{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0001","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=3。依明示多項式矩公式求從 a=2 到 b=2 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"0/1","explanation_zh":"有向矩積分=0/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087001,"parameters":{"c":3,"a":2,"b":2,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"zero_integral\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0002","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-1。依明示多項式矩公式求從 a=-11 到 b=-9 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-2/1","explanation_zh":"有向矩積分=-2/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087002,"parameters":{"c":-1,"a":-11,"b":-9,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"-1\",\"2\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0003","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=2。依明示多項式矩公式求從 a=-10 到 b=-7 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"6/1","explanation_zh":"有向矩積分=6/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087003,"parameters":{"c":2,"a":-10,"b":-7,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"2\",\"3\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0004","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-3。依明示多項式矩公式求從 a=-9 到 b=-13 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"12/1","explanation_zh":"有向矩積分=12/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087004,"parameters":{"c":-3,"a":-9,"b":-13,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"-3\",\"-4\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0005","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=4。依明示多項式矩公式求從 a=-8 到 b=-3 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"20/1","explanation_zh":"有向矩積分=20/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087005,"parameters":{"c":4,"a":-8,"b":-3,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"4\",\"5\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0006","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-5。依明示多項式矩公式求從 a=-7 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-30/1","explanation_zh":"有向矩積分=-30/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087006,"parameters":{"c":-5,"a":-7,"b":-1,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"-5\",\"6\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0007","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=6。依明示多項式矩公式求從 a=-6 到 b=-13 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-42/1","explanation_zh":"有向矩積分=-42/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087007,"parameters":{"c":6,"a":-6,"b":-13,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"6\",\"-7\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0008","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-7。依明示多項式矩公式求從 a=-5 到 b=3 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-56/1","explanation_zh":"有向矩積分=-56/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087008,"parameters":{"c":-7,"a":-5,"b":3,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"-7\",\"8\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0009","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=8。依明示多項式矩公式求從 a=-4 到 b=5 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"72/1","explanation_zh":"有向矩積分=72/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087009,"parameters":{"c":8,"a":-4,"b":5,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"8\",\"9\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0010","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-9。依明示多項式矩公式求從 a=-3 到 b=-13 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"90/1","explanation_zh":"有向矩積分=90/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087010,"parameters":{"c":-9,"a":-3,"b":-13,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"-9\",\"-10\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0011","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=10。依明示多項式矩公式求從 a=-2 到 b=9 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"110/1","explanation_zh":"有向矩積分=110/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087011,"parameters":{"c":10,"a":-2,"b":9,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"10\",\"11\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0012","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-11。依明示多項式矩公式求從 a=-1 到 b=11 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-132/1","explanation_zh":"有向矩積分=-132/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087012,"parameters":{"c":-11,"a":-1,"b":11,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"-11\",\"12\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0013","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=12。依明示多項式矩公式求從 a=0 到 b=-13 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-156/1","explanation_zh":"有向矩積分=-156/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087013,"parameters":{"c":12,"a":0,"b":-13,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"12\",\"-13\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0014","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-13。依明示多項式矩公式求從 a=1 到 b=15 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-182/1","explanation_zh":"有向矩積分=-182/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087014,"parameters":{"c":-13,"a":1,"b":15,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"-13\",\"14\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0015","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=14。依明示多項式矩公式求從 a=2 到 b=17 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"210/1","explanation_zh":"有向矩積分=210/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087015,"parameters":{"c":14,"a":2,"b":17,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"14\",\"15\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0016","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-15。