{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0001","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[-8,1],\"factor\":[1,2]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(-8, 1/2, -4)","explanation_zh":"應用基礎核對=(-8, 1/2, -4)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206001,"parameters":{"current":[-8,1],"factor":[1,2],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[-8,1],\"factor\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0002","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[-7,2],\"factor\":[2,3]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(-7/2, 2/3, -7/3)","explanation_zh":"應用基礎核對=(-7/2, 2/3, -7/3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206002,"parameters":{"current":[-7,2],"factor":[2,3],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[-7,2],\"factor\":[2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0003","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[-6,3],\"factor\":[3,4]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(-2, 3/4, -3/2)","explanation_zh":"應用基礎核對=(-2, 3/4, -3/2)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206003,"parameters":{"current":[-6,3],"factor":[3,4],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[-6,3],\"factor\":[3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0004","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[-5,4],\"factor\":[4,5]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(-5/4, 4/5, -1)","explanation_zh":"應用基礎核對=(-5/4, 4/5, -1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206004,"parameters":{"current":[-5,4],"factor":[4,5],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[-5,4],\"factor\":[4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0005","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[-4,5],\"factor\":[5,6]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(-4/5, 5/6, -2/3)","explanation_zh":"應用基礎核對=(-4/5, 5/6, -2/3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206005,"parameters":{"current":[-4,5],"factor":[5,6],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[-4,5],\"factor\":[5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0006","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[-3,1],\"factor\":[6,7]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(-3, 6/7, -18/7)","explanation_zh":"應用基礎核對=(-3, 6/7, -18/7)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206006,"parameters":{"current":[-3,1],"factor":[6,7],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[-3,1],\"factor\":[6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0007","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[-2,2],\"factor\":[7,2]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(-1, 7/2, -7/2)","explanation_zh":"應用基礎核對=(-1, 7/2, -7/2)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206007,"parameters":{"current":[-2,2],"factor":[7,2],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[-2,2],\"factor\":[7,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0008","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[-1,3],\"factor\":[1,3]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(-1/3, 1/3, -1/9)","explanation_zh":"應用基礎核對=(-1/3, 1/3, -1/9)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206008,"parameters":{"current":[-1,3],"factor":[1,3],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[-1,3],\"factor\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0009","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[0,4],\"factor\":[2,4]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/2, 0)","explanation_zh":"應用基礎核對=(0, 1/2, 0)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206009,"parameters":{"current":[0,4],"factor":[2,4],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[0,4],\"factor\":[2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0010","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[1,5],\"factor\":[3,5]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(1/5, 3/5, 3/25)","explanation_zh":"應用基礎核對=(1/5, 3/5, 3/25)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206010,"parameters":{"current":[1,5],"factor":[3,5],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[1,5],\"factor\":[3,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0011","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[2,1],\"factor\":[4,6]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(2, 2/3, 4/3)","explanation_zh":"應用基礎核對=(2, 2/3, 4/3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206011,"parameters":{"current":[2,1],"factor":[4,6],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[2,1],\"factor\":[4,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0012","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[3,2],\"factor\":[5,7]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(3/2, 5/7, 15/14)","explanation_zh":"應用基礎核對=(3/2, 5/7, 15/14)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206012,"parameters":{"current":[3,2],"factor":[5,7],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[3,2],\"factor\":[5,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0013","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[4,3],\"factor\":[6,2]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(4/3, 3, 4)","explanation_zh":"應用基礎核對=(4/3, 3, 