{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0001","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[0,5],\"rate\":[0,1],\"capacity\":[5,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 5, 0)","explanation_zh":"應用擴展核對=(0, 0, 5, 0)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217001,"parameters":{"current":[0,5],"rate":[0,1],"capacity":[5,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[5,1],\"current\":[0,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0002","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[1,5],\"rate\":[1,4],\"capacity\":[6,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(1/5, 1/4, 6, 149/600)","explanation_zh":"應用擴展核對=(1/5, 1/4, 6, 149/600)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217002,"parameters":{"current":[1,5],"rate":[1,4],"capacity":[6,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[6,1],\"current\":[1,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0003","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[2,5],\"rate\":[1,2],\"capacity\":[7,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(2/5, 1/2, 7, 103/175)","explanation_zh":"應用擴展核對=(2/5, 1/2, 7, 103/175)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217003,"parameters":{"current":[2,5],"rate":[1,2],"capacity":[7,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[7,1],\"current\":[2,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0004","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[3,5],\"rate\":[3,4],\"capacity\":[8,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(3/5, 3/4, 8, 813/800)","explanation_zh":"應用擴展核對=(3/5, 3/4, 8, 813/800)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217004,"parameters":{"current":[3,5],"rate":[3,4],"capacity":[8,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[8,1],\"current\":[3,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0005","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[4,5],\"rate\":[1,1],\"capacity\":[9,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(4/5, 1, 9, 344/225)","explanation_zh":"應用擴展核對=(4/5, 1, 9, 344/225)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217005,"parameters":{"current":[4,5],"rate":[1,1],"capacity":[9,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[9,1],\"current\":[4,5],\"rate\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0006","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[5,5],\"rate\":[0,1],\"capacity\":[5,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 5, 1)","explanation_zh":"應用擴展核對=(1, 0, 5, 1)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217006,"parameters":{"current":[5,5],"rate":[0,1],"capacity":[5,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[5,1],\"current\":[5,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0007","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[6,5],\"rate\":[1,4],\"capacity\":[6,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(6/5, 1/4, 6, 36/25)","explanation_zh":"應用擴展核對=(6/5, 1/4, 6, 36/25)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217007,"parameters":{"current":[6,5],"rate":[1,4],"capacity":[6,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[6,1],\"current\":[6,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0008","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[7,5],\"rate\":[1,2],\"capacity\":[7,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(7/5, 1/2, 7, 49/25)","explanation_zh":"應用擴展核對=(7/5, 1/2, 7, 49/25)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217008,"parameters":{"current":[7,5],"rate":[1,2],"capacity":[7,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[7,1],\"current\":[7,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0009","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[8,5],\"rate\":[3,4],\"capacity\":[8,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(8/5, 3/4, 8, 64/25)","explanation_zh":"應用擴展核對=(8/5, 3/4, 8, 64/25)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217009,"parameters":{"current":[8,5],"rate":[3,4],"capacity":[8,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[8,1],\"current\":[8,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0010","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[9,5],\"rate\":[1,1],\"capacity\":[9,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(9/5, 1, 9, 81/25)","explanation_zh":"應用擴展核對=(9/5, 1, 9, 81/25)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217010,"parameters":{"current":[9,5],"rate":[1,1],"capacity":[9,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[9,1],\"current\":[9,5],\"rate\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0011","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[10,5],\"rate\":[0,1],\"capacity\":[5,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(2, 0, 5, 2)","explanation_zh":"應用擴展核對=(2, 0, 5, 2)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217011,"parameters":{"current":[10,5],"rate":[0,1],"capacity":[5,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[5,1],\"current\":[10,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0012","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[11,5],\"rate\":[1,4],\"capacity\":[6,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(11/5, 1/4, 6, 1529/600)","explanation_zh":"應用擴展核對=(11/5, 1/4, 6, 1529/600)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217012,"parameters":{"current":[11,5],"rate":[1,4],"capacity":[6,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[6,1],\"current\":[11,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0013","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[12,5],\"rate\":[1,2],\"capacity\":[7,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(12/5, 1/2, 7, 558/175)","explanation_zh":"應用擴展核對=(12/5, 1/2, 7, 558/175)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217013,"parameters":{"current":[12,5],"rate":[1,2],"capacity":[7,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[7,1],\"current\":[12,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0014","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[13,5],\"rate\":[3,4],\"capacity\":[8,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(13/5, 