依明示多項式矩公式求從 a=3 到 b=-13 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"240/1","explanation_zh":"有向矩積分=240/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087016,"parameters":{"c":-15,"a":3,"b":-13,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"-15\",\"-16\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0017","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=16。依明示多項式矩公式求從 a=4 到 b=21 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"272/1","explanation_zh":"有向矩積分=272/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087017,"parameters":{"c":16,"a":4,"b":21,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"16\",\"17\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0018","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-17。依明示多項式矩公式求從 a=5 到 b=23 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-306/1","explanation_zh":"有向矩積分=-306/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087018,"parameters":{"c":-17,"a":5,"b":23,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"-17\",\"18\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0019","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=18。依明示多項式矩公式求從 a=6 到 b=-13 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-342/1","explanation_zh":"有向矩積分=-342/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087019,"parameters":{"c":18,"a":6,"b":-13,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"18\",\"-19\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0020","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-19。依明示多項式矩公式求從 a=7 到 b=27 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-380/1","explanation_zh":"有向矩積分=-380/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087020,"parameters":{"c":-19,"a":7,"b":27,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"-19\",\"20\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0021","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=20。依明示多項式矩公式求從 a=8 到 b=29 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"420/1","explanation_zh":"有向矩積分=420/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087021,"parameters":{"c":20,"a":8,"b":29,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"20\",\"21\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0022","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-21。依明示多項式矩公式求從 a=9 到 b=-13 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"462/1","explanation_zh":"有向矩積分=462/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087022,"parameters":{"c":-21,"a":9,"b":-13,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"-21\",\"-22\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0023","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=22。依明示多項式矩公式求從 a=10 到 b=33 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"506/1","explanation_zh":"有向矩積分=506/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087023,"parameters":{"c":22,"a":10,"b":33,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"22\",\"23\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0024","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-23。依明示多項式矩公式求從 a=11 到 b=35 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-552/1","explanation_zh":"有向矩積分=-552/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087024,"parameters":{"c":-23,"a":11,"b":35,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"-23\",\"24\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0025","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"常數式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=24。依明示多項式矩公式求從 a=12 到 b=-13 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-600/1","explanation_zh":"有向矩積分=-600/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["常數式的有向矩積分"]},"variant":{"family_id":"qf-integral-constant","seed":202610087025,"parameters":{"c":24,"a":12,"b":-13,"operation":"constant_integral","domain":"reals","semantic_template":"qf-integral-constant/{\"inputs\":[\"24\",\"-25\"],\"operation\":\"constant_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0026","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=2x^1，n為0至5。依明示多項式矩公式求從 a=3 到 b=3 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"0/1","explanation_zh":"有向矩積分=0/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087026,"parameters":{"c":2,"n":1,"a":3,"b":3,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"zero_integral\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0027","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-2x^1，n為0至5。依明示多項式矩公式求從 a=-2 到 b=0 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"4/1","explanation_zh":"有向矩積分=4/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087027,"parameters":{"c":-2,"n":1,"a":-2,"b":0,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"-2\",1,\"-2\",\"0\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0028","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=3x^2，n為0至5。依明示多項式矩公式求從 a=-3 到 b=1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"28/1","explanation_zh":"有向矩積分=28/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087028,"parameters":{"c":3,"n":2,"a":-3,"b":1,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"3\",2,\"-3\",\"1\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0029","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-4x^3，n為0至5。依明示多項式矩公式求從 a=2 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"15/1","explanation_zh":"有向矩積分=15/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087029,"parameters":{"c":-4,"n":3,"a":2,"b":-1,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"-4\",3,\"2\",\"-1\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0030","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=5x^4，n為0至5。依明示多項式矩公式求從 a=-2 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"31/1","explanation_zh":"有向矩積分=31/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087030,"parameters":{"c":5,"n":4,"a":-2,"b":-1,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"5\",4,\"-2\",\"-1\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0031","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-6x^5，n為0至5。