4)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206013,"parameters":{"current":[4,3],"factor":[6,2],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[4,3],\"factor\":[6,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0014","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[5,4],\"factor\":[7,3]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(5/4, 7/3, 35/12)","explanation_zh":"應用基礎核對=(5/4, 7/3, 35/12)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206014,"parameters":{"current":[5,4],"factor":[7,3],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[5,4],\"factor\":[7,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0015","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[6,5],\"factor\":[1,4]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(6/5, 1/4, 3/10)","explanation_zh":"應用基礎核對=(6/5, 1/4, 3/10)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206015,"parameters":{"current":[6,5],"factor":[1,4],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[6,5],\"factor\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0016","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[7,1],\"factor\":[2,5]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(7, 2/5, 14/5)","explanation_zh":"應用基礎核對=(7, 2/5, 14/5)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206016,"parameters":{"current":[7,1],"factor":[2,5],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[7,1],\"factor\":[2,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0017","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[8,2],\"factor\":[3,6]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(4, 1/2, 2)","explanation_zh":"應用基礎核對=(4, 1/2, 2)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206017,"parameters":{"current":[8,2],"factor":[3,6],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[8,2],\"factor\":[3,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0018","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[9,3],\"factor\":[4,7]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(3, 4/7, 12/7)","explanation_zh":"應用基礎核對=(3, 4/7, 12/7)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206018,"parameters":{"current":[9,3],"factor":[4,7],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[9,3],\"factor\":[4,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0019","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[10,4],\"factor\":[5,2]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(5/2, 5/2, 25/4)","explanation_zh":"應用基礎核對=(5/2, 5/2, 25/4)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206019,"parameters":{"current":[10,4],"factor":[5,2],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[10,4],\"factor\":[5,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0020","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[11,5],\"factor\":[6,3]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(11/5, 2, 22/5)","explanation_zh":"應用基礎核對=(11/5, 2, 22/5)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206020,"parameters":{"current":[11,5],"factor":[6,3],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[11,5],\"factor\":[6,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0021","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[12,1],\"factor\":[7,4]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(12, 7/4, 21)","explanation_zh":"應用基礎核對=(12, 7/4, 21)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206021,"parameters":{"current":[12,1],"factor":[7,4],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[12,1],\"factor\":[7,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0022","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[13,2],\"factor\":[1,5]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(13/2, 1/5, 13/10)","explanation_zh":"應用基礎核對=(13/2, 1/5, 13/10)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206022,"parameters":{"current":[13,2],"factor":[1,5],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[13,2],\"factor\":[1,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0023","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[14,3],\"factor\":[2,6]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(14/3, 1/3, 14/9)","explanation_zh":"應用基礎核對=(14/3, 1/3, 14/9)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206023,"parameters":{"current":[14,3],"factor":[2,6],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[14,3],\"factor\":[2,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0024","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[15,4],\"factor\":[3,7]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(15/4, 3/7, 45/28)","explanation_zh":"應用基礎核對=(15/4, 3/7, 45/28)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206024,"parameters":{"current":[15,4],"factor":[3,7],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[15,4],\"factor\":[3,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0025","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[16,5],\"factor\":[4,2]}。