3/4, 8, 3133/800)","explanation_zh":"應用擴展核對=(13/5, 3/4, 8, 3133/800)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217014,"parameters":{"current":[13,5],"rate":[3,4],"capacity":[8,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[8,1],\"current\":[13,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0015","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[14,5],\"rate\":[1,1],\"capacity\":[9,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(14/5, 1, 9, 1064/225)","explanation_zh":"應用擴展核對=(14/5, 1, 9, 1064/225)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217015,"parameters":{"current":[14,5],"rate":[1,1],"capacity":[9,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[9,1],\"current\":[14,5],\"rate\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0016","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[15,5],\"rate\":[0,1],\"capacity\":[5,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 5, 3)","explanation_zh":"應用擴展核對=(3, 0, 5, 3)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217016,"parameters":{"current":[15,5],"rate":[0,1],"capacity":[5,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[5,1],\"current\":[15,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0017","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[16,5],\"rate\":[1,4],\"capacity\":[6,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(16/5, 1/4, 6, 268/75)","explanation_zh":"應用擴展核對=(16/5, 1/4, 6, 268/75)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217017,"parameters":{"current":[16,5],"rate":[1,4],"capacity":[6,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[6,1],\"current\":[16,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0018","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[17,5],\"rate\":[1,2],\"capacity\":[7,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(17/5, 1/2, 7, 748/175)","explanation_zh":"應用擴展核對=(17/5, 1/2, 7, 748/175)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217018,"parameters":{"current":[17,5],"rate":[1,2],"capacity":[7,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[7,1],\"current\":[17,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0019","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[18,5],\"rate\":[3,4],\"capacity\":[8,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(18/5, 3/4, 8, 1017/200)","explanation_zh":"應用擴展核對=(18/5, 3/4, 8, 1017/200)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217019,"parameters":{"current":[18,5],"rate":[3,4],"capacity":[8,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[8,1],\"current\":[18,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0020","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[19,5],\"rate\":[1,1],\"capacity\":[9,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(19/5, 1, 9, 1349/225)","explanation_zh":"應用擴展核對=(19/5, 1, 9, 1349/225)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217020,"parameters":{"current":[19,5],"rate":[1,1],"capacity":[9,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[9,1],\"current\":[19,5],\"rate\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0021","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[20,5],\"rate\":[0,1],\"capacity\":[5,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 5, 4)","explanation_zh":"應用擴展核對=(4, 0, 5, 4)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217021,"parameters":{"current":[20,5],"rate":[0,1],"capacity":[5,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[5,1],\"current\":[20,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0022","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[21,5],\"rate\":[1,4],\"capacity\":[6,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(21/5, 1/4, 6, 903/200)","explanation_zh":"應用擴展核對=(21/5, 1/4, 6, 903/200)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217022,"parameters":{"current":[21,5],"rate":[1,4],"capacity":[6,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[6,1],\"current\":[21,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0023","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[22,5],\"rate\":[1,2],\"capacity\":[7,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(22/5, 1/2, 7, 913/175)","explanation_zh":"應用擴展核對=(22/5, 1/2, 7, 913/175)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217023,"parameters":{"current":[22,5],"rate":[1,2],"capacity":[7,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[7,1],\"current\":[22,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0024","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[23,5],\"rate\":[3,4],\"capacity\":[8,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(23/5, 3/4, 8, 4853/800)","explanation_zh":"應用擴展核對=(23/5, 3/4, 8, 4853/800)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217024,"parameters":{"current":[23,5],"rate":[3,4],"capacity":[8,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[8,1],\"current\":[23,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0025","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-single-step；資料={\"current\":[24,5],\"rate\":[1,1],\"capacity\":[9,1]}。請依 [current,rate,capacity,next] 欄序回答。","answer":{"kind":"expression","value":"(24/5, 1, 9, 176/25)","explanation_zh":"應用擴展核對=(24/5, 1, 9, 176/25)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-single-step","seed":202610217025,"parameters":{"current":[24,5],"rate":[1,1],"capacity":[9,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-single-step","semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[9,1],\"current\":[24,5],\"rate\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0026","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[0,25],\"rate\":[0,1],\"capacity\":[5,1],\"steps\":2}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 5, 2, [0, 0, 0])","explanation_zh":"應用擴展核對=(0, 0, 5, 2, [0, 0, 0])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217026,"parameters":{"initial":[0,25],"rate":[0,1],"capacity":[5,1],"steps":2,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[5,1],\"initial\":[0,25],\"rate\":[0,1],\"steps\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0027","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[1,25],\"rate\":[1,4],\"capacity\":[6,1],\"steps\":1}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(1/25, 