{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r06-definite-integral-0001","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[0]，a=0、b=1、c=2。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(0/1, 0/1, 0/1)","explanation_zh":"有向三段多項式矩=(0/1, 0/1, 0/1)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096001,"parameters":{"f":[0],"a":0,"b":1,"c":2,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[\"all_zero_segments\"]"}},"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-20261009-r06-definite-integral-0002","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[3]，a=3、b=-1、c=2。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(-12/1, 9/1, -3/1)","explanation_zh":"有向三段多項式矩=(-12/1, 9/1, -3/1)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096002,"parameters":{"f":[3],"a":3,"b":-1,"c":2,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[3],3,-1,2]"}},"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-20261009-r06-definite-integral-0003","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 2, 3]，a=0、b=2、c=0。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(14/1, -14/1, 0/1)","explanation_zh":"有向三段多項式矩=(14/1, -14/1, 0/1)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096003,"parameters":{"f":[1,2,3],"a":0,"b":2,"c":0,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[1,2,3],0,2,0]"}},"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-20261009-r06-definite-integral-0004","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-4, 4, 5, 1]，a=-2、b=-6、c=-11。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(160/3, 20015/12, 6885/4)","explanation_zh":"有向三段多項式矩=(160/3, 20015/12, 6885/4)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096004,"parameters":{"f":[-4,4,5,1],"a":-2,"b":-6,"c":-11,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[-4,4,5,1],-2,-6,-11]"}},"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-20261009-r06-definite-integral-0005","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-3, 5, 6, 1]，a=-1、b=0、c=6。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(-15/4, 828/1, 3297/4)","explanation_zh":"有向三段多項式矩=(-15/4, 828/1, 3297/4)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096005,"parameters":{"f":[-3,5,6,1],"a":-1,"b":0,"c":6,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[-3,5,6,1],-1,0,6]"}},"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-20261009-r06-definite-integral-0006","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-2, 6, 7, 1]，a=0、b=-6、c=-4。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(-60/1, 92/3, -88/3)","explanation_zh":"有向三段多項式矩=(-60/1, 92/3, -88/3)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096006,"parameters":{"f":[-2,6,7,1],"a":0,"b":-6,"c":-4,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[-2,6,7,1],0,-6,-4]"}},"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-20261009-r06-definite-integral-0007","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-1, 7, 8, 1]，a=-2、b=1、c=-7。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(27/4, -424/3, -1615/12)","explanation_zh":"有向三段多項式矩=(27/4, -424/3, -1615/12)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096007,"parameters":{"f":[-1,7,8,1],"a":-2,"b":1,"c":-7,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[-1,7,8,1],-2,1,-7]"}},"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-20261009-r06-definite-integral-0008","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[0, 8, 9, 1]，a=-1、b=-9、c=-5。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(-224/1, 104/1, -120/1)","explanation_zh":"有向三段多項式矩=(-224/1, 104/1, -120/1)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096008,"parameters":{"f":[0,8,9,1],"a":-1,"b":-9,"c":-5,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[0,8,9,1],-1,-9,-5]"}},"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-20261009-r06-definite-integral-0009","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 9, 10, 1]，a=0、b=1、c=6。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(109/12, 14435/12, 1212/1)","explanation_zh":"有向三段多項式矩=(109/12, 14435/12, 1212/1)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096009,"parameters":{"f":[1,9,10,1],"a":0,"b":1,"c":6,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[1,9,10,1],0,1,6]"}},"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-20261009-r06-definite-integral-0010","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[2, 10, 11, 1]，a=-2、b=-12、c=-23。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(-1340/3, 340835/12, 111825/4)","explanation_zh":"有向三段多項式矩=(-1340/3, 340835/12, 111825/4)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096010,"parameters":{"f":[2,10,11,1],"a":-2,"b":-12,"c":-23,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[2,10,11,1],-2,-12,-23]"}},"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-20261009-r06-definite-integral-0011","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[3, 11, 12, 1]，a=-1、b=2、c=4。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(261/4, 356/1, 1685/4)","explanation_zh":"有向三段多項式矩=(261/4, 356/1, 1685/4)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096011,"parameters":{"f":[3,11,12,1],"a":-1,"b":2,"c":4,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[3,11,12,1],-1,2,4]"}},"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-20261009-r06-definite-integral-0012","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[4, 12, 13, 1]，a=0、b=-12、c=-9。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(-1488/1, 1677/4, -4275/4)","explanation_zh":"有向三段多項式矩=(-1488/1, 1677/4, -4275/4)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096012,"parameters":{"f":[4,12,13,1],"a":0,"b":-12,"c":-9,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[4,12,13,1],0,-12,-9]"}},"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-20261009-r06-definite-integral-0013","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[5, 13, 14, 1]，a=-2、b=-1、c=-15。