7,"b":11,"x0":0,"steps":6,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":7,\"b\":11,\"modulus\":6,\"steps\":6,\"x0\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0033","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":7,\"a\":8,\"b\":14,\"x0\":0,\"steps\":7}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(7, 8, 14, 0, 7, [0, 0, 0, 0, 0, 0, 0, 0])","explanation_zh":"應用基礎核對=(7, 8, 14, 0, 7, [0, 0, 0, 0, 0, 0, 0, 0])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201033,"parameters":{"modulus":7,"a":8,"b":14,"x0":0,"steps":7,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":8,\"b\":14,\"modulus\":7,\"steps\":7,\"x0\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0034","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":8,\"a\":9,\"b\":17,\"x0\":0,\"steps\":8}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(8, 9, 17, 0, 8, [0, 1, 2, 3, 4, 5, 6, 7, 0])","explanation_zh":"應用基礎核對=(8, 9, 17, 0, 8, [0, 1, 2, 3, 4, 5, 6, 7, 0])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201034,"parameters":{"modulus":8,"a":9,"b":17,"x0":0,"steps":8,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":9,\"b\":17,\"modulus\":8,\"steps\":8,\"x0\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0035","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":9,\"a\":10,\"b\":20,\"x0\":0,\"steps\":0}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(9, 10, 20, 0, 0, [0])","explanation_zh":"應用基礎核對=(9, 10, 20, 0, 0, [0])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201035,"parameters":{"modulus":9,"a":10,"b":20,"x0":0,"steps":0,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":10,\"b\":20,\"modulus\":9,\"steps\":0,\"x0\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0036","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":5,\"a\":11,\"b\":23,\"x0\":0,\"steps\":1}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(5, 11, 23, 0, 1, [0, 3])","explanation_zh":"應用基礎核對=(5, 11, 23, 0, 1, [0, 3])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201036,"parameters":{"modulus":5,"a":11,"b":23,"x0":0,"steps":1,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":11,\"b\":23,\"modulus\":5,\"steps\":1,\"x0\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0037","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":6,\"a\":12,\"b\":26,\"x0\":5,\"steps\":2}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(6, 12, 26, 5, 2, [5, 2, 2])","explanation_zh":"應用基礎核對=(6, 12, 26, 5, 2, [5, 2, 2])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201037,"parameters":{"modulus":6,"a":12,"b":26,"x0":5,"steps":2,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":12,\"b\":26,\"modulus\":6,\"steps\":2,\"x0\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0038","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":7,\"a\":13,\"b\":29,\"x0\":5,\"steps\":3}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(7, 13, 29, 5, 3, [5, 3, 5, 3])","explanation_zh":"應用基礎核對=(7, 13, 29, 5, 3, [5, 3, 5, 3])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201038,"parameters":{"modulus":7,"a":13,"b":29,"x0":5,"steps":3,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":13,\"b\":29,\"modulus\":7,\"steps\":3,\"x0\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0039","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":8,\"a\":14,\"b\":32,\"x0\":5,\"steps\":4}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(8, 14, 32, 5, 4, [5, 6, 4, 0, 0])","explanation_zh":"應用基礎核對=(8, 14, 32, 5, 4, [5, 6, 4, 0, 0])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201039,"parameters":{"modulus":8,"a":14,"b":32,"x0":5,"steps":4,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":14,\"b\":32,\"modulus\":8,\"steps\":4,\"x0\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0040","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":9,\"a\":15,\"b\":35,\"x0\":5,\"steps\":5}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(9, 15, 35, 5, 5, [5, 2, 2, 2, 2, 2])","explanation_zh":"應用基礎核對=(9, 15, 35, 5, 5, [5, 2, 2, 2, 2, 2])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201040,"parameters":{"modulus":9,"a":15,"b":35,"x0":5,"steps":5,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":15,\"b\":35,\"modulus\":9,\"steps\":5,\"x0\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0041","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":5,\"a\":16,\"b\":38,\"x0\":0,\"steps\":6}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(5, 16, 38, 0, 6, [0, 3, 1, 4, 2, 0, 3])","explanation_zh":"應用基礎核對=(5, 16, 38, 0, 6, [0, 3, 1, 4, 2, 0, 3])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201041,"parameters":{"modulus":5,"a":16,"b":38,"x0":0,"steps":6,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":16,\"b\":38,\"modulus\":5,\"steps\":6,\"x0\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0042","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":6,\"a\":17,\"b\":41,\"x0\":4,\"steps\":7}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(6, 17, 41, 4, 7, [4, 1, 4, 1, 4, 1, 4, 1])","explanation_zh":"應用基礎核對=(6, 17, 41, 4, 7, [4, 1, 4, 1, 4, 1, 4, 1])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201042,"parameters":{"modulus":6,"a":17,"b":41,"x0":4,"steps":7,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":17,\"b\":41,\"modulus\":6,\"steps\":7,\"x0\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0043","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":7,\"a\":18,\"b\":44,\"x0\":3,\"steps\":8}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(7, 