1/4, 6, 1, [1/25, 749/15000])","explanation_zh":"應用擴展核對=(1/25, 1/4, 6, 1, [1/25, 749/15000])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217027,"parameters":{"initial":[1,25],"rate":[1,4],"capacity":[6,1],"steps":1,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[6,1],\"initial\":[1,25],\"rate\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0028","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[2,25],\"rate\":[1,2],\"capacity\":[7,1],\"steps\":0}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(2/25, 1/2, 7, 0, [2/25])","explanation_zh":"應用擴展核對=(2/25, 1/2, 7, 0, [2/25])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217028,"parameters":{"initial":[2,25],"rate":[1,2],"capacity":[7,1],"steps":0,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[7,1],\"initial\":[2,25],\"rate\":[1,2],\"steps\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0029","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[3,25],\"rate\":[3,4],\"capacity\":[8,1],\"steps\":1}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(3/25, 3/4, 8, 1, [3/25, 4173/20000])","explanation_zh":"應用擴展核對=(3/25, 3/4, 8, 1, [3/25, 4173/20000])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217029,"parameters":{"initial":[3,25],"rate":[3,4],"capacity":[8,1],"steps":1,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[8,1],\"initial\":[3,25],\"rate\":[3,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0030","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[4,25],\"rate\":[1,1],\"capacity\":[9,1],\"steps\":0}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(4/25, 1, 9, 0, [4/25])","explanation_zh":"應用擴展核對=(4/25, 1, 9, 0, [4/25])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217030,"parameters":{"initial":[4,25],"rate":[1,1],"capacity":[9,1],"steps":0,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[9,1],\"initial\":[4,25],\"rate\":[1,1],\"steps\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0031","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[5,25],\"rate\":[0,1],\"capacity\":[5,1],\"steps\":2}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(1/5, 0, 5, 2, [1/5, 1/5, 1/5])","explanation_zh":"應用擴展核對=(1/5, 0, 5, 2, [1/5, 1/5, 1/5])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217031,"parameters":{"initial":[5,25],"rate":[0,1],"capacity":[5,1],"steps":2,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[5,1],\"initial\":[5,25],\"rate\":[0,1],\"steps\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0032","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[6,25],\"rate\":[1,4],\"capacity\":[6,1],\"steps\":0}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(6/25, 1/4, 6, 0, [6/25])","explanation_zh":"應用擴展核對=(6/25, 1/4, 6, 0, [6/25])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217032,"parameters":{"initial":[6,25],"rate":[1,4],"capacity":[6,1],"steps":0,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[6,1],\"initial\":[6,25],\"rate\":[1,4],\"steps\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0033","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[7,25],\"rate\":[1,2],\"capacity\":[7,1],\"steps\":1}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(7/25, 1/2, 7, 1, [7/25, 259/625])","explanation_zh":"應用擴展核對=(7/25, 1/2, 7, 1, [7/25, 259/625])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217033,"parameters":{"initial":[7,25],"rate":[1,2],"capacity":[7,1],"steps":1,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[7,1],\"initial\":[7,25],\"rate\":[1,2],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0034","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[8,25],\"rate\":[3,4],\"capacity\":[8,1],\"steps\":0}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(8/25, 3/4, 8, 0, [8/25])","explanation_zh":"應用擴展核對=(8/25, 3/4, 8, 0, [8/25])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217034,"parameters":{"initial":[8,25],"rate":[3,4],"capacity":[8,1],"steps":0,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[8,1],\"initial\":[8,25],\"rate\":[3,4],\"steps\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0035","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[9,25],\"rate\":[1,1],\"capacity\":[9,1],\"steps\":1}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(9/25, 1, 9, 1, [9/25, 441/625])","explanation_zh":"應用擴展核對=(9/25, 1, 9, 1, [9/25, 441/625])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217035,"parameters":{"initial":[9,25],"rate":[1,1],"capacity":[9,1],"steps":1,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[9,1],\"initial\":[9,25],\"rate\":[1,1],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0036","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[10,25],\"rate\":[0,1],\"capacity\":[5,1],\"steps\":2}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(2/5, 0, 5, 2, [2/5, 2/5, 2/5])","explanation_zh":"應用擴展核對=(2/5, 0, 5, 2, [2/5, 2/5, 2/5])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217036,"parameters":{"initial":[10,25],"rate":[0,1],"capacity":[5,1],"steps":2,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[5,1],\"initial\":[10,25],\"rate\":[0,1],\"steps\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0037","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[11,25],\"rate\":[1,4],\"capacity\":[6,1],\"steps\":1}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(11/25, 1/4, 6, 1, [11/25, 8129/15000])","explanation_zh":"應用擴展核對=(11/25, 1/4, 6, 1, [11/25, 8129/15000])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217037,"parameters":{"initial":[11,25],"rate":[1,4],"capacity":[6,1],"steps":1,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[6,1],\"initial\":[11,25],\"rate\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0038","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[12,25],\"rate\":[1,2],\"capacity\":[7,1],\"steps\":0}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(12/25, 1/2, 7, 0, [12/25])","explanation_zh":"應用擴展核對=(12/25, 