依明示多項式矩公式求從 a=-3 到 b=0 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"729/1","explanation_zh":"有向矩積分=729/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087031,"parameters":{"c":-6,"n":5,"a":-3,"b":0,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"-6\",5,\"-3\",\"0\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0032","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=7x^0，n為0至5。依明示多項式矩公式求從 a=1 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-14/1","explanation_zh":"有向矩積分=-14/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087032,"parameters":{"c":7,"n":0,"a":1,"b":-1,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"7\",\"constant_width\",\"-2\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0033","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-8x^1，n為0至5。依明示多項式矩公式求從 a=-2 到 b=2 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"0/1","explanation_zh":"有向矩積分=0/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087033,"parameters":{"c":-8,"n":1,"a":-2,"b":2,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"-8\",1,\"-2\",\"2\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0034","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=9x^2，n為0至5。依明示多項式矩公式求從 a=-3 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"78/1","explanation_zh":"有向矩積分=78/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087034,"parameters":{"c":9,"n":2,"a":-3,"b":-1,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"9\",2,\"-3\",\"-1\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0035","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-10x^3，n為0至5。依明示多項式矩公式求從 a=0 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-5/2","explanation_zh":"有向矩積分=-5/2。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087035,"parameters":{"c":-10,"n":3,"a":0,"b":-1,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"-10\",3,\"0\",\"-1\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0036","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=11x^4，n為0至5。依明示多項式矩公式求從 a=-2 到 b=1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"363/5","explanation_zh":"有向矩積分=363/5。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087036,"parameters":{"c":11,"n":4,"a":-2,"b":1,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"11\",4,\"-2\",\"1\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0037","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-12x^5，n為0至5。依明示多項式矩公式求從 a=-3 到 b=2 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"1330/1","explanation_zh":"有向矩積分=1330/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087037,"parameters":{"c":-12,"n":5,"a":-3,"b":2,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"-12\",5,\"-3\",\"2\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0038","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=13x^0，n為0至5。依明示多項式矩公式求從 a=1 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-26/1","explanation_zh":"有向矩積分=-26/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087038,"parameters":{"c":13,"n":0,"a":1,"b":-1,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"13\",\"constant_width\",\"-2\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0039","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-14x^1，n為0至5。依明示多項式矩公式求從 a=-2 到 b=0 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"28/1","explanation_zh":"有向矩積分=28/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087039,"parameters":{"c":-14,"n":1,"a":-2,"b":0,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"-14\",1,\"-2\",\"0\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0040","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=15x^2，n為0至5。依明示多項式矩公式求從 a=-3 到 b=1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"140/1","explanation_zh":"有向矩積分=140/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087040,"parameters":{"c":15,"n":2,"a":-3,"b":1,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"15\",2,\"-3\",\"1\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0041","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-16x^3，n為0至5。依明示多項式矩公式求從 a=2 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"60/1","explanation_zh":"有向矩積分=60/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087041,"parameters":{"c":-16,"n":3,"a":2,"b":-1,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"-16\",3,\"2\",\"-1\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0042","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=17x^4，n為0至5。依明示多項式矩公式求從 a=-2 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"527/5","explanation_zh":"有向矩積分=527/5。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087042,"parameters":{"c":17,"n":4,"a":-2,"b":-1,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"17\",4,\"-2\",\"-1\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0043","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-18x^5，n為0至5。依明示多項式矩公式求從 a=-3 到 b=0 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"2187/1","explanation_zh":"有向矩積分=2187/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087043,"parameters":{"c":-18,"n":5,"a":-3,"b":0,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"-18\",5,\"-3\",\"0\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0044","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=19x^0，n為0至5。依明示多項式矩公式求從 a=1 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-38/1","explanation_zh":"有向矩積分=-38/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087044,"parameters":{"c":19,"n":0,"a":1,"b":-1,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"19\",\"constant_width\",\"-2\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0045","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-20x^1，n為0至5。依明示多項式矩公式求從 a=-2 到 b=2 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"0/1","explanation_zh":"有向矩積分=0/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087045,"parameters":{"c":-20,"n":1,"a":-2,"b":2,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"-20\",1,\"-2\",\"2\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0046","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=21x^2，n為0至5。