請依 [current,factor,next] 欄序回答。","answer":{"kind":"expression","value":"(16/5, 2, 32/5)","explanation_zh":"應用基礎核對=(16/5, 2, 32/5)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-recurrence-step","seed":202610206025,"parameters":{"current":[16,5],"factor":[4,2],"domain":"Exact_discrete_exponential_model","operation":"growth-recurrence-step","semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[16,5],\"factor\":[4,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0026","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[-8,1],\"factor\":[1,2],\"n\":0}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(-8, 1/2, 0, -8)","explanation_zh":"應用基礎核對=(-8, 1/2, 0, -8)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206026,"parameters":{"initial":[-8,1],"factor":[1,2],"n":0,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[1,2],\"initial\":[-8,1],\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0027","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[-7,2],\"factor\":[2,3],\"n\":1}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(-7/2, 2/3, 1, -7/3)","explanation_zh":"應用基礎核對=(-7/2, 2/3, 1, -7/3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206027,"parameters":{"initial":[-7,2],"factor":[2,3],"n":1,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[2,3],\"initial\":[-7,2],\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0028","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[-6,3],\"factor\":[3,4],\"n\":2}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(-2, 3/4, 2, -9/8)","explanation_zh":"應用基礎核對=(-2, 3/4, 2, -9/8)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206028,"parameters":{"initial":[-6,3],"factor":[3,4],"n":2,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[3,4],\"initial\":[-6,3],\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0029","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[-5,4],\"factor\":[4,5],\"n\":3}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(-5/4, 4/5, 3, -16/25)","explanation_zh":"應用基礎核對=(-5/4, 4/5, 3, -16/25)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206029,"parameters":{"initial":[-5,4],"factor":[4,5],"n":3,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[4,5],\"initial\":[-5,4],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0030","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[-4,5],\"factor\":[1,6],\"n\":4}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(-4/5, 1/6, 4, -1/1620)","explanation_zh":"應用基礎核對=(-4/5, 1/6, 4, -1/1620)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206030,"parameters":{"initial":[-4,5],"factor":[1,6],"n":4,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[1,6],\"initial\":[-4,5],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0031","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[-3,1],\"factor\":[2,7],\"n\":0}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(-3, 2/7, 0, -3)","explanation_zh":"應用基礎核對=(-3, 2/7, 0, -3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206031,"parameters":{"initial":[-3,1],"factor":[2,7],"n":0,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[2,7],\"initial\":[-3,1],\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0032","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[-2,2],\"factor\":[3,2],\"n\":1}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(-1, 3/2, 1, -3/2)","explanation_zh":"應用基礎核對=(-1, 3/2, 1, -3/2)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206032,"parameters":{"initial":[-2,2],"factor":[3,2],"n":1,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[3,2],\"initial\":[-2,2],\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0033","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[-1,3],\"factor\":[4,3],\"n\":2}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(-1/3, 4/3, 2, -16/27)","explanation_zh":"應用基礎核對=(-1/3, 4/3, 2, -16/27)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206033,"parameters":{"initial":[-1,3],"factor":[4,3],"n":2,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[4,3],\"initial\":[-1,3],\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0034","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[0,4],\"factor\":[1,4],\"n\":3}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/4, 3, 0)","explanation_zh":"應用基礎核對=(0, 1/4, 3, 0)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206034,"parameters":{"initial":[0,4],"factor":[1,4],"n":3,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[1,4],\"initial\":[0,4],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0035","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[1,5],\"factor\":[2,5],\"n\":4}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(1/5, 2/5, 4, 16/3125)","explanation_zh":"應用基礎核對=(1/5, 2/5, 4, 16/3125)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206035,"parameters":{"initial":[1,5],"factor":[2,5],"n":4,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[2,5],\"initial\":[1,5],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0036","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[2,1],\"factor\":[3,6],\"n\":0}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(2, 1/2, 0, 2)","explanation_zh":"應用基礎核對=(2, 1/2, 0, 2)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206036,"parameters":{"initial":[2,1],"factor":[3,6],"n":0,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[3,6],\"initial\":[2,1],\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0037","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[3,2],\"factor\":[4,7],\"n\":1}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(3/2, 4/7, 1, 6/7)","explanation_zh":"應用基礎核對=(3/2, 4/7, 1, 6/7)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206037,"parameters":{"initial":[3,2],"factor":[4,7],"n":1,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[4,7],\"initial\":[3,2],\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0038","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[4,3],\"factor\":[1,2],\"n\":2}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(4/3, 1/2, 2, 