1/2, 7, 0, [12/25])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217038,"parameters":{"initial":[12,25],"rate":[1,2],"capacity":[7,1],"steps":0,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[7,1],\"initial\":[12,25],\"rate\":[1,2],\"steps\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0039","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[13,25],\"rate\":[3,4],\"capacity\":[8,1],\"steps\":1}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(13/25, 3/4, 8, 1, [13/25, 17693/20000])","explanation_zh":"應用擴展核對=(13/25, 3/4, 8, 1, [13/25, 17693/20000])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217039,"parameters":{"initial":[13,25],"rate":[3,4],"capacity":[8,1],"steps":1,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[8,1],\"initial\":[13,25],\"rate\":[3,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0040","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[14,25],\"rate\":[1,1],\"capacity\":[9,1],\"steps\":0}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(14/25, 1, 9, 0, [14/25])","explanation_zh":"應用擴展核對=(14/25, 1, 9, 0, [14/25])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217040,"parameters":{"initial":[14,25],"rate":[1,1],"capacity":[9,1],"steps":0,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[9,1],\"initial\":[14,25],\"rate\":[1,1],\"steps\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0041","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[15,25],\"rate\":[0,1],\"capacity\":[5,1],\"steps\":2}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(3/5, 0, 5, 2, [3/5, 3/5, 3/5])","explanation_zh":"應用擴展核對=(3/5, 0, 5, 2, [3/5, 3/5, 3/5])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217041,"parameters":{"initial":[15,25],"rate":[0,1],"capacity":[5,1],"steps":2,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[5,1],\"initial\":[15,25],\"rate\":[0,1],\"steps\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0042","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[16,25],\"rate\":[1,4],\"capacity\":[6,1],\"steps\":0}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(16/25, 1/4, 6, 0, [16/25])","explanation_zh":"應用擴展核對=(16/25, 1/4, 6, 0, [16/25])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217042,"parameters":{"initial":[16,25],"rate":[1,4],"capacity":[6,1],"steps":0,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[6,1],\"initial\":[16,25],\"rate\":[1,4],\"steps\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0043","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[17,25],\"rate\":[1,2],\"capacity\":[7,1],\"steps\":1}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(17/25, 1/2, 7, 1, [17/25, 4318/4375])","explanation_zh":"應用擴展核對=(17/25, 1/2, 7, 1, [17/25, 4318/4375])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217043,"parameters":{"initial":[17,25],"rate":[1,2],"capacity":[7,1],"steps":1,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[7,1],\"initial\":[17,25],\"rate\":[1,2],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0044","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[18,25],\"rate\":[3,4],\"capacity\":[8,1],\"steps\":0}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(18/25, 3/4, 8, 0, [18/25])","explanation_zh":"應用擴展核對=(18/25, 3/4, 8, 0, [18/25])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217044,"parameters":{"initial":[18,25],"rate":[3,4],"capacity":[8,1],"steps":0,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[8,1],\"initial\":[18,25],\"rate\":[3,4],\"steps\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0045","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[19,25],\"rate\":[1,1],\"capacity\":[9,1],\"steps\":1}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(19/25, 1, 9, 1, [19/25, 8189/5625])","explanation_zh":"應用擴展核對=(19/25, 1, 9, 1, [19/25, 8189/5625])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217045,"parameters":{"initial":[19,25],"rate":[1,1],"capacity":[9,1],"steps":1,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[9,1],\"initial\":[19,25],\"rate\":[1,1],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0046","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[20,25],\"rate\":[0,1],\"capacity\":[5,1],\"steps\":2}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(4/5, 0, 5, 2, [4/5, 4/5, 4/5])","explanation_zh":"應用擴展核對=(4/5, 0, 5, 2, [4/5, 4/5, 4/5])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217046,"parameters":{"initial":[20,25],"rate":[0,1],"capacity":[5,1],"steps":2,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[5,1],\"initial\":[20,25],\"rate\":[0,1],\"steps\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0047","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[21,25],\"rate\":[1,4],\"capacity\":[6,1],\"steps\":1}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(21/25, 1/4, 6, 1, [21/25, 5103/5000])","explanation_zh":"應用擴展核對=(21/25, 1/4, 6, 1, [21/25, 5103/5000])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217047,"parameters":{"initial":[21,25],"rate":[1,4],"capacity":[6,1],"steps":1,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[6,1],\"initial\":[21,25],\"rate\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0048","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[22,25],\"rate\":[1,2],\"capacity\":[7,1],\"steps\":0}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(22/25, 1/2, 7, 0, [22/25])","explanation_zh":"應用擴展核對=(22/25, 1/2, 7, 0, [22/25])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217048,"parameters":{"initial":[22,25],"rate":[1,2],"capacity":[7,1],"steps":0,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[7,1],\"initial\":[22,25],\"rate\":[1,2],\"steps\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0049","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[23,25],\"rate\":[3,4],\"capacity\":[8,1],\"steps\":1}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(23/25, 3/4, 8, 1, [23/25, 30613/20000])","explanation_zh":"應用擴展核對=(23/25, 3/4, 8, 1, [23/25, 30613/20000])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217049,"parameters":{"initial":[23,25],"rate":[3,4],"capacity":[8,1],"steps":1,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[8,1],\"initial\":[23,25],\"rate\":[3,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0050","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-orbit-prefix；資料={\"initial\":[24,25],\"rate\":[1,1],\"capacity\":[9,1],\"steps\":0}。