依明示多項式矩公式求從 a=-3 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"182/1","explanation_zh":"有向矩積分=182/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087046,"parameters":{"c":21,"n":2,"a":-3,"b":-1,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"21\",2,\"-3\",\"-1\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0047","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-22x^3，n為0至5。依明示多項式矩公式求從 a=0 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-11/2","explanation_zh":"有向矩積分=-11/2。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087047,"parameters":{"c":-22,"n":3,"a":0,"b":-1,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"-22\",3,\"0\",\"-1\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0048","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=23x^4，n為0至5。依明示多項式矩公式求從 a=-2 到 b=1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"759/5","explanation_zh":"有向矩積分=759/5。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087048,"parameters":{"c":23,"n":4,"a":-2,"b":1,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"23\",4,\"-2\",\"1\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0049","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-24x^5，n為0至5。依明示多項式矩公式求從 a=-3 到 b=2 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"2660/1","explanation_zh":"有向矩積分=2660/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087049,"parameters":{"c":-24,"n":5,"a":-3,"b":2,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"-24\",5,\"-3\",\"2\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0050","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"單項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=25x^0，n為0至5。依明示多項式矩公式求從 a=1 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-50/1","explanation_zh":"有向矩積分=-50/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單項式的有向矩積分"]},"variant":{"family_id":"qf-integral-monomial","seed":202610087050,"parameters":{"c":25,"n":0,"a":1,"b":-1,"operation":"monomial_integral","domain":"reals","semantic_template":"qf-integral-monomial/{\"inputs\":[\"25\",\"constant_width\",\"-2\"],\"operation\":\"monomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0051","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[1, 2]。依明示多項式矩公式求從 a=0 到 b=0 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"0/1","explanation_zh":"有向矩積分=0/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087051,"parameters":{"f":[1,2],"a":0,"b":0,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[\"zero_integral\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0052","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[3]。依明示多項式矩公式求從 a=2 到 b=-2 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-12/1","explanation_zh":"有向矩積分=-12/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087052,"parameters":{"f":[3],"a":2,"b":-2,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[\"3\",\"constant_width\",\"-4\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0053","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-10, 3, 1, 3]。依明示多項式矩公式求從 a=-3 到 b=3 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-42/1","explanation_zh":"有向矩積分=-42/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087053,"parameters":{"f":[-10,3,1,3],"a":-3,"b":3,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"-10\",\"3\",\"1\",\"3\"],\"-3\",\"3\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0054","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-9, 4, -1, 4]。依明示多項式矩公式求從 a=4 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-655/3","explanation_zh":"有向矩積分=-655/3。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087054,"parameters":{"f":[-9,4,-1,4],"a":4,"b":-1,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"-9\",\"4\",\"-1\",\"4\"],\"4\",\"-1\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0055","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-8, 5, 0, 1]。依明示多項式矩公式求從 a=-2 到 b=1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-141/4","explanation_zh":"有向矩積分=-141/4。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087055,"parameters":{"f":[-8,5,0,1],"a":-2,"b":1,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"-8\",\"5\",\"0\",\"1\"],\"-2\",\"1\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0056","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-7, 6, 1, 2]。依明示多項式矩公式求從 a=2 到 b=-3 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"425/6","explanation_zh":"有向矩積分=425/6。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087056,"parameters":{"f":[-7,6,1,2],"a":2,"b":-3,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"-7\",\"6\",\"1\",\"2\"],\"2\",\"-3\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0057","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-6, 7, -1, 3]。依明示多項式矩公式求從 a=-1 到 b=3 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"164/3","explanation_zh":"有向矩積分=164/3。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087057,"parameters":{"f":[-6,7,-1,3],"a":-1,"b":3,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"-6\",\"7\",\"-1\",\"3\"],\"-1\",\"3\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0058","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-5, 8, 0, 4]。依明示多項式矩公式求從 a=4 到 b=-2 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-258/1","explanation_zh":"有向矩積分=-258/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087058,"parameters":{"f":[-5,8,0,4],"a":4,"b":-2,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"-5\",\"8\",\"0\",\"4\"],\"4\",\"-2\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0059","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-4, 9, 1, 1]。依明示多項式矩公式求從 a=-3 到 b=1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-188/3","explanation_zh":"有向矩積分=-188/3。