1/3)","explanation_zh":"應用基礎核對=(4/3, 1/2, 2, 1/3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206038,"parameters":{"initial":[4,3],"factor":[1,2],"n":2,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[1,2],\"initial\":[4,3],\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0039","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[5,4],\"factor\":[2,3],\"n\":3}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(5/4, 2/3, 3, 10/27)","explanation_zh":"應用基礎核對=(5/4, 2/3, 3, 10/27)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206039,"parameters":{"initial":[5,4],"factor":[2,3],"n":3,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[2,3],\"initial\":[5,4],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0040","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[6,5],\"factor\":[3,4],\"n\":4}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(6/5, 3/4, 4, 243/640)","explanation_zh":"應用基礎核對=(6/5, 3/4, 4, 243/640)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206040,"parameters":{"initial":[6,5],"factor":[3,4],"n":4,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[3,4],\"initial\":[6,5],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0041","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[7,1],\"factor\":[4,5],\"n\":0}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(7, 4/5, 0, 7)","explanation_zh":"應用基礎核對=(7, 4/5, 0, 7)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206041,"parameters":{"initial":[7,1],"factor":[4,5],"n":0,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[4,5],\"initial\":[7,1],\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0042","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[8,2],\"factor\":[1,6],\"n\":1}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(4, 1/6, 1, 2/3)","explanation_zh":"應用基礎核對=(4, 1/6, 1, 2/3)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206042,"parameters":{"initial":[8,2],"factor":[1,6],"n":1,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[1,6],\"initial\":[8,2],\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0043","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[9,3],\"factor\":[2,7],\"n\":2}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(3, 2/7, 2, 12/49)","explanation_zh":"應用基礎核對=(3, 2/7, 2, 12/49)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206043,"parameters":{"initial":[9,3],"factor":[2,7],"n":2,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[2,7],\"initial\":[9,3],\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0044","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[10,4],\"factor\":[3,2],\"n\":3}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(5/2, 3/2, 3, 135/16)","explanation_zh":"應用基礎核對=(5/2, 3/2, 3, 135/16)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206044,"parameters":{"initial":[10,4],"factor":[3,2],"n":3,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[3,2],\"initial\":[10,4],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0045","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[11,5],\"factor\":[4,3],\"n\":4}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(11/5, 4/3, 4, 2816/405)","explanation_zh":"應用基礎核對=(11/5, 4/3, 4, 2816/405)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206045,"parameters":{"initial":[11,5],"factor":[4,3],"n":4,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[4,3],\"initial\":[11,5],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0046","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[12,1],\"factor\":[1,4],\"n\":0}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(12, 1/4, 0, 12)","explanation_zh":"應用基礎核對=(12, 1/4, 0, 12)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206046,"parameters":{"initial":[12,1],"factor":[1,4],"n":0,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[1,4],\"initial\":[12,1],\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0047","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[13,2],\"factor\":[2,5],\"n\":1}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(13/2, 2/5, 1, 13/5)","explanation_zh":"應用基礎核對=(13/2, 2/5, 1, 13/5)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206047,"parameters":{"initial":[13,2],"factor":[2,5],"n":1,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[2,5],\"initial\":[13,2],\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0048","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[14,3],\"factor\":[3,6],\"n\":2}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(14/3, 1/2, 2, 7/6)","explanation_zh":"應用基礎核對=(14/3, 1/2, 2, 7/6)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206048,"parameters":{"initial":[14,3],"factor":[3,6],"n":2,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[3,6],\"initial\":[14,3],\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0049","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[15,4],\"factor\":[4,7],\"n\":3}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(15/4, 4/7, 3, 240/343)","explanation_zh":"應用基礎核對=(15/4, 4/7, 3, 240/343)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206049,"parameters":{"initial":[15,4],"factor":[4,7],"n":3,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[4,7],\"initial\":[15,4],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0050","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[16,5],\"factor\":[1,2],\"n\":4}。