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(173/12, -5110/3, -20267/12)","explanation_zh":"有向三段多項式矩=(173/12, -5110/3, -20267/12)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096013,"parameters":{"f":[5,13,14,1],"a":-2,"b":-1,"c":-15,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[5,13,14,1],-2,-1,-15]"}},"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-20261009-r06-definite-integral-0014","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[6, 14, 15, 1]，a=-1、b=-15、c=-10。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(-2730/1, 3495/4, -7425/4)","explanation_zh":"有向三段多項式矩=(-2730/1, 3495/4, -7425/4)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096014,"parameters":{"f":[6,14,15,1],"a":-1,"b":-15,"c":-10,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[6,14,15,1],-1,-15,-10]"}},"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-20261009-r06-definite-integral-0015","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[7, 15, 16, 1]，a=0、b=3、c=9。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(1011/4, 5946/1, 24795/4)","explanation_zh":"有向三段多項式矩=(1011/4, 5946/1, 24795/4)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096015,"parameters":{"f":[7,15,16,1],"a":0,"b":3,"c":9,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[7,15,16,1],0,3,9]"}},"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-20261009-r06-definite-integral-0016","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[8, 16, 17, 1]，a=-2、b=-18、c=-35。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(-12992/3, 1752887/12, 566973/4)","explanation_zh":"有向三段多項式矩=(-12992/3, 1752887/12, 566973/4)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096016,"parameters":{"f":[8,16,17,1],"a":-2,"b":-18,"c":-35,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[8,16,17,1],-2,-18,-35]"}},"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-20261009-r06-definite-integral-0017","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[9, 17, 18, 1]，a=-1、b=0、c=3。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(25/4, 1143/4, 292/1)","explanation_zh":"有向三段多項式矩=(25/4, 1143/4, 292/1)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096017,"parameters":{"f":[9,17,18,1],"a":-1,"b":0,"c":3,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[9,17,18,1],-1,0,3]"}},"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-20261009-r06-definite-integral-0018","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[10, 18, 19, 1]，a=0、b=-18、c=-14。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(-7956/1, 5416/3, -18452/3)","explanation_zh":"有向三段多項式矩=(-7956/1, 5416/3, -18452/3)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096018,"parameters":{"f":[10,18,19,1],"a":0,"b":-18,"c":-14,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[10,18,19,1],0,-18,-14]"}},"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-20261009-r06-definite-integral-0019","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[11, 19, 20, 1]，a=-2、b=1、c=-19。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(243/4, -29860/3, -118711/12)","explanation_zh":"有向三段多項式矩=(243/4, -29860/3, -118711/12)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096019,"parameters":{"f":[11,19,20,1],"a":-2,"b":1,"c":-19,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[11,19,20,1],-2,1,-19]"}},"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-20261009-r06-definite-integral-0020","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[12, 20, 21, 1]，a=-1、b=-21、c=-15。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(-12040/1, 3150/1, -8890/1)","explanation_zh":"有向三段多項式矩=(-12040/1, 3150/1, -8890/1)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096020,"parameters":{"f":[12,20,21,1],"a":-1,"b":-21,"c":-15,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[12,20,21,1],-1,-21,-15]"}},"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-20261009-r06-definite-integral-0021","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[13, 21, 22, 1]，a=0、b=1、c=3。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(373/12, 962/3, 1407/4)","explanation_zh":"有向三段多項式矩=(373/12, 962/3, 1407/4)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096021,"parameters":{"f":[13,21,22,1],"a":0,"b":1,"c":3,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[13,21,22,1],0,1,3]"}},"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-20261009-r06-definite-integral-0022","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[14, 22, 23, 1]，a=-2、b=-24、c=-47。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(-50996/3, 5575499/12, 1790505/4)","explanation_zh":"有向三段多項式矩=(-50996/3, 5575499/12, 1790505/4)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096022,"parameters":{"f":[14,22,23,1],"a":-2,"b":-24,"c":-47,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[14,22,23,1],-2,-24,-47]"}},"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-20261009-r06-definite-integral-0023","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[15, 23, 24, 1]，a=-1、b=2、c=6。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(621/4, 2412/1, 10269/4)","explanation_zh":"有向三段多項式矩=(621/4, 2412/1, 10269/4)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096023,"parameters":{"f":[15,23,24,1],"a":-1,"b":2,"c":6,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[15,23,24,1],-1,2,6]"}},"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-20261009-r06-definite-integral-0024","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[16, 24, 25, 1]，a=0、b=-24、c=-19。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(-25728/1, 62135/12, -246601/12)","explanation_zh":"有向三段多項式矩=(-25728/1, 62135/12, -246601/12)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096024,"parameters":{"f":[16,24,25,1],"a":0,"b":-24,"c":-19,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[16,24,25,1],0,-24,-19]"}},"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-20261009-r06-definite-integral-0025","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"三段有向多項式矩積分","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[17, 25, 26, 1]，a=-2、b=-1、c=-27。