18, 44, 3, 8, [3, 0, 2, 3, 0, 2, 3, 0, 2])","explanation_zh":"應用基礎核對=(7, 18, 44, 3, 8, [3, 0, 2, 3, 0, 2, 3, 0, 2])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201043,"parameters":{"modulus":7,"a":18,"b":44,"x0":3,"steps":8,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":18,\"b\":44,\"modulus\":7,\"steps\":8,\"x0\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0044","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":8,\"a\":19,\"b\":47,\"x0\":2,\"steps\":0}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(8, 19, 47, 2, 0, [2])","explanation_zh":"應用基礎核對=(8, 19, 47, 2, 0, [2])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201044,"parameters":{"modulus":8,"a":19,"b":47,"x0":2,"steps":0,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":19,\"b\":47,\"modulus\":8,\"steps\":0,\"x0\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0045","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":9,\"a\":20,\"b\":50,\"x0\":1,\"steps\":1}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(9, 20, 50, 1, 1, [1, 7])","explanation_zh":"應用基礎核對=(9, 20, 50, 1, 1, [1, 7])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201045,"parameters":{"modulus":9,"a":20,"b":50,"x0":1,"steps":1,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":20,\"b\":50,\"modulus\":9,\"steps\":1,\"x0\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0046","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":5,\"a\":21,\"b\":53,\"x0\":0,\"steps\":2}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(5, 21, 53, 0, 2, [0, 3, 1])","explanation_zh":"應用基礎核對=(5, 21, 53, 0, 2, [0, 3, 1])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201046,"parameters":{"modulus":5,"a":21,"b":53,"x0":0,"steps":2,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":21,\"b\":53,\"modulus\":5,\"steps\":2,\"x0\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0047","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":6,\"a\":22,\"b\":56,\"x0\":3,\"steps\":3}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(6, 22, 56, 3, 3, [3, 2, 4, 0])","explanation_zh":"應用基礎核對=(6, 22, 56, 3, 3, [3, 2, 4, 0])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201047,"parameters":{"modulus":6,"a":22,"b":56,"x0":3,"steps":3,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":22,\"b\":56,\"modulus\":6,\"steps\":3,\"x0\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0048","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":7,\"a\":23,\"b\":59,\"x0\":1,\"steps\":4}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(7, 23, 59, 1, 4, [1, 5, 6, 1, 5])","explanation_zh":"應用基礎核對=(7, 23, 59, 1, 4, [1, 5, 6, 1, 5])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201048,"parameters":{"modulus":7,"a":23,"b":59,"x0":1,"steps":4,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":23,\"b\":59,\"modulus\":7,\"steps\":4,\"x0\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0049","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":8,\"a\":24,\"b\":62,\"x0\":7,\"steps\":5}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(8, 24, 62, 7, 5, [7, 6, 6, 6, 6, 6])","explanation_zh":"應用基礎核對=(8, 24, 62, 7, 5, [7, 6, 6, 6, 6, 6])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201049,"parameters":{"modulus":8,"a":24,"b":62,"x0":7,"steps":5,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":24,\"b\":62,\"modulus\":8,\"steps\":5,\"x0\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0050","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-orbit-prefix；資料={\"modulus\":9,\"a\":25,\"b\":65,\"x0\":6,\"steps\":6}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(9, 25, 65, 6, 6, [6, 8, 4, 3, 5, 1, 0])","explanation_zh":"應用基礎核對=(9, 25, 65, 6, 6, [6, 8, 4, 3, 5, 1, 0])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201050,"parameters":{"modulus":9,"a":25,"b":65,"x0":6,"steps":6,"domain":"Fin_discrete_dynamics","operation":"dynamics-orbit-prefix","semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":25,\"b\":65,\"modulus\":9,\"steps\":6,\"x0\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0051","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":5,\"a\":-4,\"b\":1}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(5, -4, 1, [])","explanation_zh":"應用基礎核對=(5, -4, 1, [])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201051,"parameters":{"modulus":5,"a":-4,"b":1,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":-4,\"b\":1,\"modulus\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0052","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":6,\"a\":-3,\"b\":3}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(6, -3, 3, [])","explanation_zh":"應用基礎核對=(6, -3, 3, [])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201052,"parameters":{"modulus":6,"a":-3,"b":3,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":-3,\"b\":3,\"modulus\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0053","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":7,\"a\":-2,\"b\":5}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(7, -2, 5, [4])","explanation_zh":"應用基礎核對=(7, -2, 5, [4])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201053,"parameters":{"modulus":7,"a":-2,"b":5,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":-2,\"b\":5,\"modulus\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0054","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":8,\"a\":-1,\"b\":7}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(8, -1, 7, [])","explanation_zh":"應用基礎核對=(8, -1, 