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087059,"parameters":{"f":[-4,9,1,1],"a":-3,"b":1,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"-4\",\"9\",\"1\",\"1\"],\"-3\",\"1\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0060","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-3, 10, -1, 2]。依明示多項式矩公式求從 a=2 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-21/2","explanation_zh":"有向矩積分=-21/2。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087060,"parameters":{"f":[-3,10,-1,2],"a":2,"b":-1,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"-3\",\"10\",\"-1\",\"2\"],\"2\",\"-1\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0061","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-2, 11, 0, 3]。依明示多項式矩公式求從 a=-2 到 b=3 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"265/4","explanation_zh":"有向矩積分=265/4。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087061,"parameters":{"f":[-2,11,0,3],"a":-2,"b":3,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"-2\",\"11\",\"0\",\"3\"],\"-2\",\"3\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0062","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-1, 12, 1, 4]。依明示多項式矩公式求從 a=4 到 b=-3 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-721/3","explanation_zh":"有向矩積分=-721/3。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087062,"parameters":{"f":[-1,12,1,4],"a":4,"b":-3,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"-1\",\"12\",\"1\",\"4\"],\"4\",\"-3\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0063","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[0, 13, -1, 1]。依明示多項式矩公式求從 a=-1 到 b=1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-2/3","explanation_zh":"有向矩積分=-2/3。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087063,"parameters":{"f":[0,13,-1,1],"a":-1,"b":1,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"0\",\"13\",\"-1\",\"1\"],\"-1\",\"1\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0064","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[1, 14, 0, 2]。依明示多項式矩公式求從 a=2 到 b=-2 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-4/1","explanation_zh":"有向矩積分=-4/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087064,"parameters":{"f":[1,14,0,2],"a":2,"b":-2,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"1\",\"14\",\"0\",\"2\"],\"2\",\"-2\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0065","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[2, 15, 1, 3]。依明示多項式矩公式求從 a=-3 到 b=3 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"30/1","explanation_zh":"有向矩積分=30/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087065,"parameters":{"f":[2,15,1,3],"a":-3,"b":3,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"2\",\"15\",\"1\",\"3\"],\"-3\",\"3\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0066","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[3, 16, -1, 4]。依明示多項式矩公式求從 a=4 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-1105/3","explanation_zh":"有向矩積分=-1105/3。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087066,"parameters":{"f":[3,16,-1,4],"a":4,"b":-1,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"3\",\"16\",\"-1\",\"4\"],\"4\",\"-1\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0067","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[4, 17, 0, 1]。依明示多項式矩公式求從 a=-2 到 b=1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-69/4","explanation_zh":"有向矩積分=-69/4。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087067,"parameters":{"f":[4,17,0,1],"a":-2,"b":1,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"4\",\"17\",\"0\",\"1\"],\"-2\",\"1\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0068","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[5, 18, 1, 2]。依明示多項式矩公式求從 a=2 到 b=-3 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"245/6","explanation_zh":"有向矩積分=245/6。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087068,"parameters":{"f":[5,18,1,2],"a":2,"b":-3,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"5\",\"18\",\"1\",\"2\"],\"2\",\"-3\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0069","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[6, 19, -1, 3]。依明示多項式矩公式求從 a=-1 到 b=3 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"452/3","explanation_zh":"有向矩積分=452/3。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087069,"parameters":{"f":[6,19,-1,3],"a":-1,"b":3,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"6\",\"19\",\"-1\",\"3\"],\"-1\",\"3\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0070","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[7, 20, 0, 4]。依明示多項式矩公式求從 a=4 到 b=-2 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-402/1","explanation_zh":"有向矩積分=-402/1。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087070,"parameters":{"f":[7,20,0,4],"a":4,"b":-2,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"7\",\"20\",\"0\",\"4\"],\"4\",\"-2\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0071","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[8, 21, 1, 1]。依明示多項式矩公式求從 a=-3 到 b=1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-188/3","explanation_zh":"有向矩積分=-188/3。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087071,"parameters":{"f":[8,21,1,1],"a":-3,"b":1,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"8\",\"21\",\"1\",\"1\"],\"-3\",\"1\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0072","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[9, 22, -1, 2]。依明示多項式矩公式求從 a=2 到 b=-1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-129/2","explanation_zh":"有向矩積分=-129/2。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087072,"parameters":{"f":[9,22,-1,2],"a":2,"b":-1,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"9\",\"22\",\"-1\",\"2\"],\"2\",\"-1\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0073","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[10, 23, 0, 3]。依明示多項式矩公式求從 a=-2 到 b=3 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"625/4","explanation_zh":"有向矩積分=625/4。