按明示多項式矩模型求 (I(a,b), I(b,c), I(a,c))，端點不預設排序。","answer":{"kind":"expression","value":"(437/12, -87178/3, -348275/12)","explanation_zh":"有向三段多項式矩=(437/12, -87178/3, -348275/12)。產生端逐段以canonical primitive端點差另算，保留方向及符號；只核對此多項式模型。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三段有向多項式矩積分"]},"variant":{"family_id":"qf-density-integral-additivity-triple","seed":202610096025,"parameters":{"f":[17,25,26,1],"a":-2,"b":-1,"c":-27,"operation":"density_integral_additivity_triple","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-additivity-triple/[[17,25,26,1],-2,-1,-27]"}},"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-20261009-r06-definite-integral-0026","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[0, 2, 0, 1]，r=2≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"奇函數資格與對稱多項式矩=(是, 0/1)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096026,"parameters":{"f":[0,2,0,1],"r":2,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[\"odd_symmetric_zero\"]"}},"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-20261009-r06-definite-integral-0027","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-3, 0, 1]，r=3≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 0/1)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 0/1)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096027,"parameters":{"f":[-3,0,1],"r":3,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[-3,0,1],3]"}},"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-20261009-r06-definite-integral-0028","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 0, 2]，r=0≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 0/1)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 0/1)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096028,"parameters":{"f":[1,0,2],"r":0,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[\"nonodd_zero_radius\"]"}},"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-20261009-r06-definite-integral-0029","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-9, 4, 4, -1]，r=4≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 296/3)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 296/3)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096029,"parameters":{"f":[-9,4,4,-1],"r":4,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[-9,0,4],4]"}},"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-20261009-r06-definite-integral-0030","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-8, 1, 5, -1]，r=5≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 1010/3)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 1010/3)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096030,"parameters":{"f":[-8,1,5,-1],"r":5,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[-8,0,5],5]"}},"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-20261009-r06-definite-integral-0031","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-7, 2, 6, -1]，r=1≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, -10/1)","explanation_zh":"奇函數資格與對稱多項式矩=(否, -10/1)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096031,"parameters":{"f":[-7,2,6,-1],"r":1,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[-7,0,6],1]"}},"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-20261009-r06-definite-integral-0032","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-6, 3, 7, -1]，r=2≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 40/3)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 40/3)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096032,"parameters":{"f":[-6,3,7,-1],"r":2,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[-6,0,7],2]"}},"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-20261009-r06-definite-integral-0033","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-5, 4, 8, -1]，r=3≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 114/1)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 114/1)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096033,"parameters":{"f":[-5,4,8,-1],"r":3,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[-5,0,8],3]"}},"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-20261009-r06-definite-integral-0034","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-4, 1, 9, -1]，r=4≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 352/1)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 352/1)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096034,"parameters":{"f":[-4,1,9,-1],"r":4,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[-4,0,9],4]"}},"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-20261009-r06-definite-integral-0035","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-3, 2, 10, -1]，r=5≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 2410/3)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 2410/3)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096035,"parameters":{"f":[-3,2,10,-1],"r":5,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[-3,0,10],5]"}},"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-20261009-r06-definite-integral-0036","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-2, 3, 11, -1]，r=1≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 10/3)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 10/3)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096036,"parameters":{"f":[-2,3,11,-1],"r":1,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[-2,0,11],1]"}},"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-20261009-r06-definite-integral-0037","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-1, 4, 12, -1]，r=2≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 60/1)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 