7, [])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201054,"parameters":{"modulus":8,"a":-1,"b":7,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":-1,\"b\":7,\"modulus\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0055","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":9,\"a\":0,\"b\":9}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(9, 0, 9, [0])","explanation_zh":"應用基礎核對=(9, 0, 9, [0])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201055,"parameters":{"modulus":9,"a":0,"b":9,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":0,\"b\":9,\"modulus\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0056","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":10,\"a\":1,\"b\":11}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(10, 1, 11, [])","explanation_zh":"應用基礎核對=(10, 1, 11, [])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201056,"parameters":{"modulus":10,"a":1,"b":11,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":1,\"b\":11,\"modulus\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0057","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":5,\"a\":2,\"b\":13}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(5, 2, 13, [2])","explanation_zh":"應用基礎核對=(5, 2, 13, [2])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201057,"parameters":{"modulus":5,"a":2,"b":13,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":2,\"b\":13,\"modulus\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0058","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":6,\"a\":3,\"b\":15}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, 15, [])","explanation_zh":"應用基礎核對=(6, 3, 15, [])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201058,"parameters":{"modulus":6,"a":3,"b":15,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":3,\"b\":15,\"modulus\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0059","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":7,\"a\":4,\"b\":17}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(7, 4, 17, [6])","explanation_zh":"應用基礎核對=(7, 4, 17, [6])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201059,"parameters":{"modulus":7,"a":4,"b":17,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":4,\"b\":17,\"modulus\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0060","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":8,\"a\":5,\"b\":19}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(8, 5, 19, [])","explanation_zh":"應用基礎核對=(8, 5, 19, [])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201060,"parameters":{"modulus":8,"a":5,"b":19,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":5,\"b\":19,\"modulus\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0061","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":9,\"a\":6,\"b\":21}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(9, 6, 21, [3])","explanation_zh":"應用基礎核對=(9, 6, 21, [3])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201061,"parameters":{"modulus":9,"a":6,"b":21,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":6,\"b\":21,\"modulus\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0062","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":10,\"a\":7,\"b\":23}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(10, 7, 23, [])","explanation_zh":"應用基礎核對=(10, 7, 23, [])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201062,"parameters":{"modulus":10,"a":7,"b":23,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":7,\"b\":23,\"modulus\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0063","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":5,\"a\":8,\"b\":25}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(5, 8, 25, [0])","explanation_zh":"應用基礎核對=(5, 8, 25, [0])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201063,"parameters":{"modulus":5,"a":8,"b":25,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":8,\"b\":25,\"modulus\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0064","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":6,\"a\":9,\"b\":27}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(6, 9, 27, [])","explanation_zh":"應用基礎核對=(6, 9, 27, [])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201064,"parameters":{"modulus":6,"a":9,"b":27,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":9,\"b\":27,\"modulus\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0065","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":7,\"a\":10,\"b\":29}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(7, 10, 29, [3])","explanation_zh":"應用基礎核對=(7, 10, 29, [3])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201065,"parameters":{"modulus":7,"a":10,"b":29,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":10,\"b\":29,\"modulus\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0066","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":8,\"a\":11,\"b\":31}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(8, 11, 31, [])","explanation_zh":"應用基礎核對=(8, 11, 31, [])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201066,"parameters":{"modulus":8,"a":11,"b":31,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":11,\"b\":31,\"modulus\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0067","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":9,\"a\":12,\"b\":33}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(9, 12, 33, [6])","explanation_zh":"應用基礎核對=(9, 12, 