請依 [initial,factor,n,term] 欄序回答。","answer":{"kind":"expression","value":"(16/5, 1/2, 4, 1/5)","explanation_zh":"應用基礎核對=(16/5, 1/2, 4, 1/5)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-nth-term","seed":202610206050,"parameters":{"initial":[16,5],"factor":[1,2],"n":4,"domain":"Exact_discrete_exponential_model","operation":"growth-nth-term","semantic_template":"qf-app20-growth-nth-term/{\"factor\":[1,2],\"initial\":[16,5],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0051","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[-8,1],\"factor\":[1,2],\"n\":0,\"candidate\":[-7,1]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(-8, 1/2, 0, -8, -7, 否)","explanation_zh":"應用基礎核對=(-8, 1/2, 0, -8, -7, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206051,"parameters":{"initial":[-8,1],"factor":[1,2],"n":0,"candidate":[-7,1],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[-7,1],\"factor\":[1,2],\"initial\":[-8,1],\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0052","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[-7,2],\"factor\":[2,3],\"n\":1,\"candidate\":[-7,3]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(-7/2, 2/3, 1, -7/3, -7/3, 是)","explanation_zh":"應用基礎核對=(-7/2, 2/3, 1, -7/3, -7/3, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206052,"parameters":{"initial":[-7,2],"factor":[2,3],"n":1,"candidate":[-7,3],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[-7,3],\"factor\":[2,3],\"initial\":[-7,2],\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0053","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[-2,1],\"factor\":[3,4],\"n\":2,\"candidate\":[-9,8]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(-2, 3/4, 2, -9/8, -9/8, 是)","explanation_zh":"應用基礎核對=(-2, 3/4, 2, -9/8, -9/8, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206053,"parameters":{"initial":[-2,1],"factor":[3,4],"n":2,"candidate":[-9,8],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[-9,8],\"factor\":[3,4],\"initial\":[-2,1],\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0054","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[-5,4],\"factor\":[4,5],\"n\":3,\"candidate\":[-15,25]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(-5/4, 4/5, 3, -16/25, -3/5, 否)","explanation_zh":"應用基礎核對=(-5/4, 4/5, 3, -16/25, -3/5, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206054,"parameters":{"initial":[-5,4],"factor":[4,5],"n":3,"candidate":[-15,25],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[-15,25],\"factor\":[4,5],\"initial\":[-5,4],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0055","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[-4,5],\"factor\":[1,6],\"n\":4,\"candidate\":[-1,1620]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(-4/5, 1/6, 4, -1/1620, -1/1620, 是)","explanation_zh":"應用基礎核對=(-4/5, 1/6, 4, -1/1620, -1/1620, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206055,"parameters":{"initial":[-4,5],"factor":[1,6],"n":4,"candidate":[-1,1620],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[-1,1620],\"factor\":[1,6],\"initial\":[-4,5],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0056","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[-3,1],\"factor\":[2,7],\"n\":0,\"candidate\":[-3,1]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(-3, 2/7, 0, -3, -3, 是)","explanation_zh":"應用基礎核對=(-3, 2/7, 0, -3, -3, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206056,"parameters":{"initial":[-3,1],"factor":[2,7],"n":0,"candidate":[-3,1],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[-3,1],\"factor\":[2,7],\"initial\":[-3,1],\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0057","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[-1,1],\"factor\":[3,2],\"n\":1,\"candidate\":[-2,2]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(-1, 3/2, 1, -3/2, -1, 否)","explanation_zh":"應用基礎核對=(-1, 3/2, 1, -3/2, -1, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206057,"parameters":{"initial":[-1,1],"factor":[3,2],"n":1,"candidate":[-2,2],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[-2,2],\"factor\":[3,2],\"initial\":[-1,1],\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0058","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[-1,3],\"factor\":[4,3],\"n\":2,\"candidate\":[-16,27]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(-1/3, 4/3, 2, -16/27, -16/27, 是)","explanation_zh":"應用基礎核對=(-1/3, 4/3, 2, -16/27, -16/27, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206058,"parameters":{"initial":[-1,3],"factor":[4,3],"n":2,"candidate":[-16,27],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[-16,27],\"factor\":[4,3],\"initial\":[-1,3],\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0059","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[0,1],\"factor\":[1,4],\"n\":3,\"candidate\":[0,1]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/4, 3, 0, 0, 是)","explanation_zh":"應用基礎核對=(0, 1/4, 3, 0, 0, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206059,"parameters":{"initial":[0,1],"factor":[1,4],"n":3,"candidate":[0,1],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[0,1],\"factor\":[1,4],\"initial\":[0,1],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0060","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[1,5],\"factor\":[2,5],\"n\":4,\"candidate\":[17,3125]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(1/5, 2/5, 4, 16/3125, 17/3125, 否)","explanation_zh":"應用基礎核對=(1/5, 2/5, 4, 16/3125, 