60/1)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096037,"parameters":{"f":[-1,4,12,-1],"r":2,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[-1,0,12],2]"}},"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-20261009-r06-definite-integral-0038","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[0, 1, 13, -1]，r=3≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 234/1)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 234/1)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096038,"parameters":{"f":[0,1,13,-1],"r":3,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[0,0,13],3]"}},"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-20261009-r06-definite-integral-0039","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 2, 14, -1]，r=4≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 1816/3)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 1816/3)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096039,"parameters":{"f":[1,2,14,-1],"r":4,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[1,0,14],4]"}},"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-20261009-r06-definite-integral-0040","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[2, 3, 15, -1]，r=5≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 1270/1)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 1270/1)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096040,"parameters":{"f":[2,3,15,-1],"r":5,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[2,0,15],5]"}},"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-20261009-r06-definite-integral-0041","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[3, 4, 16, -1]，r=1≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 50/3)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 50/3)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096041,"parameters":{"f":[3,4,16,-1],"r":1,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[3,0,16],1]"}},"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-20261009-r06-definite-integral-0042","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[4, 1, 17, -1]，r=2≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 320/3)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 320/3)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096042,"parameters":{"f":[4,1,17,-1],"r":2,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[4,0,17],2]"}},"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-20261009-r06-definite-integral-0043","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[5, 2, 18, -1]，r=3≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 354/1)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 354/1)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096043,"parameters":{"f":[5,2,18,-1],"r":3,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[5,0,18],3]"}},"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-20261009-r06-definite-integral-0044","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[6, 3, 19, -1]，r=4≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 2576/3)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 2576/3)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096044,"parameters":{"f":[6,3,19,-1],"r":4,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[6,0,19],4]"}},"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-20261009-r06-definite-integral-0045","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[7, 4, 20, -1]，r=5≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 5210/3)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 5210/3)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096045,"parameters":{"f":[7,4,20,-1],"r":5,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[7,0,20],5]"}},"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-20261009-r06-definite-integral-0046","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[8, 1, 21, -1]，r=1≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 30/1)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 30/1)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096046,"parameters":{"f":[8,1,21,-1],"r":1,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[8,0,21],1]"}},"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-20261009-r06-definite-integral-0047","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[9, 2, 22, -1]，r=2≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 460/3)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 460/3)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096047,"parameters":{"f":[9,2,22,-1],"r":2,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[9,0,22],2]"}},"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-20261009-r06-definite-integral-0048","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[10, 3, 23, -1]，r=3≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 474/1)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 474/1)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096048,"parameters":{"f":[10,3,23,-1],"r":3,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[10,0,23],3]"}},"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-20261009-r06-definite-integral-0049","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[11, 4, 24, -1]，r=4≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 1112/1)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 1112/1)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096049,"parameters":{"f":[11,4,24,-1],"r":4,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[11,0,24],4]"}},"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-20261009-r06-definite-integral-0050","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"奇函數資格與對稱積分成對","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[12, 1, 25, -1]，r=5≥0。列 (f是否為奇函數, I(−r,r))，第二項採最簡分數。","answer":{"kind":"expression","value":"(否, 6610/3)","explanation_zh":"奇函數資格與對稱多項式矩=(否, 6610/3)。多項式的全部偶次係數為零才是奇函數；對稱積分0不推出奇函數，亦可能來自零半徑或偶次項抵消。