33, [6])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201067,"parameters":{"modulus":9,"a":12,"b":33,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":12,\"b\":33,\"modulus\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0068","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":10,\"a\":13,\"b\":35}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(10, 13, 35, [])","explanation_zh":"應用基礎核對=(10, 13, 35, [])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201068,"parameters":{"modulus":10,"a":13,"b":35,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":13,\"b\":35,\"modulus\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0069","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":5,\"a\":14,\"b\":37}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(5, 14, 37, [1])","explanation_zh":"應用基礎核對=(5, 14, 37, [1])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201069,"parameters":{"modulus":5,"a":14,"b":37,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":14,\"b\":37,\"modulus\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0070","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":6,\"a\":15,\"b\":39}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(6, 15, 39, [])","explanation_zh":"應用基礎核對=(6, 15, 39, [])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201070,"parameters":{"modulus":6,"a":15,"b":39,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":15,\"b\":39,\"modulus\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0071","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":7,\"a\":16,\"b\":41}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(7, 16, 41, [1])","explanation_zh":"應用基礎核對=(7, 16, 41, [1])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201071,"parameters":{"modulus":7,"a":16,"b":41,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":16,\"b\":41,\"modulus\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0072","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":8,\"a\":17,\"b\":43}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(8, 17, 43, [])","explanation_zh":"應用基礎核對=(8, 17, 43, [])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201072,"parameters":{"modulus":8,"a":17,"b":43,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":17,\"b\":43,\"modulus\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0073","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":9,\"a\":18,\"b\":45}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(9, 18, 45, [0])","explanation_zh":"應用基礎核對=(9, 18, 45, [0])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201073,"parameters":{"modulus":9,"a":18,"b":45,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":18,\"b\":45,\"modulus\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0074","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":10,\"a\":19,\"b\":47}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(10, 19, 47, [])","explanation_zh":"應用基礎核對=(10, 19, 47, [])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201074,"parameters":{"modulus":10,"a":19,"b":47,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":19,\"b\":47,\"modulus\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0075","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-fixed-point-inventory；資料={\"modulus\":5,\"a\":20,\"b\":49}。請依 [modulus,a,b,fixed_points] 欄序回答。","answer":{"kind":"expression","value":"(5, 20, 49, [4])","explanation_zh":"應用基礎核對=(5, 20, 49, [4])；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201075,"parameters":{"modulus":5,"a":20,"b":49,"domain":"Fin_discrete_dynamics","operation":"dynamics-fixed-point-inventory","semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":20,\"b\":49,\"modulus\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0076","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":5,\"a\":1,\"b\":-3,\"subset\":[0,1]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(5, 1, -3, [0, 1], [2, 3], 否)","explanation_zh":"應用基礎核對=(5, 1, -3, [0, 1], [2, 3], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201076,"parameters":{"modulus":5,"a":1,"b":-3,"subset":[0,1],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":1,\"b\":-3,\"modulus\":5,\"subset\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0077","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":6,\"a\":2,\"b\":-2,\"subset\":[2]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(6, 2, -2, [2], [2], 是)","explanation_zh":"應用基礎核對=(6, 2, -2, [2], [2], 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201077,"parameters":{"modulus":6,"a":2,"b":-2,"subset":[2],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":2,\"b\":-2,\"modulus\":6,\"subset\":[2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0078","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":7,\"a\":3,\"b\":-1,\"subset\":[0,2]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(7, 3, -1, [0, 2], [5, 6], 否)","explanation_zh":"應用基礎核對=(7, 3, -1, [0, 2], [5, 6], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201078,"parameters":{"modulus":7,"a":3,"b":-1,"subset":[0,2],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":3,\"b\":-1,\"modulus\":7,\"subset\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0079","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":8,\"a\":4,\"b\":0,\"subset\":[1,2]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(8, 4, 0, [1, 2], [0, 4], 否)","explanation_zh":"應用基礎核對=(8, 4, 0, [1, 2], [0, 4], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201079,"parameters":{"modulus":8,"a":4,"b":0,"subset":[1,2],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":4,\"b\":0,\"modulus\":8,\"subset\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0080","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":9,\"a\":5,\"b\":1,\"subset\":[0,1,2]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(9, 5, 1, [0, 1, 2], [1, 2, 6], 否)","explanation_zh":"應用基礎核對=(9, 5, 1, [0, 1, 2], [1, 2, 6], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201080,"parameters":{"modulus":9,"a":5,"b":1,"subset":[0,1,2],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":5,\"b\":1,\"modulus\":9,\"subset\":[0,1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0081","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":5,\"a\":6,\"b\":2,\"subset\":[3]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 2, [3], [0], 否)","explanation_zh":"應用基礎核對=(5, 6, 2, [3], [0], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201081,"parameters":{"modulus":5,"a":6,"b":2,"subset":[3],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":6,\"b\":2,\"modulus\":5,\"subset\":[3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0082","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":6,\"a\":7,\"b\":3,\"subset\":[0,3]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(6, 7, 3, [0, 3], [0, 3], 是)","explanation_zh":"應用基礎核對=(6, 7, 3, [0, 3], [0, 3], 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201082,"parameters":{"modulus":6,"a":7,"b":3,"subset":[0,3],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":7,\"b\":3,\"modulus\":6,\"subset\":[0,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0083","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":7,\"a\":8,\"b\":4,\"subset\":[1,3]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(7, 8, 4, [1, 3], [0, 5], 否)","explanation_zh":"應用基礎核對=(7, 8, 4, [1, 3], [0, 5], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201083,"parameters":{"modulus":7,"a":8,"b":4,"subset":[1,3],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":8,\"b\":4,\"modulus\":7,\"subset\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0084","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":8,\"a\":9,\"b\":5,\"subset\":[0,1,3]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(8, 9, 5, [0, 1, 3], [0, 5, 6], 否)","explanation_zh":"應用基礎核對=(8, 9, 5, [0, 1, 3], [0, 5, 6], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201084,"parameters":{"modulus":8,"a":9,"b":5,"subset":[0,1,3],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":9,\"b\":5,\"modulus\":8,\"subset\":[0,1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0085","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":9,\"a\":10,\"b\":6,\"subset\":[2,3]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(9, 10, 6, [2, 3], [0, 8], 否)","explanation_zh":"應用基礎核對=(9, 10, 6, [2, 3], [0, 8], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201085,"parameters":{"modulus":9,"a":10,"b":6,"subset":[2,3],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":10,\"b\":6,\"modulus\":9,\"subset\":[2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0086","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":5,\"a\":11,\"b\":7,\"subset\":[0,2,3]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(5, 11, 7, [0, 2, 3], [0, 2, 4], 否)","explanation_zh":"應用基礎核對=(5, 11, 7, [0, 2, 3], [0, 2, 4], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201086,"parameters":{"modulus":5,"a":11,"b":7,"subset":[0,2,3],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":11,\"b\":7,\"modulus\":5,\"subset\":[0,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0087","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":6,\"a\":12,\"b\":8,\"subset\":[1,2,3]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(6, 12, 8, [1, 2, 3], [2], 是)","explanation_zh":"應用基礎核對=(6, 12, 8, [1, 2, 3], [2], 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201087,"parameters":{"modulus":6,"a":12,"b":8,"subset":[1,2,3],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":12,\"b\":8,\"modulus\":6,\"subset\":[1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0088","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":7,\"a\":13,\"b\":9,\"subset\":[0,1,2,3]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(7, 13, 9, [0, 1, 2, 3], [0, 1, 2, 6], 否)","explanation_zh":"應用基礎核對=(7, 13, 9, [0, 1, 2, 3], [0, 1, 2, 6], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201088,"parameters":{"modulus":7,"a":13,"b":9,"subset":[0,1,2,3],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":13,\"b\":9,\"modulus\":7,\"subset\":[0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0089","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":8,\"a\":14,\"b\":10,\"subset\":[4]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(8, 14, 10, [4], [2], 否)","explanation_zh":"應用基礎核對=(8, 14, 10, [4], [2], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201089,"parameters":{"modulus":8,"a":14,"b":10,"subset":[4],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":14,\"b\":10,\"modulus\":8,\"subset\":[4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0090","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":9,\"a\":15,\"b\":11,\"subset\":[0,4]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(9, 15, 11, [0, 4], [2, 8], 否)","explanation_zh":"應用基礎核對=(9, 15, 11, [0, 4], [2, 8], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201090,"parameters":{"modulus":9,"a":15,"b":11,"subset":[0,4],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":15,\"b\":11,\"modulus\":9,\"subset\":[0,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0091","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":5,\"a\":16,\"b\":12,\"subset\":[1,4]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(5, 