{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r03-orbit-fixed-periodic-0001","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":5,\"a\":-7,\"b\":-9,\"x\":0}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(5, -7, -9, 0, 1, 否)","explanation_zh":"應用擴展核對=(5, -7, -9, 0, 1, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213001,"parameters":{"modulus":5,"a":-7,"b":-9,"x":0,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":-7,\"b\":-9,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0002","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":6,\"a\":-6,\"b\":-7,\"x\":1}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(6, -6, -7, 1, 5, 否)","explanation_zh":"應用擴展核對=(6, -6, -7, 1, 5, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213002,"parameters":{"modulus":6,"a":-6,"b":-7,"x":1,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":-6,\"b\":-7,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0003","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":7,\"a\":-5,\"b\":-5,\"x\":2}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(7, -5, -5, 2, 6, 否)","explanation_zh":"應用擴展核對=(7, -5, -5, 2, 6, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213003,"parameters":{"modulus":7,"a":-5,"b":-5,"x":2,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":-5,\"b\":-5,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0004","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":8,\"a\":-4,\"b\":-3,\"x\":3}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(8, -4, -3, 3, 1, 否)","explanation_zh":"應用擴展核對=(8, -4, -3, 3, 1, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213004,"parameters":{"modulus":8,"a":-4,"b":-3,"x":3,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":-4,\"b\":-3,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0005","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":9,\"a\":-3,\"b\":-1,\"x\":4}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(9, -3, -1, 4, 5, 否)","explanation_zh":"應用擴展核對=(9, -3, -1, 4, 5, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213005,"parameters":{"modulus":9,"a":-3,"b":-1,"x":4,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":-3,\"b\":-1,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0006","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":10,\"a\":-2,\"b\":1,\"x\":5}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(10, -2, 1, 5, 1, 否)","explanation_zh":"應用擴展核對=(10, -2, 1, 5, 1, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213006,"parameters":{"modulus":10,"a":-2,"b":1,"x":5,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0007","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":5,\"a\":-1,\"b\":3,\"x\":1}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(5, -1, 3, 1, 2, 否)","explanation_zh":"應用擴展核對=(5, -1, 3, 1, 2, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213007,"parameters":{"modulus":5,"a":-1,"b":3,"x":1,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":-1,\"b\":3,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0008","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":6,\"a\":0,\"b\":5,\"x\":1}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 5, 1, 5, 否)","explanation_zh":"應用擴展核對=(6, 0, 5, 1, 5, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213008,"parameters":{"modulus":6,"a":0,"b":5,"x":1,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":0,\"b\":5,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0009","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":7,\"a\":1,\"b\":7,\"x\":1}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(7, 1, 7, 1, 1, 是)","explanation_zh":"應用擴展核對=(7, 1, 7, 1, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213009,"parameters":{"modulus":7,"a":1,"b":7,"x":1,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":1,\"b\":7,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0010","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":8,\"a\":2,\"b\":9,\"x\":1}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(8, 2, 9, 1, 3, 否)","explanation_zh":"應用擴展核對=(8, 2, 9, 1, 3, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213010,"parameters":{"modulus":8,"a":2,"b":9,"x":1,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":2,\"b\":9,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0011","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":9,\"a\":3,\"b\":11,\"x\":1}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(9, 3, 11, 1, 5, 否)","explanation_zh":"應用擴展核對=(9, 3, 11, 1, 5, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213011,"parameters":{"modulus":9,"a":3,"b":11,"x":1,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":3,\"b\":11,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0012","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":10,\"a\":4,\"b\":13,\"x\":1}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(10, 4, 13, 1, 7, 否)","explanation_zh":"應用擴展核對=(10, 4, 13, 1, 7, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213012,"parameters":{"modulus":10,"a":4,"b":13,"x":1,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":4,\"b\":13,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0013","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":5,\"a\":5,\"b\":15,\"x\":2}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(5, 5, 15, 2, 0, 否)","explanation_zh":"應用擴展核對=(5, 5, 15, 2, 0, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213013,"parameters":{"modulus":5,"a":5,"b":15,"x":2,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":5,\"b\":15,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0014","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":6,\"a\":6,\"b\":17,\"x\":1}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(6, 6, 17, 1, 5, 否)","explanation_zh":"應用擴展核對=(6, 6, 17, 1, 5, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213014,"parameters":{"modulus":6,"a":6,"b":17,"x":1,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":6,\"b\":17,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0015","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":7,\"a\":7,\"b\":19,\"x\":0}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(7, 7, 19, 0, 5, 否)","explanation_zh":"應用擴展核對=(7, 7, 19, 0, 5, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213015,"parameters":{"modulus":7,"a":7,"b":19,"x":0,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":7,\"b\":19,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0016","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":8,\"a\":8,\"b\":21,\"x\":7}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(8, 8, 21, 7, 5, 否)","explanation_zh":"應用擴展核對=(8, 8, 21, 7, 5, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213016,"parameters":{"modulus":8,"a":8,"b":21,"x":7,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":8,\"b\":21,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0017","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":9,\"a\":9,\"b\":23,\"x\":7}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(9, 9, 23, 7, 5, 否)","explanation_zh":"應用擴展核對=(9, 9, 23, 7, 5, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213017,"parameters":{"modulus":9,"a":9,"b":23,"x":7,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":9,\"b\":23,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0018","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":10,\"a\":10,\"b\":25,\"x\":7}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(10, 10, 25, 7, 5, 否)","explanation_zh":"應用擴展核對=(10, 10, 25, 7, 5, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213018,"parameters":{"modulus":10,"a":10,"b":25,"x":7,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":10,\"b\":25,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0019","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":5,\"a\":11,\"b\":27,\"x\":3}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(5, 11, 27, 3, 0, 否)","explanation_zh":"應用擴展核對=(5, 11, 27, 3, 0, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213019,"parameters":{"modulus":5,"a":11,"b":27,"x":3,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":11,\"b\":27,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0020","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":6,\"a\":12,\"b\":29,\"x\":1}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(6, 12, 29, 1, 5, 否)","explanation_zh":"應用擴展核對=(6, 12, 29, 1, 5, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213020,"parameters":{"modulus":6,"a":12,"b":29,"x":1,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":12,\"b\":29,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0021","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":7,\"a\":13,\"b\":31,\"x\":6}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(7, 13, 31, 6, 4, 否)","explanation_zh":"應用擴展核對=(7, 13, 31, 6, 4, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213021,"parameters":{"modulus":7,"a":13,"b":31,"x":6,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":13,\"b\":31,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0022","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":8,\"a\":14,\"b\":33,\"x\":5}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(8, 14, 33, 5, 7, 否)","explanation_zh":"應用擴展核對=(8, 14, 33, 5, 7, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213022,"parameters":{"modulus":8,"a":14,"b":33,"x":5,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":14,\"b\":33,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0023","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":9,\"a\":15,\"b\":35,\"x\":4}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(9, 15, 35, 4, 5, 否)","explanation_zh":"應用擴展核對=(9, 15, 35, 4, 5, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213023,"parameters":{"modulus":9,"a":15,"b":35,"x":4,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":15,\"b\":35,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0024","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":10,\"a\":16,\"b\":37,\"x\":3}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(10, 16, 37, 3, 5, 否)","explanation_zh":"應用擴展核對=(10, 16, 37, 3, 5, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213024,"parameters":{"modulus":10,"a":16,"b":37,"x":3,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":16,\"b\":37,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0025","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-fixed-audit；資料={\"modulus\":5,\"a\":17,\"b\":39,\"x\":4}。請依 [modulus,a,b,x,T_x,fixed] 欄序回答。","answer":{"kind":"expression","value":"(5, 17, 39, 4, 2, 否)","explanation_zh":"應用擴展核對=(5, 17, 39, 4, 2, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-fixed-audit","seed":202610213025,"parameters":{"modulus":5,"a":17,"b":39,"x":4,"domain":"Finite_orbit_periodicity","operation":"orbit-fixed-audit","semantic_template":"qf-app21-orbit-fixed-audit/{\"a\":17,\"b\":39,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0026","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":5,\"a\":-3,\"b\":-8,\"x\":0,\"p\":1}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(5, -3, -8, 0, 1, 2, 否, 4, 否)","explanation_zh":"應用擴展核對=(5, -3, -8, 0, 1, 2, 否, 4, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213026,"parameters":{"modulus":5,"a":-3,"b":-8,"x":0,"p":1,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":-3,\"b\":-8,\"modulus\":5,\"p\":1,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0027","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":6,\"a\":-2,\"b\":-7,\"x\":1,\"p\":2}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(6, -2, -7, 1, 2, 5, 否, 3, 否)","explanation_zh":"應用擴展核對=(6, -2, -7, 1, 2, 5, 否, 3, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213027,"parameters":{"modulus":6,"a":-2,"b":-7,"x":1,"p":2,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":-2,\"b\":-7,\"modulus\":6,\"p\":2,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0028","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":7,\"a\":-1,\"b\":-6,\"x\":2,\"p\":3}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(7, -1, -6, 2, 3, 6, 否, 2, 否)","explanation_zh":"應用擴展核對=(7, -1, -6, 2, 3, 6, 否, 2, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213028,"parameters":{"modulus":7,"a":-1,"b":-6,"x":2,"p":3,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":-1,\"b\":-6,\"modulus\":7,\"p\":3,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0029","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":8,\"a\":0,\"b\":-5,\"x\":3,\"p\":4}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, -5, 3, 4, 3, 是, 1, 否)","explanation_zh":"應用擴展核對=(8, 0, -5, 3, 4, 3, 是, 1, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213029,"parameters":{"modulus":8,"a":0,"b":-5,"x":3,"p":4,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":0,\"b\":-5,\"modulus\":8,\"p\":4,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0030","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":9,\"a\":1,\"b\":-4,\"x\":4,\"p\":5}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(9, 1, -4, 4, 5, 2, 否, 9, 否)","explanation_zh":"應用擴展核對=(9, 1, -4, 4, 5, 2, 否, 9, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213030,"parameters":{"modulus":9,"a":1,"b":-4,"x":4,"p":5,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":1,\"b\":-4,\"modulus\":9,\"p\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0031","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":10,\"a\":2,\"b\":-3,\"x\":5,\"p\":6}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(10, 2, -3, 5, 6, 1, 否, 4, 否)","explanation_zh":"應用擴展核對=(10, 2, -3, 5, 6, 1, 否, 4, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213031,"parameters":{"modulus":10,"a":2,"b":-3,"x":5,"p":6,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":2,\"b\":-3,\"modulus\":10,\"p\":6,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0032","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":5,\"a\":3,\"b\":-2,\"x\":1,\"p\":7}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(5, 3, -2, 1, 7, 1, 是, 1, 否)","explanation_zh":"應用擴展核對=(5, 3, -2, 1, 7, 1, 是, 1, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213032,"parameters":{"modulus":5,"a":3,"b":-2,"x":1,"p":7,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":3,\"b\":-2,\"modulus\":5,\"p\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0033","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":6,\"a\":-3,\"b\":-1,\"x\":1,\"p\":8}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(6, -3, -1, 1, 8, 5, 否, null, 否)","explanation_zh":"應用擴展核對=(6, -3, -1, 1, 8, 5, 否, null, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213033,"parameters":{"modulus":6,"a":-3,"b":-1,"x":1,"p":8,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":-3,\"b\":-1,\"modulus\":6,\"p\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0034","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":7,\"a\":-2,\"b\":0,\"x\":1,\"p\":1}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(7, -2, 0, 1, 1, 5, 否, 6, 否)","explanation_zh":"應用擴展核對=(7, -2, 0, 1, 1, 5, 否, 6, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213034,"parameters":{"modulus":7,"a":-2,"b":0,"x":1,"p":1,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":-2,\"b\":0,\"modulus\":7,\"p\":1,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0035","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":8,\"a\":-1,\"b\":1,\"x\":1,\"p\":2}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(8, -1, 1, 1, 2, 1, 是, 2, 是)","explanation_zh":"應用擴展核對=(8, -1, 1, 1, 2, 1, 是, 2, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213035,"parameters":{"modulus":8,"a":-1,"b":1,"x":1,"p":2,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":-1,\"b\":1,\"modulus\":8,\"p\":2,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0036","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":9,\"a\":0,\"b\":2,\"x\":1,\"p\":3}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(9, 0, 2, 1, 3, 2, 否, null, 否)","explanation_zh":"應用擴展核對=(9, 0, 2, 1, 3, 2, 否, null, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213036,"parameters":{"modulus":9,"a":0,"b":2,"x":1,"p":3,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":0,\"b\":2,\"modulus\":9,\"p\":3,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0037","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":10,\"a\":1,\"b\":3,\"x\":1,\"p\":4}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(10, 1, 3, 1, 4, 3, 否, 10, 否)","explanation_zh":"應用擴展核對=(10, 1, 3, 1, 4, 3, 否, 10, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213037,"parameters":{"modulus":10,"a":1,"b":3,"x":1,"p":4,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":1,\"b\":3,\"modulus\":10,\"p\":4,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0038","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":5,\"a\":2,\"b\":4,\"x\":2,\"p\":5}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(5, 2, 4, 2, 5, 3, 否, 4, 否)","explanation_zh":"應用擴展核對=(5, 2, 4, 2, 5, 3, 否, 4, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213038,"parameters":{"modulus":5,"a":2,"b":4,"x":2,"p":5,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":2,\"b\":4,\"modulus\":5,\"p\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0039","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":6,\"a\":3,\"b\":5,\"x\":1,\"p\":6}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, 5, 1, 6, 5, 否, null, 否)","explanation_zh":"應用擴展核對=(6, 3, 5, 1, 6, 5, 否, null, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213039,"parameters":{"modulus":6,"a":3,"b":5,"x":1,"p":6,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":3,\"b\":5,\"modulus\":6,\"p\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0040","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":7,\"a\":-3,\"b\":6,\"x\":0,\"p\":7}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(7, -3, 6, 0, 7, 6, 否, 3, 否)","explanation_zh":"應用擴展核對=(7, -3, 6, 0, 7, 6, 否, 3, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213040,"parameters":{"modulus":7,"a":-3,"b":6,"x":0,"p":7,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":-3,\"b\":6,\"modulus\":7,\"p\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0041","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":8,\"a\":-2,\"b\":7,\"x\":7,\"p\":8}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(8, -2, 7, 7, 8, 5, 否, null, 否)","explanation_zh":"應用擴展核對=(8, -2, 7, 7, 8, 5, 否, null, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213041,"parameters":{"modulus":8,"a":-2,"b":7,"x":7,"p":8,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":-2,\"b\":7,\"modulus\":8,\"p\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0042","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":9,\"a\":-1,\"b\":8,\"x\":7,\"p\":1}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(9, -1, 8, 7, 1, 1, 否, 2, 否)","explanation_zh":"應用擴展核對=(9, -1, 8, 7, 1, 1, 否, 2, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213042,"parameters":{"modulus":9,"a":-1,"b":8,"x":7,"p":1,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":-1,\"b\":8,\"modulus\":9,\"p\":1,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0043","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":10,\"a\":0,\"b\":9,\"x\":7,\"p\":2}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(10, 