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087073,"parameters":{"f":[10,23,0,3],"a":-2,"b":3,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"10\",\"23\",\"0\",\"3\"],\"-2\",\"3\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0074","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[11, 24, 1, 4]。依明示多項式矩公式求從 a=4 到 b=-3 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"-1099/3","explanation_zh":"有向矩積分=-1099/3。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087074,"parameters":{"f":[11,24,1,4],"a":4,"b":-3,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"11\",\"24\",\"1\",\"4\"],\"4\",\"-3\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0075","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"多項式的有向矩積分","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[12, 25, -1, 1]。依明示多項式矩公式求從 a=-1 到 b=1 的有向積分，採最簡正分母分數。","answer":{"kind":"expression","value":"70/3","explanation_zh":"有向矩積分=70/3。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式的有向矩積分"]},"variant":{"family_id":"qf-integral-polynomial","seed":202610087075,"parameters":{"f":[12,25,-1,1],"a":-1,"b":1,"operation":"polynomial_integral","domain":"reals","representation":"ascending_powers","semantic_template":"qf-integral-polynomial/{\"inputs\":[[\"12\",\"25\",\"-1\",\"1\"],\"-1\",\"1\"],\"operation\":\"polynomial_integral\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0076","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=2x+(1)，區間 [0,2] 分為 N=2 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"4/1","explanation_zh":"左端有限和=4/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087076,"parameters":{"m":2,"c":1,"a":0,"b":2,"N":2,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"2\",\"1\",\"2\",\"2\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0077","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=0x+(2)，區間 [-1,2] 分為 N=3 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"6/1","explanation_zh":"左端有限和=6/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087077,"parameters":{"m":0,"c":2,"a":-1,"b":2,"N":3,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"constant_sum\",\"2\",\"3\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0078","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=3x+(-2)，區間 [1,1] 分為 N=4 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"0/1","explanation_zh":"左端有限和=0/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087078,"parameters":{"m":3,"c":-2,"a":1,"b":1,"N":4,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"zero_width\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0079","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-4x+(-9)，區間 [1,5] 分為 N=1 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"-52/1","explanation_zh":"左端有限和=-52/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087079,"parameters":{"m":-4,"c":-9,"a":1,"b":5,"N":1,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"-4\",\"-13\",\"4\",\"1\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0080","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=5x+(-8)，區間 [-2,3] 分為 N=5 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"-40/1","explanation_zh":"左端有限和=-40/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087080,"parameters":{"m":5,"c":-8,"a":-2,"b":3,"N":5,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"5\",\"-18\",\"5\",\"5\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0081","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-6x+(-7)，區間 [-1,0] 分為 N=6 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"-7/2","explanation_zh":"左端有限和=-7/2。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087081,"parameters":{"m":-6,"c":-7,"a":-1,"b":0,"N":6,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"-6\",\"-1\",\"1\",\"6\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0082","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=7x+(-6)，區間 [0,2] 分為 N=7 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"0/1","explanation_zh":"左端有限和=0/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087082,"parameters":{"m":7,"c":-6,"a":0,"b":2,"N":7,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"7\",\"-6\",\"2\",\"7\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0083","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-8x+(-5)，區間 [1,4] 分為 N=8 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"-141/2","explanation_zh":"左端有限和=-141/2。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087083,"parameters":{"m":-8,"c":-5,"a":1,"b":4,"N":8,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"-8\",\"-13\",\"3\",\"8\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0084","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=9x+(-4)，區間 [-2,2] 分為 N=9 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"-24/1","explanation_zh":"左端有限和=-24/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087084,"parameters":{"m":9,"c":-4,"a":-2,"b":2,"N":9,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"9\",\"-22\",\"4\",\"9\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0085","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-10x+(-3)，區間 [-1,4] 分為 N=10 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"-155/2","explanation_zh":"左端有限和=-155/2。