16, 12, [1, 4], [1, 3], 否)","explanation_zh":"應用基礎核對=(5, 16, 12, [1, 4], [1, 3], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201091,"parameters":{"modulus":5,"a":16,"b":12,"subset":[1,4],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":16,\"b\":12,\"modulus\":5,\"subset\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0092","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":6,\"a\":17,\"b\":13,\"subset\":[0,1,4]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(6, 17, 13, [0, 1, 4], [0, 1, 3], 否)","explanation_zh":"應用基礎核對=(6, 17, 13, [0, 1, 4], [0, 1, 3], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201092,"parameters":{"modulus":6,"a":17,"b":13,"subset":[0,1,4],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":17,\"b\":13,\"modulus\":6,\"subset\":[0,1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0093","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":7,\"a\":18,\"b\":14,\"subset\":[2,4]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(7, 18, 14, [2, 4], [1, 2], 否)","explanation_zh":"應用基礎核對=(7, 18, 14, [2, 4], [1, 2], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201093,"parameters":{"modulus":7,"a":18,"b":14,"subset":[2,4],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":18,\"b\":14,\"modulus\":7,\"subset\":[2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0094","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":8,\"a\":19,\"b\":15,\"subset\":[0,2,4]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(8, 19, 15, [0, 2, 4], [3, 5, 7], 否)","explanation_zh":"應用基礎核對=(8, 19, 15, [0, 2, 4], [3, 5, 7], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201094,"parameters":{"modulus":8,"a":19,"b":15,"subset":[0,2,4],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":19,\"b\":15,\"modulus\":8,\"subset\":[0,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0095","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":9,\"a\":20,\"b\":16,\"subset\":[1,2,4]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(9, 20, 16, [1, 2, 4], [0, 2, 6], 否)","explanation_zh":"應用基礎核對=(9, 20, 16, [1, 2, 4], [0, 2, 6], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201095,"parameters":{"modulus":9,"a":20,"b":16,"subset":[1,2,4],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":20,\"b\":16,\"modulus\":9,\"subset\":[1,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0096","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":5,\"a\":21,\"b\":17,\"subset\":[0,1,2,4]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(5, 21, 17, [0, 1, 2, 4], [1, 2, 3, 4], 否)","explanation_zh":"應用基礎核對=(5, 21, 17, [0, 1, 2, 4], [1, 2, 3, 4], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201096,"parameters":{"modulus":5,"a":21,"b":17,"subset":[0,1,2,4],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":21,\"b\":17,\"modulus\":5,\"subset\":[0,1,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0097","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":6,\"a\":22,\"b\":18,\"subset\":[3,4]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(6, 22, 18, [3, 4], [0, 4], 否)","explanation_zh":"應用基礎核對=(6, 22, 18, [3, 4], [0, 4], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201097,"parameters":{"modulus":6,"a":22,"b":18,"subset":[3,4],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":22,\"b\":18,\"modulus\":6,\"subset\":[3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0098","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":7,\"a\":23,\"b\":19,\"subset\":[0,3,4]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(7, 23, 19, [0, 3, 4], [4, 5, 6], 否)","explanation_zh":"應用基礎核對=(7, 23, 19, [0, 3, 4], [4, 5, 6], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201098,"parameters":{"modulus":7,"a":23,"b":19,"subset":[0,3,4],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":23,\"b\":19,\"modulus\":7,\"subset\":[0,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0099","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":8,\"a\":24,\"b\":20,\"subset\":[1,3,4]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(8, 24, 20, [1, 3, 4], [4], 是)","explanation_zh":"應用基礎核對=(8, 24, 20, [1, 3, 4], [4], 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201099,"parameters":{"modulus":8,"a":24,"b":20,"subset":[1,3,4],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":24,\"b\":20,\"modulus\":8,\"subset\":[1,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0100","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-dynamical-system"],"title_zh":"有限離散動力核對","stem_zh":"明示有限狀態自映射。操作=dynamics-invariant-subset-audit；資料={\"modulus\":9,\"a\":25,\"b\":21,\"subset\":[0,1,3,4]}。請依 [modulus,a,b,subset,image,invariant] 欄序回答。","answer":{"kind":"expression","value":"(9, 25, 21, [0, 1, 3, 4], [1, 3, 4, 6], 否)","explanation_zh":"應用基礎核對=(9, 25, 21, [0, 1, 3, 4], [1, 3, 4, 6], 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；有限軌道不代表一般動力行為；不宣稱穩定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限離散動力核對"]},"variant":{"family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201100,"parameters":{"modulus":9,"a":25,"b":21,"subset":[0,1,3,4],"domain":"Fin_discrete_dynamics","operation":"dynamics-invariant-subset-audit","semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":25,\"b\":21,\"modulus\":9,\"subset\":[0,1,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}