0, 9, 7, 2, 9, 否, null, 否)","explanation_zh":"應用擴展核對=(10, 0, 9, 7, 2, 9, 否, null, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213043,"parameters":{"modulus":10,"a":0,"b":9,"x":7,"p":2,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":0,\"b\":9,\"modulus\":10,\"p\":2,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0044","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":5,\"a\":1,\"b\":10,\"x\":3,\"p\":3}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(5, 1, 10, 3, 3, 3, 是, 1, 否)","explanation_zh":"應用擴展核對=(5, 1, 10, 3, 3, 3, 是, 1, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213044,"parameters":{"modulus":5,"a":1,"b":10,"x":3,"p":3,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":1,\"b\":10,\"modulus\":5,\"p\":3,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0045","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":6,\"a\":2,\"b\":11,\"x\":1,\"p\":4}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(6, 2, 11, 1, 4, 1, 是, 1, 否)","explanation_zh":"應用擴展核對=(6, 2, 11, 1, 4, 1, 是, 1, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213045,"parameters":{"modulus":6,"a":2,"b":11,"x":1,"p":4,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":2,\"b\":11,\"modulus\":6,\"p\":4,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0046","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":7,\"a\":3,\"b\":12,\"x\":6,\"p\":5}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(7, 3, 12, 6, 5, 5, 否, 6, 否)","explanation_zh":"應用擴展核對=(7, 3, 12, 6, 5, 5, 否, 6, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213046,"parameters":{"modulus":7,"a":3,"b":12,"x":6,"p":5,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":3,\"b\":12,\"modulus\":7,\"p\":5,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0047","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":8,\"a\":-3,\"b\":13,\"x\":5,\"p\":6}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(8, -3, 13, 5, 6, 7, 否, 8, 否)","explanation_zh":"應用擴展核對=(8, -3, 13, 5, 6, 7, 否, 8, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213047,"parameters":{"modulus":8,"a":-3,"b":13,"x":5,"p":6,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":-3,\"b\":13,\"modulus\":8,\"p\":6,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0048","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":9,\"a\":-2,\"b\":14,\"x\":4,\"p\":7}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(9, -2, 14, 4, 7, 0, 否, 9, 否)","explanation_zh":"應用擴展核對=(9, -2, 14, 4, 7, 0, 否, 9, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213048,"parameters":{"modulus":9,"a":-2,"b":14,"x":4,"p":7,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":-2,\"b\":14,\"modulus\":9,\"p\":7,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0049","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":10,\"a\":-1,\"b\":15,\"x\":3,\"p\":8}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(10, -1, 15, 3, 8, 3, 是, 2, 否)","explanation_zh":"應用擴展核對=(10, -1, 15, 3, 8, 3, 是, 2, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213049,"parameters":{"modulus":10,"a":-1,"b":15,"x":3,"p":8,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":-1,\"b\":15,\"modulus\":10,\"p\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0050","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-return-time-audit；資料={\"modulus\":5,\"a\":0,\"b\":16,\"x\":4,\"p\":1}。請依 [modulus,a,b,x,p,T_power_x,returns,least_period,period_exactly_p] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 16, 4, 1, 1, 否, null, 否)","explanation_zh":"應用擴展核對=(5, 0, 16, 4, 1, 1, 否, null, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-return-time-audit","seed":202610213050,"parameters":{"modulus":5,"a":0,"b":16,"x":4,"p":1,"domain":"Finite_orbit_periodicity","operation":"orbit-return-time-audit","semantic_template":"qf-app21-orbit-return-time-audit/{\"a\":0,\"b\":16,\"modulus\":5,\"p\":1,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0051","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":5,\"a\":-3,\"b\":-8,\"x\":1}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(5, -3, -8, 1, [1, 4, 0, 2], 1, 0, 4)","explanation_zh":"應用擴展核對=(5, -3, -8, 1, [1, 4, 0, 2], 1, 0, 4)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213051,"parameters":{"modulus":5,"a":-3,"b":-8,"x":1,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":-3,\"b\":-8,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0052","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":6,\"a\":-2,\"b\":-7,\"x\":3}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(6, -2, -7, 3, [3, 5, 1], 3, 0, 3)","explanation_zh":"應用擴展核對=(6, -2, -7, 3, [3, 5, 1], 3, 0, 3)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213052,"parameters":{"modulus":6,"a":-2,"b":-7,"x":3,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":-2,\"b\":-7,\"modulus\":6,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0053","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":7,\"a\":-1,\"b\":-6,\"x\":5}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(7, -1, -6, 5, [5, 3], 5, 0, 2)","explanation_zh":"應用擴展核對=(7, -1, -6, 5, [5, 3], 5, 0, 2)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213053,"parameters":{"modulus":7,"a":-1,"b":-6,"x":5,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":-1,\"b\":-6,\"modulus\":7,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0054","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":8,\"a\":0,\"b\":-5,\"x\":7}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, -5, 7, [7, 3], 3, 1, 非週期點)","explanation_zh":"應用擴展核對=(8, 0, -5, 7, [7, 3], 3, 1, 非週期點)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213054,"parameters":{"modulus":8,"a":0,"b":-5,"x":7,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":0,\"b\":-5,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0055","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":9,\"a\":1,\"b\":-4,\"x\":0}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(9, 1, -4, 0, [0, 5, 1, 6, 2, 7, 3, 8, 4], 0, 0, 9)","explanation_zh":"應用擴展核對=(9, 1, -4, 0, [0, 5, 1, 6, 2, 7, 3, 8, 4], 0, 0, 