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["奇函數資格與對稱積分成對"]},"variant":{"family_id":"qf-density-integral-parity-symmetric-pair","seed":202610096050,"parameters":{"f":[12,1,25,-1],"r":5,"operation":"density_integral_parity_symmetric_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-integral-parity-symmetric-pair/[[12,0,25],5]"}},"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-20261009-r06-definite-integral-0051","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=0x+(2)，其升冪係數=[2, 0]；a=0、b=2，N=3等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"0/1","explanation_zh":"多項式矩減左端有限和=0/1。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096051,"parameters":{"m":0,"c":2,"a":0,"b":2,"N":3,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[\"zero_error\"]"}},"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-20261009-r06-definite-integral-0052","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=-2x+(-11)，其升冪係數=[-11, -2]；a=-1、b=1，N=1等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"-4/1","explanation_zh":"多項式矩減左端有限和=-4/1。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096052,"parameters":{"m":-2,"c":-11,"a":-1,"b":1,"N":1,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[-2,\"2\",1]"}},"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-20261009-r06-definite-integral-0053","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=3x+(-10)，其升冪係數=[-10, 3]；a=0、b=3，N=3等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"9/2","explanation_zh":"多項式矩減左端有限和=9/2。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096053,"parameters":{"m":3,"c":-10,"a":0,"b":3,"N":3,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[3,\"3\",3]"}},"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-20261009-r06-definite-integral-0054","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=-4x+(-9)，其升冪係數=[-9, -4]；a=1、b=5，N=4等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"-8/1","explanation_zh":"多項式矩減左端有限和=-8/1。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096054,"parameters":{"m":-4,"c":-9,"a":1,"b":5,"N":4,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[-4,\"4\",4]"}},"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-20261009-r06-definite-integral-0055","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=5x+(-8)，其升冪係數=[-8, 5]；a=-2、b=3，N=5等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"25/2","explanation_zh":"多項式矩減左端有限和=25/2。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096055,"parameters":{"m":5,"c":-8,"a":-2,"b":3,"N":5,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[5,\"5\",5]"}},"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-20261009-r06-definite-integral-0056","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=-6x+(-7)，其升冪係數=[-7, -6]；a=-1、b=5，N=6等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"-18/1","explanation_zh":"多項式矩減左端有限和=-18/1。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096056,"parameters":{"m":-6,"c":-7,"a":-1,"b":5,"N":6,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[-6,\"6\",6]"}},"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-20261009-r06-definite-integral-0057","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=7x+(-6)，其升冪係數=[-6, 7]；a=0、b=1，N=7等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"1/2","explanation_zh":"多項式矩減左端有限和=1/2。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096057,"parameters":{"m":7,"c":-6,"a":0,"b":1,"N":7,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[7,\"1\",7]"}},"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-20261009-r06-definite-integral-0058","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=-8x+(-5)，其升冪係數=[-5, -8]；a=1、b=3，N=8等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"-2/1","explanation_zh":"多項式矩減左端有限和=-2/1。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096058,"parameters":{"m":-8,"c":-5,"a":1,"b":3,"N":8,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[-8,\"2\",8]"}},"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-20261009-r06-definite-integral-0059","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=9x+(-4)，其升冪係數=[-4, 9]；a=-2、b=1，N=9等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"9/2","explanation_zh":"多項式矩減左端有限和=9/2。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096059,"parameters":{"m":9,"c":-4,"a":-2,"b":1,"N":9,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[9,\"3\",9]"}},"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-20261009-r06-definite-integral-0060","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=-10x+(-3)，其升冪係數=[-3, -10]；a=-1、b=3，N=10等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"-8/1","explanation_zh":"多項式矩減左端有限和=-8/1。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096060,"parameters":{"m":-10,"c":-3,"a":-1,"b":3,"N":10,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[-10,\"4\",10]"}},"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-20261009-r06-definite-integral-0061","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=11x+(-2)，其升冪係數=[-2, 11]；a=0、b=5，N=11等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"25/2","explanation_zh":"多項式矩減左端有限和=25/2。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096061,"parameters":{"m":11,"c":-2,"a":0,"b":5,"N":11,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[11,\"5\",11]"}},"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-20261009-r06-definite-integral-0062","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=-12x+(-1)，其升冪係數=[-1, -12]；a=1、b=7，N=12等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"-18/1","explanation_zh":"多項式矩減左端有限和=-18/1。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096062,"parameters":{"m":-12,"c":-1,"a":1,"b":7,"N":12,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[-12,\"6\",12]"}},"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-20261009-r06-definite-integral-0063","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=13x+(0)，其升冪係數=[0, 13]；a=-2、b=-1，N=13等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"1/2","explanation_zh":"多項式矩減左端有限和=1/2。