17/3125, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206060,"parameters":{"initial":[1,5],"factor":[2,5],"n":4,"candidate":[17,3125],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[17,3125],\"factor\":[2,5],\"initial\":[1,5],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0061","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[2,1],\"factor\":[1,2],\"n\":0,\"candidate\":[2,1]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(2, 1/2, 0, 2, 2, 是)","explanation_zh":"應用基礎核對=(2, 1/2, 0, 2, 2, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206061,"parameters":{"initial":[2,1],"factor":[1,2],"n":0,"candidate":[2,1],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[2,1],\"factor\":[1,2],\"initial\":[2,1],\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0062","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[3,2],\"factor\":[4,7],\"n\":1,\"candidate\":[6,7]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(3/2, 4/7, 1, 6/7, 6/7, 是)","explanation_zh":"應用基礎核對=(3/2, 4/7, 1, 6/7, 6/7, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206062,"parameters":{"initial":[3,2],"factor":[4,7],"n":1,"candidate":[6,7],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[6,7],\"factor\":[4,7],\"initial\":[3,2],\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0063","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[4,3],\"factor\":[1,2],\"n\":2,\"candidate\":[2,3]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(4/3, 1/2, 2, 1/3, 2/3, 否)","explanation_zh":"應用基礎核對=(4/3, 1/2, 2, 1/3, 2/3, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206063,"parameters":{"initial":[4,3],"factor":[1,2],"n":2,"candidate":[2,3],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[2,3],\"factor\":[1,2],\"initial\":[4,3],\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0064","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[5,4],\"factor\":[2,3],\"n\":3,\"candidate\":[10,27]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(5/4, 2/3, 3, 10/27, 10/27, 是)","explanation_zh":"應用基礎核對=(5/4, 2/3, 3, 10/27, 10/27, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206064,"parameters":{"initial":[5,4],"factor":[2,3],"n":3,"candidate":[10,27],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[10,27],\"factor\":[2,3],\"initial\":[5,4],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0065","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[6,5],\"factor\":[3,4],\"n\":4,\"candidate\":[243,640]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(6/5, 3/4, 4, 243/640, 243/640, 是)","explanation_zh":"應用基礎核對=(6/5, 3/4, 4, 243/640, 243/640, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206065,"parameters":{"initial":[6,5],"factor":[3,4],"n":4,"candidate":[243,640],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[243,640],\"factor\":[3,4],\"initial\":[6,5],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0066","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[7,1],\"factor\":[4,5],\"n\":0,\"candidate\":[8,1]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(7, 4/5, 0, 7, 8, 否)","explanation_zh":"應用基礎核對=(7, 4/5, 0, 7, 8, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206066,"parameters":{"initial":[7,1],"factor":[4,5],"n":0,"candidate":[8,1],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[8,1],\"factor\":[4,5],\"initial\":[7,1],\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0067","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[4,1],\"factor\":[1,6],\"n\":1,\"candidate\":[2,3]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(4, 1/6, 1, 2/3, 2/3, 是)","explanation_zh":"應用基礎核對=(4, 1/6, 1, 2/3, 2/3, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206067,"parameters":{"initial":[4,1],"factor":[1,6],"n":1,"candidate":[2,3],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[2,3],\"factor\":[1,6],\"initial\":[4,1],\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0068","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[3,1],\"factor\":[2,7],\"n\":2,\"candidate\":[12,49]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(3, 2/7, 2, 12/49, 12/49, 是)","explanation_zh":"應用基礎核對=(3, 2/7, 2, 12/49, 12/49, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206068,"parameters":{"initial":[3,1],"factor":[2,7],"n":2,"candidate":[12,49],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[12,49],\"factor\":[2,7],\"initial\":[3,1],\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0069","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[5,2],\"factor\":[3,2],\"n\":3,\"candidate\":[136,16]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(5/2, 3/2, 3, 135/16, 17/2, 否)","explanation_zh":"應用基礎核對=(5/2, 3/2, 3, 135/16, 17/2, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206069,"parameters":{"initial":[5,2],"factor":[3,2],"n":3,"candidate":[136,16],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[136,16],\"factor\":[3,2],\"initial\":[5,2],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0070","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[11,5],\"factor\":[4,3],\"n\":4,\"candidate\":[2816,405]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(11/5, 4/3, 4, 2816/405, 2816/405, 是)","explanation_zh":"應用基礎核對=(11/5, 4/3, 4, 2816/405, 2816/405, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206070,"parameters":{"initial":[11,5],"factor":[4,3],"n":4,"candidate":[2816,405],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[2816,405],\"factor\":[4,3],\"initial\":[11,5],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0071","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[12,1],\"factor\":[1,4],\"n\":0,\"candidate\":[12,1]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(12, 