9)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213055,"parameters":{"modulus":9,"a":1,"b":-4,"x":0,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":1,\"b\":-4,\"modulus\":9,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0056","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":10,\"a\":2,\"b\":-3,\"x\":1}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(10, 2, -3, 1, [1, 9, 5, 7], 1, 0, 4)","explanation_zh":"應用擴展核對=(10, 2, -3, 1, [1, 9, 5, 7], 1, 0, 4)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213056,"parameters":{"modulus":10,"a":2,"b":-3,"x":1,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":2,\"b\":-3,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0057","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":5,\"a\":3,\"b\":-2,\"x\":3}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(5, 3, -2, 3, [3, 2, 4, 0], 3, 0, 4)","explanation_zh":"應用擴展核對=(5, 3, -2, 3, [3, 2, 4, 0], 3, 0, 4)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213057,"parameters":{"modulus":5,"a":3,"b":-2,"x":3,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":3,\"b\":-2,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0058","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":6,\"a\":-3,\"b\":-1,\"x\":3}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(6, -3, -1, 3, [3, 2, 5], 2, 1, 非週期點)","explanation_zh":"應用擴展核對=(6, -3, -1, 3, [3, 2, 5], 2, 1, 非週期點)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213058,"parameters":{"modulus":6,"a":-3,"b":-1,"x":3,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":-3,\"b\":-1,\"modulus\":6,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0059","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":7,\"a\":-2,\"b\":0,\"x\":3}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(7, -2, 0, 3, [3, 1, 5, 4, 6, 2], 3, 0, 6)","explanation_zh":"應用擴展核對=(7, -2, 0, 3, [3, 1, 5, 4, 6, 2], 3, 0, 6)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213059,"parameters":{"modulus":7,"a":-2,"b":0,"x":3,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":-2,\"b\":0,\"modulus\":7,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0060","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":8,\"a\":-1,\"b\":1,\"x\":3}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(8, -1, 1, 3, [3, 6], 3, 0, 2)","explanation_zh":"應用擴展核對=(8, -1, 1, 3, [3, 6], 3, 0, 2)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213060,"parameters":{"modulus":8,"a":-1,"b":1,"x":3,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":-1,\"b\":1,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0061","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":9,\"a\":0,\"b\":2,\"x\":3}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(9, 0, 2, 3, [3, 2], 2, 1, 非週期點)","explanation_zh":"應用擴展核對=(9, 0, 2, 3, [3, 2], 2, 1, 非週期點)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213061,"parameters":{"modulus":9,"a":0,"b":2,"x":3,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":0,\"b\":2,\"modulus\":9,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0062","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":10,\"a\":1,\"b\":3,\"x\":3}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(10, 1, 3, 3, [3, 6, 9, 2, 5, 8, 1, 4, 7, 0], 3, 0, 10)","explanation_zh":"應用擴展核對=(10, 1, 3, 3, [3, 6, 9, 2, 5, 8, 1, 4, 7, 0], 3, 0, 10)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213062,"parameters":{"modulus":10,"a":1,"b":3,"x":3,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":1,\"b\":3,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0063","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":5,\"a\":2,\"b\":4,\"x\":0}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(5, 2, 4, 0, [0, 4, 2, 3], 0, 0, 4)","explanation_zh":"應用擴展核對=(5, 2, 4, 0, [0, 4, 2, 3], 0, 0, 4)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213063,"parameters":{"modulus":5,"a":2,"b":4,"x":0,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":2,\"b\":4,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0064","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":6,\"a\":3,\"b\":5,\"x\":3}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, 5, 3, [3, 2, 5], 2, 1, 非週期點)","explanation_zh":"應用擴展核對=(6, 3, 5, 3, [3, 2, 5], 2, 1, 非週期點)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213064,"parameters":{"modulus":6,"a":3,"b":5,"x":3,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":3,\"b\":5,\"modulus\":6,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0065","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":7,\"a\":-3,\"b\":6,\"x\":1}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(7, -3, 6, 1, [1, 3, 4], 1, 0, 3)","explanation_zh":"應用擴展核對=(7, -3, 6, 1, [1, 3, 4], 1, 0, 3)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213065,"parameters":{"modulus":7,"a":-3,"b":6,"x":1,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":-3,\"b\":6,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0066","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":8,\"a\":-2,\"b\":7,\"x\":7}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(8, -2, 7, 7, [7, 1, 5], 5, 2, 非週期點)","explanation_zh":"應用擴展核對=(8, -2, 7, 7, [7, 1, 5], 5, 2, 非週期點)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213066,"parameters":{"modulus":8,"a":-2,"b":7,"x":7,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":-2,\"b\":7,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0067","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":9,\"a\":-1,\"b\":8,\"x\":6}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(9, -1, 8, 6, [6, 2], 6, 0, 2)","explanation_zh":"應用擴展核對=(9, -1, 8, 6, [6, 2], 6, 0, 2)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213067,"parameters":{"modulus":9,"a":-1,"b":8,"x":6,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":-1,\"b\":8,\"modulus\":9,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0068","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":10,\"a\":0,\"b\":9,\"x\":5}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(10, 0, 9, 5, [5, 9], 9, 1, 非週期點)","explanation_zh":"應用擴展核對=(10, 0, 9, 5, [5, 9], 9, 1, 非週期點)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213068,"parameters":{"modulus":10,"a":0,"b":9,"x":5,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":0,\"b\":9,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0069","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":5,\"a\":1,\"b\":10,\"x\":2}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(5, 1, 10, 2, [2], 2, 0, 