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096063,"parameters":{"m":13,"c":0,"a":-2,"b":-1,"N":13,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[13,\"1\",13]"}},"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-20261009-r06-definite-integral-0064","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=-14x+(1)，其升冪係數=[1, -14]；a=-1、b=1，N=14等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"-2/1","explanation_zh":"多項式矩減左端有限和=-2/1。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096064,"parameters":{"m":-14,"c":1,"a":-1,"b":1,"N":14,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[-14,\"2\",14]"}},"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-20261009-r06-definite-integral-0065","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=15x+(2)，其升冪係數=[2, 15]；a=0、b=3，N=15等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"9/2","explanation_zh":"多項式矩減左端有限和=9/2。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096065,"parameters":{"m":15,"c":2,"a":0,"b":3,"N":15,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[15,\"3\",15]"}},"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-20261009-r06-definite-integral-0066","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=-16x+(3)，其升冪係數=[3, -16]；a=1、b=5，N=16等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"-8/1","explanation_zh":"多項式矩減左端有限和=-8/1。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096066,"parameters":{"m":-16,"c":3,"a":1,"b":5,"N":16,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[-16,\"4\",16]"}},"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-20261009-r06-definite-integral-0067","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=17x+(4)，其升冪係數=[4, 17]；a=-2、b=3，N=17等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"25/2","explanation_zh":"多項式矩減左端有限和=25/2。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096067,"parameters":{"m":17,"c":4,"a":-2,"b":3,"N":17,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[17,\"5\",17]"}},"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-20261009-r06-definite-integral-0068","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=-18x+(5)，其升冪係數=[5, -18]；a=-1、b=5，N=18等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"-18/1","explanation_zh":"多項式矩減左端有限和=-18/1。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096068,"parameters":{"m":-18,"c":5,"a":-1,"b":5,"N":18,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[-18,\"6\",18]"}},"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-20261009-r06-definite-integral-0069","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=19x+(6)，其升冪係數=[6, 19]；a=0、b=1，N=19等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"1/2","explanation_zh":"多項式矩減左端有限和=1/2。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096069,"parameters":{"m":19,"c":6,"a":0,"b":1,"N":19,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[19,\"1\",19]"}},"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-20261009-r06-definite-integral-0070","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=-20x+(7)，其升冪係數=[7, -20]；a=1、b=3，N=20等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"-2/1","explanation_zh":"多項式矩減左端有限和=-2/1。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096070,"parameters":{"m":-20,"c":7,"a":1,"b":3,"N":20,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[-20,\"2\",20]"}},"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-20261009-r06-definite-integral-0071","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=21x+(8)，其升冪係數=[8, 21]；a=-2、b=1，N=21等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"9/2","explanation_zh":"多項式矩減左端有限和=9/2。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096071,"parameters":{"m":21,"c":8,"a":-2,"b":1,"N":21,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[21,\"3\",21]"}},"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-20261009-r06-definite-integral-0072","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=-22x+(9)，其升冪係數=[9, -22]；a=-1、b=3，N=22等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"-8/1","explanation_zh":"多項式矩減左端有限和=-8/1。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096072,"parameters":{"m":-22,"c":9,"a":-1,"b":3,"N":22,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[-22,\"4\",22]"}},"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-20261009-r06-definite-integral-0073","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=23x+(10)，其升冪係數=[10, 23]；a=0、b=5，N=23等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"25/2","explanation_zh":"多項式矩減左端有限和=25/2。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096073,"parameters":{"m":23,"c":10,"a":0,"b":5,"N":23,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[23,\"5\",23]"}},"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-20261009-r06-definite-integral-0074","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=-24x+(11)，其升冪係數=[11, -24]；a=1、b=7，N=24等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"-18/1","explanation_zh":"多項式矩減左端有限和=-18/1。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096074,"parameters":{"m":-24,"c":11,"a":1,"b":7,"N":24,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[-24,\"6\",24]"}},"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-20261009-r06-definite-integral-0075","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"精確矩積分減左端有限和","stem_zh":"全實數論域；升冪係數的受限多項式模型。f(x)=25x+(12)，其升冪係數=[12, 25]；a=-2、b=-1，N=50等份左端和S_N。求 I(a,b)−S_N。","answer":{"kind":"expression","value":"1/4","explanation_zh":"多項式矩減左端有限和=1/4。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["精確矩積分減左端有限和"]},"variant":{"family_id":"qf-density-integral-left-error","seed":202610096075,"parameters":{"m":25,"c":12,"a":-2,"b":-1,"N":50,"operation":"density_integral_left_error","domain":"reals","semantic_template":"qf-density-integral-left-error/[25,\"1\",50]"}},"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-20261009-r06-definite-integral-0076","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=1，升冪係數=[1]。