1/4, 0, 12, 12, 是)","explanation_zh":"應用基礎核對=(12, 1/4, 0, 12, 12, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206071,"parameters":{"initial":[12,1],"factor":[1,4],"n":0,"candidate":[12,1],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[12,1],\"factor\":[1,4],\"initial\":[12,1],\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0072","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[13,2],\"factor\":[2,5],\"n\":1,\"candidate\":[14,5]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(13/2, 2/5, 1, 13/5, 14/5, 否)","explanation_zh":"應用基礎核對=(13/2, 2/5, 1, 13/5, 14/5, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206072,"parameters":{"initial":[13,2],"factor":[2,5],"n":1,"candidate":[14,5],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[14,5],\"factor\":[2,5],\"initial\":[13,2],\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0073","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[14,3],\"factor\":[1,2],\"n\":2,\"candidate\":[7,6]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(14/3, 1/2, 2, 7/6, 7/6, 是)","explanation_zh":"應用基礎核對=(14/3, 1/2, 2, 7/6, 7/6, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206073,"parameters":{"initial":[14,3],"factor":[1,2],"n":2,"candidate":[7,6],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[7,6],\"factor\":[1,2],\"initial\":[14,3],\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0074","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[15,4],\"factor\":[4,7],\"n\":3,\"candidate\":[240,343]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(15/4, 4/7, 3, 240/343, 240/343, 是)","explanation_zh":"應用基礎核對=(15/4, 4/7, 3, 240/343, 240/343, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206074,"parameters":{"initial":[15,4],"factor":[4,7],"n":3,"candidate":[240,343],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[240,343],\"factor\":[4,7],\"initial\":[15,4],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0075","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-closed-form-audit；資料={\"initial\":[16,5],\"factor\":[1,2],\"n\":4,\"candidate\":[2,5]}。請依 [initial,factor,n,term,candidate,matches] 欄序回答。","answer":{"kind":"expression","value":"(16/5, 1/2, 4, 1/5, 2/5, 否)","explanation_zh":"應用基礎核對=(16/5, 1/2, 4, 1/5, 2/5, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-closed-form-audit","seed":202610206075,"parameters":{"initial":[16,5],"factor":[1,2],"n":4,"candidate":[2,5],"domain":"Exact_discrete_exponential_model","operation":"growth-closed-form-audit","semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[2,5],\"factor\":[1,2],\"initial\":[16,5],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0076","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-factor-regime；資料={\"factor\":[0,1]}。請依 [factor,regime] 欄序回答。","answer":{"kind":"expression","value":"(0, 因子介於0與1)","explanation_zh":"應用基礎核對=(0, 因子介於0與1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-factor-regime","seed":202610206076,"parameters":{"factor":[0,1],"domain":"Exact_discrete_exponential_model","operation":"growth-factor-regime","semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0077","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-factor-regime；資料={\"factor\":[1,2]}。請依 [factor,regime] 欄序回答。","answer":{"kind":"expression","value":"(1/2, 因子介於0與1)","explanation_zh":"應用基礎核對=(1/2, 因子介於0與1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-factor-regime","seed":202610206077,"parameters":{"factor":[1,2],"domain":"Exact_discrete_exponential_model","operation":"growth-factor-regime","semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0078","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-factor-regime；資料={\"factor\":[2,3]}。請依 [factor,regime] 欄序回答。","answer":{"kind":"expression","value":"(2/3, 因子介於0與1)","explanation_zh":"應用基礎核對=(2/3, 因子介於0與1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-factor-regime","seed":202610206078,"parameters":{"factor":[2,3],"domain":"Exact_discrete_exponential_model","operation":"growth-factor-regime","semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0079","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-factor-regime；資料={\"factor\":[3,4]}。請依 [factor,regime] 欄序回答。","answer":{"kind":"expression","value":"(3/4, 因子介於0與1)","explanation_zh":"應用基礎核對=(3/4, 因子介於0與1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-factor-regime","seed":202610206079,"parameters":{"factor":[3,4],"domain":"Exact_discrete_exponential_model","operation":"growth-factor-regime","semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0080","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-factor-regime；資料={\"factor\":[4,5]}。請依 [factor,regime] 欄序回答。","answer":{"kind":"expression","value":"(4/5, 因子介於0與1)","explanation_zh":"應用基礎核對=(4/5, 因子介於0與1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-factor-regime","seed":202610206080,"parameters":{"factor":[4,5],"domain":"Exact_discrete_exponential_model","operation":"growth-factor-regime","semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0081","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-factor-regime；資料={\"factor\":[5,6]}。