1)","explanation_zh":"應用擴展核對=(5, 1, 10, 2, [2], 2, 0, 1)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213069,"parameters":{"modulus":5,"a":1,"b":10,"x":2,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":1,\"b\":10,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0070","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":6,\"a\":2,\"b\":11,\"x\":3}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(6, 2, 11, 3, [3, 5], 3, 0, 2)","explanation_zh":"應用擴展核對=(6, 2, 11, 3, [3, 5], 3, 0, 2)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213070,"parameters":{"modulus":6,"a":2,"b":11,"x":3,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":2,\"b\":11,\"modulus\":6,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0071","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":7,\"a\":3,\"b\":12,\"x\":6}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(7, 3, 12, 6, [6, 2, 4, 3, 0, 5], 6, 0, 6)","explanation_zh":"應用擴展核對=(7, 3, 12, 6, [6, 2, 4, 3, 0, 5], 6, 0, 6)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213071,"parameters":{"modulus":7,"a":3,"b":12,"x":6,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":3,\"b\":12,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0072","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":8,\"a\":-3,\"b\":13,\"x\":3}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(8, -3, 13, 3, [3, 4, 1, 2, 7, 0, 5, 6], 3, 0, 8)","explanation_zh":"應用擴展核對=(8, -3, 13, 3, [3, 4, 1, 2, 7, 0, 5, 6], 3, 0, 8)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213072,"parameters":{"modulus":8,"a":-3,"b":13,"x":3,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":-3,\"b\":13,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0073","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":9,\"a\":-2,\"b\":14,\"x\":0}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(9, -2, 14, 0, [0, 5, 4, 6, 2, 1, 3, 8, 7], 0, 0, 9)","explanation_zh":"應用擴展核對=(9, -2, 14, 0, [0, 5, 4, 6, 2, 1, 3, 8, 7], 0, 0, 9)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213073,"parameters":{"modulus":9,"a":-2,"b":14,"x":0,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":-2,\"b\":14,\"modulus\":9,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0074","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":10,\"a\":-1,\"b\":15,\"x\":7}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(10, -1, 15, 7, [7, 8], 7, 0, 2)","explanation_zh":"應用擴展核對=(10, -1, 15, 7, [7, 8], 7, 0, 2)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213074,"parameters":{"modulus":10,"a":-1,"b":15,"x":7,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":-1,\"b\":15,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0075","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":5,\"a\":0,\"b\":16,\"x\":4}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 16, 4, [4, 1], 1, 1, 非週期點)","explanation_zh":"應用擴展核對=(5, 0, 16, 4, [4, 1], 1, 1, 非週期點)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-least-period","seed":202610213075,"parameters":{"modulus":5,"a":0,"b":16,"x":4,"domain":"Finite_orbit_periodicity","operation":"orbit-least-period","semantic_template":"qf-app21-orbit-least-period/{\"a\":0,\"b\":16,\"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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0076","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":5,\"a\":-3,\"b\":-8,\"period_bound\":1}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(5, -3, -8, 1, [[0, 4], [1, 4], [2, 4], [3, 1], [4, 4]], [3])","explanation_zh":"應用擴展核對=(5, -3, -8, 1, [[0, 4], [1, 4], [2, 4], [3, 1], [4, 4]], [3])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213076,"parameters":{"modulus":5,"a":-3,"b":-8,"period_bound":1,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":-3,\"b\":-8,\"modulus\":5,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0077","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":6,\"a\":-2,\"b\":-7,\"period_bound\":2}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(6, -2, -7, 2, [[0, 非週期點], [1, 3], [2, 非週期點], [3, 3], [4, 非週期點], [5, 3]], [])","explanation_zh":"應用擴展核對=(6, -2, -7, 2, [[0, 非週期點], [1, 3], [2, 非週期點], [3, 3], [4, 非週期點], [5, 3]], [])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213077,"parameters":{"modulus":6,"a":-2,"b":-7,"period_bound":2,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":-2,\"b\":-7,\"modulus\":6,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0078","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":7,\"a\":-1,\"b\":-6,\"period_bound\":3}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(7, -1, -6, 3, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2]], [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"應用擴展核對=(7, -1, -6, 3, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2]], [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213078,"parameters":{"modulus":7,"a":-1,"b":-6,"period_bound":3,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":-1,\"b\":-6,\"modulus\":7,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0079","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":8,\"a\":0,\"b\":-5,\"period_bound\":4}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(8, 0, -5, 4, [[0, 非週期點], [1, 非週期點], [2, 非週期點], [3, 1], [4, 非週期點], [5, 非週期點], [6, 非週期點], [7, 非週期點]], [3])","explanation_zh":"應用擴展核對=(8, 0, -5, 4, [[0, 非週期點], [1, 非週期點], [2, 非週期點], [3, 1], [4, 非週期點], [5, 非週期點], [6, 非週期點], [7, 非週期點]], [3])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213079,"parameters":{"modulus":8,"a":0,"b":-5,"period_bound":4,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":0,\"b\":-5,\"modulus\":8,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0080","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":9,\"a\":1,\"b\":-4,\"period_bound\":5}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(9, 1, -4, 5, [[0, 9], [1, 9], [2, 9], [3, 9], [4, 9], [5, 9], [6, 9], [7, 9], [8, 9]], [])","explanation_zh":"應用擴展核對=(9, 1, -4, 5, [[0, 9], [1, 9], [2, 9], [3, 9], [4, 9], [5, 9], [6, 9], [7, 9], [8, 9]], [])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213080,"parameters":{"modulus":9,"a":1,"b":-4,"period_bound":5,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":1,\"b\":-4,\"modulus\":9,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0081","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":10,\"a\":2,\"b\":-3,\"period_bound\":6}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(10, 