起點a=3，I(a,b)=0/1，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"3/1","explanation_zh":"有向終點=3/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096076,"parameters":{"c":1,"a":3,"target_n":0,"target_d":1,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[\"zero_target\",3]"}},"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-20261009-r06-definite-integral-0077","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=-2，升冪係數=[-2]。起點a=-11，I(a,b)=-8/2，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"-9/1","explanation_zh":"有向終點=-9/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096077,"parameters":{"c":-2,"a":-11,"target_n":-8,"target_d":2,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[-2,-11,\"-4/1\"]"}},"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-20261009-r06-definite-integral-0078","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=3，升冪係數=[3]。起點a=-10，I(a,b)=27/3，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"-7/1","explanation_zh":"有向終點=-7/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096078,"parameters":{"c":3,"a":-10,"target_n":27,"target_d":3,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[3,-10,\"9/1\"]"}},"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-20261009-r06-definite-integral-0079","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=-4，升冪係數=[-4]。起點a=-9，I(a,b)=16/1，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"-13/1","explanation_zh":"有向終點=-13/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096079,"parameters":{"c":-4,"a":-9,"target_n":16,"target_d":1,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[-4,-9,\"16/1\"]"}},"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-20261009-r06-definite-integral-0080","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=5，升冪係數=[5]。起點a=-8，I(a,b)=50/2，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"-3/1","explanation_zh":"有向終點=-3/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096080,"parameters":{"c":5,"a":-8,"target_n":50,"target_d":2,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[5,-8,\"25/1\"]"}},"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-20261009-r06-definite-integral-0081","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=-6，升冪係數=[-6]。起點a=-7，I(a,b)=-108/3，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"-1/1","explanation_zh":"有向終點=-1/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096081,"parameters":{"c":-6,"a":-7,"target_n":-108,"target_d":3,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[-6,-7,\"-36/1\"]"}},"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-20261009-r06-definite-integral-0082","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=7，升冪係數=[7]。起點a=-6，I(a,b)=-49/1，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"-13/1","explanation_zh":"有向終點=-13/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096082,"parameters":{"c":7,"a":-6,"target_n":-49,"target_d":1,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[7,-6,\"-49/1\"]"}},"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-20261009-r06-definite-integral-0083","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=-8，升冪係數=[-8]。起點a=-5，I(a,b)=-128/2，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"3/1","explanation_zh":"有向終點=3/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096083,"parameters":{"c":-8,"a":-5,"target_n":-128,"target_d":2,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[-8,-5,\"-64/1\"]"}},"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-20261009-r06-definite-integral-0084","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=9，升冪係數=[9]。起點a=-4，I(a,b)=243/3，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"5/1","explanation_zh":"有向終點=5/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096084,"parameters":{"c":9,"a":-4,"target_n":243,"target_d":3,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[9,-4,\"81/1\"]"}},"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-20261009-r06-definite-integral-0085","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=-10，升冪係數=[-10]。起點a=-3，I(a,b)=100/1，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"-13/1","explanation_zh":"有向終點=-13/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096085,"parameters":{"c":-10,"a":-3,"target_n":100,"target_d":1,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[-10,-3,\"100/1\"]"}},"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-20261009-r06-definite-integral-0086","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=11，升冪係數=[11]。起點a=-2，I(a,b)=242/2，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"9/1","explanation_zh":"有向終點=9/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096086,"parameters":{"c":11,"a":-2,"target_n":242,"target_d":2,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[11,-2,\"121/1\"]"}},"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-20261009-r06-definite-integral-0087","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=-12，升冪係數=[-12]。起點a=-1，I(a,b)=-432/3，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"11/1","explanation_zh":"有向終點=11/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096087,"parameters":{"c":-12,"a":-1,"target_n":-432,"target_d":3,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[-12,-1,\"-144/1\"]"}},"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-20261009-r06-definite-integral-0088","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=13，升冪係數=[13]。起點a=0，I(a,b)=-169/1，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"-13/1","explanation_zh":"有向終點=-13/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096088,"parameters":{"c":13,"a":0,"target_n":-169,"target_d":1,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[13,0,\"-169/1\"]"}},"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-20261009-r06-definite-integral-0089","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=-14，升冪係數=[-14]。