請依 [factor,regime] 欄序回答。","answer":{"kind":"expression","value":"(5/6, 因子介於0與1)","explanation_zh":"應用基礎核對=(5/6, 因子介於0與1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-factor-regime","seed":202610206081,"parameters":{"factor":[5,6],"domain":"Exact_discrete_exponential_model","operation":"growth-factor-regime","semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0090","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-factor-regime；資料={\"factor\":[5,1]}。請依 [factor,regime] 欄序回答。","answer":{"kind":"expression","value":"(5, 倍增因子大於1)","explanation_zh":"應用基礎核對=(5, 倍增因子大於1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-factor-regime","seed":202610206090,"parameters":{"factor":[5,1],"domain":"Exact_discrete_exponential_model","operation":"growth-factor-regime","semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0092","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-factor-regime；資料={\"factor\":[7,3]}。請依 [factor,regime] 欄序回答。","answer":{"kind":"expression","value":"(7/3, 倍增因子大於1)","explanation_zh":"應用基礎核對=(7/3, 倍增因子大於1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-factor-regime","seed":202610206092,"parameters":{"factor":[7,3],"domain":"Exact_discrete_exponential_model","operation":"growth-factor-regime","semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[7,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0093","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-factor-regime；資料={\"factor\":[8,4]}。請依 [factor,regime] 欄序回答。","answer":{"kind":"expression","value":"(2, 倍增因子大於1)","explanation_zh":"應用基礎核對=(2, 倍增因子大於1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-factor-regime","seed":202610206093,"parameters":{"factor":[8,4],"domain":"Exact_discrete_exponential_model","operation":"growth-factor-regime","semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[8,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0094","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-factor-regime；資料={\"factor\":[0,5]}。請依 [factor,regime] 欄序回答。","answer":{"kind":"expression","value":"(0, 因子介於0與1)","explanation_zh":"應用基礎核對=(0, 因子介於0與1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-factor-regime","seed":202610206094,"parameters":{"factor":[0,5],"domain":"Exact_discrete_exponential_model","operation":"growth-factor-regime","semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[0,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0095","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-factor-regime；資料={\"factor\":[1,6]}。請依 [factor,regime] 欄序回答。","answer":{"kind":"expression","value":"(1/6, 因子介於0與1)","explanation_zh":"應用基礎核對=(1/6, 因子介於0與1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-factor-regime","seed":202610206095,"parameters":{"factor":[1,6],"domain":"Exact_discrete_exponential_model","operation":"growth-factor-regime","semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[1,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0096","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-factor-regime；資料={\"factor\":[2,7]}。請依 [factor,regime] 欄序回答。","answer":{"kind":"expression","value":"(2/7, 因子介於0與1)","explanation_zh":"應用基礎核對=(2/7, 因子介於0與1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-factor-regime","seed":202610206096,"parameters":{"factor":[2,7],"domain":"Exact_discrete_exponential_model","operation":"growth-factor-regime","semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[2,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0097","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-factor-regime；資料={\"factor\":[3,1]}。請依 [factor,regime] 欄序回答。","answer":{"kind":"expression","value":"(3, 倍增因子大於1)","explanation_zh":"應用基礎核對=(3, 倍增因子大於1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-factor-regime","seed":202610206097,"parameters":{"factor":[3,1],"domain":"Exact_discrete_exponential_model","operation":"growth-factor-regime","semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0098","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-factor-regime；資料={\"factor\":[4,2]}。請依 [factor,regime] 欄序回答。","answer":{"kind":"expression","value":"(2, 倍增因子大於1)","explanation_zh":"應用基礎核對=(2, 倍增因子大於1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-factor-regime","seed":202610206098,"parameters":{"factor":[4,2],"domain":"Exact_discrete_exponential_model","operation":"growth-factor-regime","semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[4,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0099","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-factor-regime；資料={\"factor\":[5,3]}。請依 [factor,regime] 欄序回答。","answer":{"kind":"expression","value":"(5/3, 倍增因子大於1)","explanation_zh":"應用基礎核對=(5/3, 倍增因子大於1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-factor-regime","seed":202610206099,"parameters":{"factor":[5,3],"domain":"Exact_discrete_exponential_model","operation":"growth-factor-regime","semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[5,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0100","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-exponential-growth"],"title_zh":"離散固定因子模型","stem_zh":"固定時間步與精確固定因子。操作=growth-factor-regime；資料={\"factor\":[6,4]}。請依 [factor,regime] 欄序回答。","answer":{"kind":"expression","value":"(3/2, 倍增因子大於1)","explanation_zh":"應用基礎核對=(3/2, 倍增因子大於1)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["離散固定因子模型"]},"variant":{"family_id":"qf-app20-growth-factor-regime","seed":202610206100,"parameters":{"factor":[6,4],"domain":"Exact_discrete_exponential_model","operation":"growth-factor-regime","semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[6,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}