請依 [initial,rate,capacity,steps,orbit] 欄序回答。","answer":{"kind":"expression","value":"(24/25, 1, 9, 0, [24/25])","explanation_zh":"應用擴展核對=(24/25, 1, 9, 0, [24/25])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-orbit-prefix","seed":202610217050,"parameters":{"initial":[24,25],"rate":[1,1],"capacity":[9,1],"steps":0,"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-orbit-prefix","semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[9,1],\"initial\":[24,25],\"rate\":[1,1],\"steps\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0051","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[0,5],\"rate\":[1,2],\"capacity\":[5,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/2, 5, 0, 是)","explanation_zh":"應用擴展核對=(0, 1/2, 5, 0, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217051,"parameters":{"current":[0,5],"rate":[1,2],"capacity":[5,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[5,1],\"current\":[0,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0052","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[1,5],\"rate\":[3,4],\"capacity\":[6,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(1/5, 3/4, 6, 69/200, 是)","explanation_zh":"應用擴展核對=(1/5, 3/4, 6, 69/200, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217052,"parameters":{"current":[1,5],"rate":[3,4],"capacity":[6,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[6,1],\"current\":[1,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0053","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[2,5],\"rate\":[1,1],\"capacity\":[7,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(2/5, 1, 7, 136/175, 是)","explanation_zh":"應用擴展核對=(2/5, 1, 7, 136/175, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217053,"parameters":{"current":[2,5],"rate":[1,1],"capacity":[7,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[7,1],\"current\":[2,5],\"rate\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0054","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[3,5],\"rate\":[0,1],\"capacity\":[8,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(3/5, 0, 8, 3/5, 是)","explanation_zh":"應用擴展核對=(3/5, 0, 8, 3/5, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217054,"parameters":{"current":[3,5],"rate":[0,1],"capacity":[8,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[8,1],\"current\":[3,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0055","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[4,5],\"rate\":[1,4],\"capacity\":[9,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(4/5, 1/4, 9, 221/225, 是)","explanation_zh":"應用擴展核對=(4/5, 1/4, 9, 221/225, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217055,"parameters":{"current":[4,5],"rate":[1,4],"capacity":[9,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[9,1],\"current\":[4,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0056","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[5,5],\"rate\":[1,2],\"capacity\":[5,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/2, 5, 7/5, 是)","explanation_zh":"應用擴展核對=(1, 1/2, 5, 7/5, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217056,"parameters":{"current":[5,5],"rate":[1,2],"capacity":[5,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[5,1],\"current\":[5,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0057","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[6,5],\"rate\":[3,4],\"capacity\":[6,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(6/5, 3/4, 6, 48/25, 是)","explanation_zh":"應用擴展核對=(6/5, 3/4, 6, 48/25, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217057,"parameters":{"current":[6,5],"rate":[3,4],"capacity":[6,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[6,1],\"current\":[6,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0058","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[7,5],\"rate\":[1,1],\"capacity\":[7,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(7/5, 1, 7, 63/25, 是)","explanation_zh":"應用擴展核對=(7/5, 1, 7, 63/25, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217058,"parameters":{"current":[7,5],"rate":[1,1],"capacity":[7,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[7,1],\"current\":[7,5],\"rate\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0059","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[8,5],\"rate\":[0,1],\"capacity\":[8,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(8/5, 0, 8, 8/5, 是)","explanation_zh":"應用擴展核對=(8/5, 0, 8, 8/5, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217059,"parameters":{"current":[8,5],"rate":[0,1],"capacity":[8,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[8,1],\"current\":[8,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0060","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[9,5],\"rate\":[1,4],\"capacity\":[9,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(9/5, 1/4, 9, 54/25, 是)","explanation_zh":"應用擴展核對=(9/5, 1/4, 9, 54/25, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217060,"parameters":{"current":[9,5],"rate":[1,4],"capacity":[9,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[9,1],\"current\":[9,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0061","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[10,5],\"rate\":[1,2],\"capacity\":[5,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(2, 1/2, 5, 13/5, 是)","explanation_zh":"應用擴展核對=(2, 1/2, 5, 13/5, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217061,"parameters":{"current":[10,5],"rate":[1,2],"capacity":[5,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[5,1],\"current\":[10,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0062","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[11,5],\"rate\":[3,4],\"capacity\":[6,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(11/5, 3/4, 6, 649/200, 是)","explanation_zh":"應用擴展核對=(11/5, 3/4, 6, 