起點a=1，I(a,b)=-392/2，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"15/1","explanation_zh":"有向終點=15/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096089,"parameters":{"c":-14,"a":1,"target_n":-392,"target_d":2,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[-14,1,\"-196/1\"]"}},"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-20261009-r06-definite-integral-0090","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=15，升冪係數=[15]。起點a=2，I(a,b)=675/3，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"17/1","explanation_zh":"有向終點=17/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096090,"parameters":{"c":15,"a":2,"target_n":675,"target_d":3,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[15,2,\"225/1\"]"}},"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-20261009-r06-definite-integral-0091","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=-16，升冪係數=[-16]。起點a=3，I(a,b)=256/1，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"-13/1","explanation_zh":"有向終點=-13/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096091,"parameters":{"c":-16,"a":3,"target_n":256,"target_d":1,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[-16,3,\"256/1\"]"}},"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-20261009-r06-definite-integral-0092","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=17，升冪係數=[17]。起點a=4，I(a,b)=578/2，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"21/1","explanation_zh":"有向終點=21/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096092,"parameters":{"c":17,"a":4,"target_n":578,"target_d":2,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[17,4,\"289/1\"]"}},"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-20261009-r06-definite-integral-0093","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=-18，升冪係數=[-18]。起點a=5，I(a,b)=-972/3，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"23/1","explanation_zh":"有向終點=23/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096093,"parameters":{"c":-18,"a":5,"target_n":-972,"target_d":3,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[-18,5,\"-324/1\"]"}},"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-20261009-r06-definite-integral-0094","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=19，升冪係數=[19]。起點a=6，I(a,b)=-361/1，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"-13/1","explanation_zh":"有向終點=-13/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096094,"parameters":{"c":19,"a":6,"target_n":-361,"target_d":1,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[19,6,\"-361/1\"]"}},"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-20261009-r06-definite-integral-0095","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=-20，升冪係數=[-20]。起點a=7，I(a,b)=-800/2，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"27/1","explanation_zh":"有向終點=27/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096095,"parameters":{"c":-20,"a":7,"target_n":-800,"target_d":2,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[-20,7,\"-400/1\"]"}},"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-20261009-r06-definite-integral-0096","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=21，升冪係數=[21]。起點a=8，I(a,b)=1323/3，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"29/1","explanation_zh":"有向終點=29/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096096,"parameters":{"c":21,"a":8,"target_n":1323,"target_d":3,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[21,8,\"441/1\"]"}},"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-20261009-r06-definite-integral-0097","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=-22，升冪係數=[-22]。起點a=9，I(a,b)=484/1，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"-13/1","explanation_zh":"有向終點=-13/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096097,"parameters":{"c":-22,"a":9,"target_n":484,"target_d":1,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[-22,9,\"484/1\"]"}},"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-20261009-r06-definite-integral-0098","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=23，升冪係數=[23]。起點a=10，I(a,b)=1058/2，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"33/1","explanation_zh":"有向終點=33/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096098,"parameters":{"c":23,"a":10,"target_n":1058,"target_d":2,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[23,10,\"529/1\"]"}},"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-20261009-r06-definite-integral-0099","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=-24，升冪係數=[-24]。起點a=11，I(a,b)=-1728/3，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"35/1","explanation_zh":"有向終點=35/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096099,"parameters":{"c":-24,"a":11,"target_n":-1728,"target_d":3,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[-24,11,\"-576/1\"]"}},"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-20261009-r06-definite-integral-0100","batch_id":"batch-20261007-advance-20261009-r06-definite-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-definite-integral"],"title_zh":"非零常數積分的整數終點反解","stem_zh":"全實數論域；升冪係數的受限多項式模型。非零常數f(x)=25，升冪係數=[25]。起點a=12，I(a,b)=-625/1，本輪保證b為整數。求有向終點b，仍保留 /1。","answer":{"kind":"expression","value":"-13/1","explanation_zh":"有向終點=-13/1。用a加目標積分除以非零c求終點，不先排序或取絕對值；目標零時b=a，保持既有整數端點切片。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非零常數積分的整數終點反解"]},"variant":{"family_id":"qf-density-integral-endpoint-recovery","seed":202610096100,"parameters":{"c":25,"a":12,"target_n":-625,"target_d":1,"operation":"density_integral_endpoint_recovery","domain":"reals","semantic_template":"qf-density-integral-endpoint-recovery/[25,12,\"-625/1\"]"}},"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"}}