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087085,"parameters":{"m":-10,"c":-3,"a":-1,"b":4,"N":10,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"-10\",\"7\",\"5\",\"10\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0086","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=11x+(-2)，區間 [0,1] 分為 N=11 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"3/1","explanation_zh":"左端有限和=3/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087086,"parameters":{"m":11,"c":-2,"a":0,"b":1,"N":11,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"11\",\"-2\",\"1\",\"11\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0087","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-12x+(-1)，區間 [1,3] 分為 N=12 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"-48/1","explanation_zh":"左端有限和=-48/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087087,"parameters":{"m":-12,"c":-1,"a":1,"b":3,"N":12,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"-12\",\"-13\",\"2\",\"12\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0088","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=13x+(0)，區間 [-2,1] 分為 N=13 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"-24/1","explanation_zh":"左端有限和=-24/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087088,"parameters":{"m":13,"c":0,"a":-2,"b":1,"N":13,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"13\",\"-26\",\"3\",\"13\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0089","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-14x+(1)，區間 [-1,3] 分為 N=14 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"-44/1","explanation_zh":"左端有限和=-44/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087089,"parameters":{"m":-14,"c":1,"a":-1,"b":3,"N":14,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"-14\",\"15\",\"4\",\"14\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0090","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=15x+(2)，區間 [0,5] 分為 N=15 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"185/1","explanation_zh":"左端有限和=185/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087090,"parameters":{"m":15,"c":2,"a":0,"b":5,"N":15,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"15\",\"2\",\"5\",\"15\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0091","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-16x+(3)，區間 [1,2] 分為 N=16 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"-41/2","explanation_zh":"左端有限和=-41/2。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087091,"parameters":{"m":-16,"c":3,"a":1,"b":2,"N":16,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"-16\",\"-13\",\"1\",\"16\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0092","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=17x+(4)，區間 [-2,0] 分為 N=17 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"-28/1","explanation_zh":"左端有限和=-28/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087092,"parameters":{"m":17,"c":4,"a":-2,"b":0,"N":17,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"17\",\"-30\",\"2\",\"17\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0093","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-18x+(5)，區間 [-1,2] 分為 N=18 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"-15/2","explanation_zh":"左端有限和=-15/2。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087093,"parameters":{"m":-18,"c":5,"a":-1,"b":2,"N":18,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"-18\",\"23\",\"3\",\"18\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0094","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=19x+(6)，區間 [0,4] 分為 N=19 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"168/1","explanation_zh":"左端有限和=168/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087094,"parameters":{"m":19,"c":6,"a":0,"b":4,"N":19,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"19\",\"6\",\"4\",\"19\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0095","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-20x+(7)，區間 [1,6] 分為 N=20 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"-605/2","explanation_zh":"左端有限和=-605/2。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087095,"parameters":{"m":-20,"c":7,"a":1,"b":6,"N":20,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"-20\",\"-13\",\"5\",\"20\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0096","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=21x+(8)，區間 [-2,-1] 分為 N=21 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"-24/1","explanation_zh":"左端有限和=-24/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087096,"parameters":{"m":21,"c":8,"a":-2,"b":-1,"N":21,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"21\",\"-34\",\"1\",\"21\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0097","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-22x+(9)，區間 [-1,1] 分為 N=22 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"20/1","explanation_zh":"左端有限和=20/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087097,"parameters":{"m":-22,"c":9,"a":-1,"b":1,"N":22,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"-22\",\"31\",\"2\",\"22\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0098","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=23x+(10)，區間 [0,3] 分為 N=23 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"129/1","explanation_zh":"左端有限和=129/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087098,"parameters":{"m":23,"c":10,"a":0,"b":3,"N":23,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"23\",\"10\",\"3\",\"23\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0099","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-24x+(11)，區間 [1,5] 分為 N=24 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"-236/1","explanation_zh":"左端有限和=-236/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087099,"parameters":{"m":-24,"c":11,"a":1,"b":5,"N":24,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"-24\",\"-13\",\"4\",\"24\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r07-definite-integral-0100","batch_id":"batch-20261007-advance-20261008-r07-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"仿射式的左端有限和","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。f(x)=25x+(12)，區間 [-2,3] 分為 N=50 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。","answer":{"kind":"expression","value":"465/4","explanation_zh":"左端有限和=465/4。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射式的左端有限和"]},"variant":{"family_id":"qf-integral-affine-left-sum","seed":202610087100,"parameters":{"m":25,"c":12,"a":-2,"b":3,"N":50,"operation":"affine_left_sum","domain":"reals","semantic_template":"qf-integral-affine-left-sum/{\"inputs\":[\"25\",\"-38\",\"5\",\"50\"],\"operation\":\"affine_left_sum\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}