2, -3, 6, [[0, 非週期點], [1, 4], [2, 非週期點], [3, 1], [4, 非週期點], [5, 4], [6, 非週期點], [7, 4], [8, 非週期點], [9, 4]], [1, 3, 5, 7, 9])","explanation_zh":"應用擴展核對=(10, 2, -3, 6, [[0, 非週期點], [1, 4], [2, 非週期點], [3, 1], [4, 非週期點], [5, 4], [6, 非週期點], [7, 4], [8, 非週期點], [9, 4]], [1, 3, 5, 7, 9])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213081,"parameters":{"modulus":10,"a":2,"b":-3,"period_bound":6,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":2,\"b\":-3,\"modulus\":10,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0082","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":5,\"a\":3,\"b\":-2,\"period_bound\":1}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(5, 3, -2, 1, [[0, 4], [1, 1], [2, 4], [3, 4], [4, 4]], [1])","explanation_zh":"應用擴展核對=(5, 3, -2, 1, [[0, 4], [1, 1], [2, 4], [3, 4], [4, 4]], [1])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213082,"parameters":{"modulus":5,"a":3,"b":-2,"period_bound":1,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":3,\"b\":-2,\"modulus\":5,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0083","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":6,\"a\":-3,\"b\":-1,\"period_bound\":2}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(6, -3, -1, 2, [[0, 非週期點], [1, 非週期點], [2, 2], [3, 非週期點], [4, 非週期點], [5, 2]], [2, 5])","explanation_zh":"應用擴展核對=(6, -3, -1, 2, [[0, 非週期點], [1, 非週期點], [2, 2], [3, 非週期點], [4, 非週期點], [5, 2]], [2, 5])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213083,"parameters":{"modulus":6,"a":-3,"b":-1,"period_bound":2,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":-3,\"b\":-1,\"modulus\":6,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0084","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":7,\"a\":-2,\"b\":0,\"period_bound\":3}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(7, -2, 0, 3, [[0, 1], [1, 6], [2, 6], [3, 6], [4, 6], [5, 6], [6, 6]], [0])","explanation_zh":"應用擴展核對=(7, -2, 0, 3, [[0, 1], [1, 6], [2, 6], [3, 6], [4, 6], [5, 6], [6, 6]], [0])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213084,"parameters":{"modulus":7,"a":-2,"b":0,"period_bound":3,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":-2,\"b\":0,\"modulus\":7,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0085","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":8,\"a\":-1,\"b\":1,\"period_bound\":4}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(8, -1, 1, 4, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], [0, 1, 2, 3, 4, 5, 6, 7])","explanation_zh":"應用擴展核對=(8, -1, 1, 4, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], [0, 1, 2, 3, 4, 5, 6, 7])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213085,"parameters":{"modulus":8,"a":-1,"b":1,"period_bound":4,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":-1,\"b\":1,\"modulus\":8,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0086","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":9,\"a\":0,\"b\":2,\"period_bound\":5}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(9, 0, 2, 5, [[0, 非週期點], [1, 非週期點], [2, 1], [3, 非週期點], [4, 非週期點], [5, 非週期點], [6, 非週期點], [7, 非週期點], [8, 非週期點]], [2])","explanation_zh":"應用擴展核對=(9, 0, 2, 5, [[0, 非週期點], [1, 非週期點], [2, 1], [3, 非週期點], [4, 非週期點], [5, 非週期點], [6, 非週期點], [7, 非週期點], [8, 非週期點]], [2])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213086,"parameters":{"modulus":9,"a":0,"b":2,"period_bound":5,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":0,\"b\":2,\"modulus\":9,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0087","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":10,\"a\":1,\"b\":3,\"period_bound\":6}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(10, 1, 3, 6, [[0, 10], [1, 10], [2, 10], [3, 10], [4, 10], [5, 10], [6, 10], [7, 10], [8, 10], [9, 10]], [])","explanation_zh":"應用擴展核對=(10, 1, 3, 6, [[0, 10], [1, 10], [2, 10], [3, 10], [4, 10], [5, 10], [6, 10], [7, 10], [8, 10], [9, 10]], [])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213087,"parameters":{"modulus":10,"a":1,"b":3,"period_bound":6,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":1,\"b\":3,\"modulus\":10,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0088","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":5,\"a\":2,\"b\":4,\"period_bound\":1}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(5, 2, 4, 1, [[0, 4], [1, 1], [2, 4], [3, 4], [4, 4]], [1])","explanation_zh":"應用擴展核對=(5, 2, 4, 1, [[0, 4], [1, 1], [2, 4], [3, 4], [4, 4]], [1])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213088,"parameters":{"modulus":5,"a":2,"b":4,"period_bound":1,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":2,\"b\":4,\"modulus\":5,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0089","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":6,\"a\":3,\"b\":5,\"period_bound\":2}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(6, 3, 5, 2, [[0, 非週期點], [1, 非週期點], [2, 2], [3, 非週期點], [4, 非週期點], [5, 2]], [2, 5])","explanation_zh":"應用擴展核對=(6, 3, 5, 2, [[0, 非週期點], [1, 非週期點], [2, 2], [3, 非週期點], [4, 非週期點], [5, 2]], [2, 5])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213089,"parameters":{"modulus":6,"a":3,"b":5,"period_bound":2,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":3,\"b\":5,\"modulus\":6,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0090","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":7,\"a\":-3,\"b\":6,\"period_bound\":3}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(7, -3, 6, 3, [[0, 3], [1, 3], [2, 3], [3, 3], [4, 3], [5, 1], [6, 3]], [0, 1, 2, 3, 4, 5, 6])","explanation_zh":"應用擴展核對=(7, -3, 6, 3, [[0, 3], [1, 3], [2, 3], [3, 3], [4, 3], [5, 1], [6, 3]], [0, 1, 2, 3, 4, 5, 6])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213090,"parameters":{"modulus":7,"a":-3,"b":6,"period_bound":3,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":-3,\"b\":6,\"modulus\":7,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0091","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":8,\"a\":-2,\"b\":7,\"period_bound\":4}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(8, -2, 7, 4, [[0, 非週期點], [1, 非週期點], [2, 非週期點], [3, 非週期點], [4, 非週期點], [5, 1], [6, 非週期點], [7, 非週期點]], [5])","explanation_zh":"應用擴展核對=(8, -2, 7, 4, [[0, 非週期點], [1, 非週期點], [2, 非週期點], [3, 非週期點], [4, 非週期點], [5, 