649/200, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217062,"parameters":{"current":[11,5],"rate":[3,4],"capacity":[6,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[6,1],\"current\":[11,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0063","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[12,5],\"rate\":[1,1],\"capacity\":[7,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(12/5, 1, 7, 696/175, 是)","explanation_zh":"應用擴展核對=(12/5, 1, 7, 696/175, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217063,"parameters":{"current":[12,5],"rate":[1,1],"capacity":[7,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[7,1],\"current\":[12,5],\"rate\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0064","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[13,5],\"rate\":[0,1],\"capacity\":[8,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(13/5, 0, 8, 13/5, 是)","explanation_zh":"應用擴展核對=(13/5, 0, 8, 13/5, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217064,"parameters":{"current":[13,5],"rate":[0,1],"capacity":[8,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[8,1],\"current\":[13,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0065","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[14,5],\"rate\":[1,4],\"capacity\":[9,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(14/5, 1/4, 9, 1477/450, 是)","explanation_zh":"應用擴展核對=(14/5, 1/4, 9, 1477/450, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217065,"parameters":{"current":[14,5],"rate":[1,4],"capacity":[9,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[9,1],\"current\":[14,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0066","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[15,5],\"rate\":[1,2],\"capacity\":[5,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(3, 1/2, 5, 18/5, 是)","explanation_zh":"應用擴展核對=(3, 1/2, 5, 18/5, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217066,"parameters":{"current":[15,5],"rate":[1,2],"capacity":[5,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[5,1],\"current\":[15,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0067","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[16,5],\"rate\":[3,4],\"capacity\":[6,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(16/5, 3/4, 6, 108/25, 是)","explanation_zh":"應用擴展核對=(16/5, 3/4, 6, 108/25, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217067,"parameters":{"current":[16,5],"rate":[3,4],"capacity":[6,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[6,1],\"current\":[16,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0068","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[17,5],\"rate\":[1,1],\"capacity\":[7,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(17/5, 1, 7, 901/175, 是)","explanation_zh":"應用擴展核對=(17/5, 1, 7, 901/175, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217068,"parameters":{"current":[17,5],"rate":[1,1],"capacity":[7,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[7,1],\"current\":[17,5],\"rate\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0069","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[18,5],\"rate\":[0,1],\"capacity\":[8,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(18/5, 0, 8, 18/5, 是)","explanation_zh":"應用擴展核對=(18/5, 0, 8, 18/5, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217069,"parameters":{"current":[18,5],"rate":[0,1],"capacity":[8,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[8,1],\"current\":[18,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0070","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[19,5],\"rate\":[1,4],\"capacity\":[9,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(19/5, 1/4, 9, 1957/450, 是)","explanation_zh":"應用擴展核對=(19/5, 1/4, 9, 1957/450, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217070,"parameters":{"current":[19,5],"rate":[1,4],"capacity":[9,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[9,1],\"current\":[19,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0071","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[20,5],\"rate\":[1,2],\"capacity\":[5,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(4, 1/2, 5, 22/5, 是)","explanation_zh":"應用擴展核對=(4, 1/2, 5, 22/5, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217071,"parameters":{"current":[20,5],"rate":[1,2],"capacity":[5,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[5,1],\"current\":[20,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0072","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[21,5],\"rate\":[3,4],\"capacity\":[6,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(21/5, 3/4, 6, 1029/200, 是)","explanation_zh":"應用擴展核對=(21/5, 3/4, 6, 1029/200, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217072,"parameters":{"current":[21,5],"rate":[3,4],"capacity":[6,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[6,1],\"current\":[21,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0073","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[22,5],\"rate\":[1,1],\"capacity\":[7,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(22/5, 1, 7, 1056/175, 是)","explanation_zh":"應用擴展核對=(22/5, 1, 7, 1056/175, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217073,"parameters":{"current":[22,5],"rate":[1,1],"capacity":[7,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[7,1],\"current\":[22,5],\"rate\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0074","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[23,5],\"rate\":[0,1],\"capacity\":[8,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(23/5, 0, 8, 23/5, 是)","explanation_zh":"應用擴展核對=(23/5, 0, 8, 23/5, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217074,"parameters":{"current":[23,5],"rate":[0,1],"capacity":[8,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[8,1],\"current\":[23,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0075","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-capacity-interval-audit；資料={\"current\":[24,5],\"rate\":[1,4],\"capacity\":[9,1]}。