1], [6, 非週期點], [7, 非週期點]], [5])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213091,"parameters":{"modulus":8,"a":-2,"b":7,"period_bound":4,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":-2,\"b\":7,\"modulus\":8,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0092","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":9,\"a\":-1,\"b\":8,\"period_bound\":5}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(9, -1, 8, 5, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2], [7, 2], [8, 2]], [0, 1, 2, 3, 4, 5, 6, 7, 8])","explanation_zh":"應用擴展核對=(9, -1, 8, 5, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2], [7, 2], [8, 2]], [0, 1, 2, 3, 4, 5, 6, 7, 8])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213092,"parameters":{"modulus":9,"a":-1,"b":8,"period_bound":5,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":-1,\"b\":8,\"modulus\":9,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0093","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":10,\"a\":0,\"b\":9,\"period_bound\":6}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(10, 0, 9, 6, [[0, 非週期點], [1, 非週期點], [2, 非週期點], [3, 非週期點], [4, 非週期點], [5, 非週期點], [6, 非週期點], [7, 非週期點], [8, 非週期點], [9, 1]], [9])","explanation_zh":"應用擴展核對=(10, 0, 9, 6, [[0, 非週期點], [1, 非週期點], [2, 非週期點], [3, 非週期點], [4, 非週期點], [5, 非週期點], [6, 非週期點], [7, 非週期點], [8, 非週期點], [9, 1]], [9])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213093,"parameters":{"modulus":10,"a":0,"b":9,"period_bound":6,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":0,\"b\":9,\"modulus\":10,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0094","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":5,\"a\":1,\"b\":10,\"period_bound\":1}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(5, 1, 10, 1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1]], [0, 1, 2, 3, 4])","explanation_zh":"應用擴展核對=(5, 1, 10, 1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1]], [0, 1, 2, 3, 4])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213094,"parameters":{"modulus":5,"a":1,"b":10,"period_bound":1,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":1,\"b\":10,\"modulus\":5,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0095","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":6,\"a\":2,\"b\":11,\"period_bound\":2}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(6, 2, 11, 2, [[0, 非週期點], [1, 1], [2, 非週期點], [3, 2], [4, 非週期點], [5, 2]], [1, 3, 5])","explanation_zh":"應用擴展核對=(6, 2, 11, 2, [[0, 非週期點], [1, 1], [2, 非週期點], [3, 2], [4, 非週期點], [5, 2]], [1, 3, 5])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213095,"parameters":{"modulus":6,"a":2,"b":11,"period_bound":2,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":2,\"b\":11,\"modulus\":6,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0096","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":7,\"a\":3,\"b\":12,\"period_bound\":3}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(7, 3, 12, 3, [[0, 6], [1, 1], [2, 6], [3, 6], [4, 6], [5, 6], [6, 6]], [1])","explanation_zh":"應用擴展核對=(7, 3, 12, 3, [[0, 6], [1, 1], [2, 6], [3, 6], [4, 6], [5, 6], [6, 6]], [1])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213096,"parameters":{"modulus":7,"a":3,"b":12,"period_bound":3,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":3,\"b\":12,\"modulus\":7,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0097","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":8,\"a\":-3,\"b\":13,\"period_bound\":4}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(8, -3, 13, 4, [[0, 8], [1, 8], [2, 8], [3, 8], [4, 8], [5, 8], [6, 8], [7, 8]], [])","explanation_zh":"應用擴展核對=(8, -3, 13, 4, [[0, 8], [1, 8], [2, 8], [3, 8], [4, 8], [5, 8], [6, 8], [7, 8]], [])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213097,"parameters":{"modulus":8,"a":-3,"b":13,"period_bound":4,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":-3,\"b\":13,\"modulus\":8,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0098","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":9,\"a\":-2,\"b\":14,\"period_bound\":5}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(9, -2, 14, 5, [[0, 9], [1, 9], [2, 9], [3, 9], [4, 9], [5, 9], [6, 9], [7, 9], [8, 9]], [])","explanation_zh":"應用擴展核對=(9, -2, 14, 5, [[0, 9], [1, 9], [2, 9], [3, 9], [4, 9], [5, 9], [6, 9], [7, 9], [8, 9]], [])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213098,"parameters":{"modulus":9,"a":-2,"b":14,"period_bound":5,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":-2,\"b\":14,\"modulus\":9,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0099","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":10,\"a\":-1,\"b\":15,\"period_bound\":6}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(10, -1, 15, 6, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9])","explanation_zh":"應用擴展核對=(10, -1, 15, 6, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213099,"parameters":{"modulus":10,"a":-1,"b":15,"period_bound":6,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":-1,\"b\":15,\"modulus\":10,\"period_bound\":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、真實學生診斷、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-20261021-r03-orbit-fixed-periodic-0100","batch_id":"batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations","language":"zh-Hant","source_mko_ids":["mko-orbit-fixed-periodic"],"title_zh":"有限軌道與週期","stem_zh":"完整有限狀態自映射。操作=orbit-periodic-inventory；資料={\"modulus\":5,\"a\":0,\"b\":16,\"period_bound\":1}。請依 [modulus,a,b,period_bound,point_period_rows,selected_points] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 16, 1, [[0, 非週期點], [1, 1], [2, 非週期點], [3, 非週期點], [4, 非週期點]], [1])","explanation_zh":"應用擴展核對=(5, 0, 16, 1, [[0, 非週期點], [1, 1], [2, 非週期點], [3, 非週期點], [4, 非週期點]], [1])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；固定點與最小正週期分開；只分類指定有限模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限軌道與週期"]},"variant":{"family_id":"qf-app21-orbit-periodic-inventory","seed":202610213100,"parameters":{"modulus":5,"a":0,"b":16,"period_bound":1,"domain":"Finite_orbit_periodicity","operation":"orbit-periodic-inventory","semantic_template":"qf-app21-orbit-periodic-inventory/{\"a\":0,\"b\":16,\"modulus\":5,\"period_bound\":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、真實學生診斷、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"}}