請依 [current,rate,capacity,next,inside_interval] 欄序回答。","answer":{"kind":"expression","value":"(24/5, 1/4, 9, 134/25, 是)","explanation_zh":"應用擴展核對=(24/5, 1/4, 9, 134/25, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217075,"parameters":{"current":[24,5],"rate":[1,4],"capacity":[9,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-capacity-interval-audit","semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[9,1],\"current\":[24,5],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0076","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[0,1],\"rate\":[1,4],\"capacity\":[5,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/4, 5, 0, 是)","explanation_zh":"應用擴展核對=(0, 1/4, 5, 0, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217076,"parameters":{"candidate":[0,1],"rate":[1,4],"capacity":[5,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[5,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0077","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[6,1],\"rate\":[1,2],\"capacity\":[6,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(6, 1/2, 6, 6, 是)","explanation_zh":"應用擴展核對=(6, 1/2, 6, 6, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217077,"parameters":{"candidate":[6,1],"rate":[1,2],"capacity":[6,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[6,1],\"capacity\":[6,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0078","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[1,1],\"rate\":[3,4],\"capacity\":[7,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(1, 3/4, 7, 23/14, 否)","explanation_zh":"應用擴展核對=(1, 3/4, 7, 23/14, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217078,"parameters":{"candidate":[1,1],"rate":[3,4],"capacity":[7,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[1,1],\"capacity\":[7,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0079","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[0,1],\"rate\":[1,1],\"capacity\":[8,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 8, 0, 是)","explanation_zh":"應用擴展核對=(0, 1, 8, 0, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217079,"parameters":{"candidate":[0,1],"rate":[1,1],"capacity":[8,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[8,1],\"rate\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0080","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[9,1],\"rate\":[0,1],\"capacity\":[9,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(9, 0, 9, 9, 是)","explanation_zh":"應用擴展核對=(9, 0, 9, 9, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217080,"parameters":{"candidate":[9,1],"rate":[0,1],"capacity":[9,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[9,1],\"capacity\":[9,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0081","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[1,1],\"rate\":[1,4],\"capacity\":[10,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/4, 10, 49/40, 否)","explanation_zh":"應用擴展核對=(1, 1/4, 10, 49/40, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217081,"parameters":{"candidate":[1,1],"rate":[1,4],"capacity":[10,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[1,1],\"capacity\":[10,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0082","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[0,1],\"rate\":[1,2],\"capacity\":[11,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/2, 11, 0, 是)","explanation_zh":"應用擴展核對=(0, 1/2, 11, 0, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217082,"parameters":{"candidate":[0,1],"rate":[1,2],"capacity":[11,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[11,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0083","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[12,1],\"rate\":[3,4],\"capacity\":[12,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(12, 3/4, 12, 12, 是)","explanation_zh":"應用擴展核對=(12, 3/4, 12, 12, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217083,"parameters":{"candidate":[12,1],"rate":[3,4],"capacity":[12,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[12,1],\"capacity\":[12,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0084","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[1,1],\"rate\":[1,1],\"capacity\":[13,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 13, 25/13, 否)","explanation_zh":"應用擴展核對=(1, 1, 13, 25/13, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217084,"parameters":{"candidate":[1,1],"rate":[1,1],"capacity":[13,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[1,1],\"capacity\":[13,1],\"rate\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0085","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[0,1],\"rate\":[0,1],\"capacity\":[14,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 14, 0, 是)","explanation_zh":"應用擴展核對=(0, 0, 14, 0, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217085,"parameters":{"candidate":[0,1],"rate":[0,1],"capacity":[14,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[14,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0086","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[15,1],\"rate\":[1,4],\"capacity\":[15,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(15, 1/4, 15, 15, 是)","explanation_zh":"應用擴展核對=(15, 1/4, 15, 15, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217086,"parameters":{"candidate":[15,1],"rate":[1,4],"capacity":[15,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[15,1],\"capacity\":[15,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0087","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[1,1],\"rate\":[1,2],\"capacity\":[16,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/2, 16, 47/32, 否)","explanation_zh":"應用擴展核對=(1, 1/2, 16, 47/32, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217087,"parameters":{"candidate":[1,1],"rate":[1,2],"capacity":[16,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[1,1],\"capacity\":[16,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0088","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[0,1],\"rate\":[3,4],\"capacity\":[17,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(0, 3/4, 17, 0, 是)","explanation_zh":"應用擴展核對=(0, 3/4, 17, 0, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217088,"parameters":{"candidate":[0,1],"rate":[3,4],"capacity":[17,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[17,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0089","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[18,1],\"rate\":[1,1],\"capacity\":[18,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(18, 1, 18, 18, 是)","explanation_zh":"應用擴展核對=(18, 1, 18, 18, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217089,"parameters":{"candidate":[18,1],"rate":[1,1],"capacity":[18,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[18,1],\"capacity\":[18,1],\"rate\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0090","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[1,1],\"rate\":[0,1],\"capacity\":[19,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 19, 1, 是)","explanation_zh":"應用擴展核對=(1, 0, 19, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217090,"parameters":{"candidate":[1,1],"rate":[0,1],"capacity":[19,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[1,1],\"capacity\":[19,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0091","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[0,1],\"rate\":[1,4],\"capacity\":[20,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/4, 20, 0, 是)","explanation_zh":"應用擴展核對=(0, 1/4, 20, 0, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217091,"parameters":{"candidate":[0,1],"rate":[1,4],"capacity":[20,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[20,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0092","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[21,1],\"rate\":[1,2],\"capacity\":[21,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(21, 1/2, 21, 21, 是)","explanation_zh":"應用擴展核對=(21, 1/2, 21, 21, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217092,"parameters":{"candidate":[21,1],"rate":[1,2],"capacity":[21,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[21,1],\"capacity\":[21,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0093","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[1,1],\"rate\":[3,4],\"capacity\":[22,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(1, 3/4, 22, 151/88, 否)","explanation_zh":"應用擴展核對=(1, 3/4, 22, 151/88, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217093,"parameters":{"candidate":[1,1],"rate":[3,4],"capacity":[22,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[1,1],\"capacity\":[22,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0094","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[0,1],\"rate\":[1,1],\"capacity\":[23,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 23, 0, 是)","explanation_zh":"應用擴展核對=(0, 1, 23, 0, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217094,"parameters":{"candidate":[0,1],"rate":[1,1],"capacity":[23,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[23,1],\"rate\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0095","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[24,1],\"rate\":[0,1],\"capacity\":[24,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(24, 0, 24, 24, 是)","explanation_zh":"應用擴展核對=(24, 0, 24, 24, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217095,"parameters":{"candidate":[24,1],"rate":[0,1],"capacity":[24,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[24,1],\"capacity\":[24,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0096","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[1,1],\"rate\":[1,4],\"capacity\":[25,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/4, 25, 31/25, 否)","explanation_zh":"應用擴展核對=(1, 1/4, 25, 31/25, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217096,"parameters":{"candidate":[1,1],"rate":[1,4],"capacity":[25,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[1,1],\"capacity\":[25,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0097","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[0,1],\"rate\":[1,2],\"capacity\":[26,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/2, 26, 0, 是)","explanation_zh":"應用擴展核對=(0, 1/2, 26, 0, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217097,"parameters":{"candidate":[0,1],"rate":[1,2],"capacity":[26,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[26,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0098","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[27,1],\"rate\":[3,4],\"capacity\":[27,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(27, 3/4, 27, 27, 是)","explanation_zh":"應用擴展核對=(27, 3/4, 27, 27, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217098,"parameters":{"candidate":[27,1],"rate":[3,4],"capacity":[27,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[27,1],\"capacity\":[27,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0099","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[1,1],\"rate\":[1,1],\"capacity\":[28,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 28, 55/28, 否)","explanation_zh":"應用擴展核對=(1, 1, 28, 55/28, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217099,"parameters":{"candidate":[1,1],"rate":[1,1],"capacity":[28,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[1,1],\"capacity\":[28,1],\"rate\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0100","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","language":"zh-Hant","source_mko_ids":["mko-discrete-logistic-growth"],"title_zh":"離散logistic增量","stem_zh":"固定離散增量模型與0≤r≤1、0≤x≤K。操作=logistic-equilibrium-audit；資料={\"candidate\":[0,1],\"rate\":[0,1],\"capacity\":[29,1]}。請依 [candidate,rate,capacity,next,fixed] 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 29, 0, 是)","explanation_zh":"應用擴展核對=(0, 0, 29, 0, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；有限軌道不是穩定性證明；不是經驗預測或模型適配證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["離散logistic增量"]},"variant":{"family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217100,"parameters":{"candidate":[0,1],"rate":[0,1],"capacity":[29,1],"domain":"Exact_discrete_logistic_increment_model","operation":"logistic-equilibrium-audit","semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[29,1],\"rate\":[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、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}