{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0001","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[1,2,3],\"center\":-12,\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, 2, [1, 2, 3], 3, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(-12, 2, [1, 2, 3], 3, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122001,"parameters":{"derivatives":[1,2,3],"center":-12,"order":2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-12,\"derivatives\":[1,2,3],\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0002","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[2,4,6,8],\"center\":-11,\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, 3, [2, 4, 6, 8], 4, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(-11, 3, [2, 4, 6, 8], 4, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122002,"parameters":{"derivatives":[2,4,6,8],"center":-11,"order":3,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-11,\"derivatives\":[2,4,6,8],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0003","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[3,6,9,12,15],\"center\":-10,\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, 4, [3, 6, 9, 12, 15], 5, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(-10, 4, [3, 6, 9, 12, 15], 5, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122003,"parameters":{"derivatives":[3,6,9,12,15],"center":-10,"order":4,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-10,\"derivatives\":[3,6,9,12,15],\"order\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0004","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[4,8,12,16,20,24],\"center\":-9,\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, 5, [4, 8, 12, 16, 20, 24], 6, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(-9, 5, [4, 8, 12, 16, 20, 24], 6, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122004,"parameters":{"derivatives":[4,8,12,16,20,24],"center":-9,"order":5,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-9,\"derivatives\":[4,8,12,16,20,24],\"order\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0005","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[5,10,15],\"center\":-8,\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, 2, [5, 10, 15], 3, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(-8, 2, [5, 10, 15], 3, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122005,"parameters":{"derivatives":[5,10,15],"center":-8,"order":2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-8,\"derivatives\":[5,10,15],\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0006","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[6,12,18,24],\"center\":-7,\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, 3, [6, 12, 18, 24], 4, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(-7, 3, [6, 12, 18, 24], 4, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122006,"parameters":{"derivatives":[6,12,18,24],"center":-7,"order":3,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-7,\"derivatives\":[6,12,18,24],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0007","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[7,14,21,28,35],\"center\":-6,\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, 4, [7, 14, 21, 28, 35], 5, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(-6, 4, [7, 14, 21, 28, 35], 5, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122007,"parameters":{"derivatives":[7,14,21,28,35],"center":-6,"order":4,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-6,\"derivatives\":[7,14,21,28,35],\"order\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0008","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[8,16,24,32,40,48],\"center\":-5,\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, 5, [8, 16, 24, 32, 40, 48], 6, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(-5, 5, [8, 16, 24, 32, 40, 48], 6, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122008,"parameters":{"derivatives":[8,16,24,32,40,48],"center":-5,"order":5,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-5,\"derivatives\":[8,16,24,32,40,48],\"order\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0009","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[9,18,27],\"center\":-4,\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, 2, [9, 18, 27], 3, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(-4, 2, [9, 18, 27], 3, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122009,"parameters":{"derivatives":[9,18,27],"center":-4,"order":2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-4,\"derivatives\":[9,18,27],\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0010","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[10,20,30,40],\"center\":-3,\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 3, [10, 20, 30, 40], 4, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(-3, 3, [10, 20, 30, 40], 4, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122010,"parameters":{"derivatives":[10,20,30,40],"center":-3,"order":3,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-3,\"derivatives\":[10,20,30,40],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0011","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[11,22,33,44,55],\"center\":-2,\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 4, [11, 22, 33, 44, 55], 5, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(-2, 4, [11, 22, 33, 44, 55], 5, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122011,"parameters":{"derivatives":[11,22,33,44,55],"center":-2,"order":4,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-2,\"derivatives\":[11,22,33,44,55],\"order\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0012","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[12,24,36,48,60,72],\"center\":-1,\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, 5, [12, 24, 36, 48, 60, 72], 6, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(-1, 5, [12, 24, 36, 48, 60, 72], 6, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122012,"parameters":{"derivatives":[12,24,36,48,60,72],"center":-1,"order":5,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":-1,\"derivatives\":[12,24,36,48,60,72],\"order\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0013","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[13,26,39],\"center\":0,\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, [13, 26, 39], 3, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(0, 2, [13, 26, 39], 3, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122013,"parameters":{"derivatives":[13,26,39],"center":0,"order":2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":0,\"derivatives\":[13,26,39],\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0014","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[14,28,42,56],\"center\":1,\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 3, [14, 28, 42, 56], 4, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(1, 3, [14, 28, 42, 56], 4, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122014,"parameters":{"derivatives":[14,28,42,56],"center":1,"order":3,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":1,\"derivatives\":[14,28,42,56],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0015","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[15,30,45,60,75],\"center\":2,\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 4, [15, 30, 45, 60, 75], 5, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(2, 4, [15, 30, 45, 60, 75], 5, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122015,"parameters":{"derivatives":[15,30,45,60,75],"center":2,"order":4,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":2,\"derivatives\":[15,30,45,60,75],\"order\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0016","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[16,32,48,64,80,96],\"center\":3,\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, [16, 32, 48, 64, 80, 96], 6, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(3, 5, [16, 32, 48, 64, 80, 96], 6, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122016,"parameters":{"derivatives":[16,32,48,64,80,96],"center":3,"order":5,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":3,\"derivatives\":[16,32,48,64,80,96],\"order\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0017","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[17,34,51],\"center\":4,\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 2, [17, 34, 51], 3, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(4, 2, [17, 34, 51], 3, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122017,"parameters":{"derivatives":[17,34,51],"center":4,"order":2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":4,\"derivatives\":[17,34,51],\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0018","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[18,36,54,72],\"center\":5,\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 3, [18, 36, 54, 72], 4, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(5, 3, [18, 36, 54, 72], 4, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122018,"parameters":{"derivatives":[18,36,54,72],"center":5,"order":3,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":5,\"derivatives\":[18,36,54,72],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0019","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[19,38,57,76,95],\"center\":6,\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 4, [19, 38, 57, 76, 95], 5, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(6, 4, [19, 38, 57, 76, 95], 5, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122019,"parameters":{"derivatives":[19,38,57,76,95],"center":6,"order":4,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":6,\"derivatives\":[19,38,57,76,95],\"order\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0020","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[20,40,60,80,100,120],\"center\":7,\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 5, [20, 40, 60, 80, 100, 120], 6, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(7, 5, [20, 40, 60, 80, 100, 120], 6, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122020,"parameters":{"derivatives":[20,40,60,80,100,120],"center":7,"order":5,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":7,\"derivatives\":[20,40,60,80,100,120],\"order\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0021","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[21,42,63],\"center\":8,\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 2, [21, 42, 63], 3, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(8, 2, [21, 42, 63], 3, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122021,"parameters":{"derivatives":[21,42,63],"center":8,"order":2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":8,\"derivatives\":[21,42,63],\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0022","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[22,44,66,88],\"center\":9,\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 3, [22, 44, 66, 88], 4, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(9, 3, [22, 44, 66, 88], 4, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122022,"parameters":{"derivatives":[22,44,66,88],"center":9,"order":3,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":9,\"derivatives\":[22,44,66,88],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0023","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[23,46,69,92,115],\"center\":10,\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 4, [23, 46, 69, 92, 115], 5, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(10, 4, [23, 46, 69, 92, 115], 5, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122023,"parameters":{"derivatives":[23,46,69,92,115],"center":10,"order":4,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":10,\"derivatives\":[23,46,69,92,115],\"order\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0024","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[24,48,72,96,120,144],\"center\":11,\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 5, [24, 48, 72, 96, 120, 144], 6, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(11, 5, [24, 48, 72, 96, 120, 144], 6, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122024,"parameters":{"derivatives":[24,48,72,96,120,144],"center":11,"order":5,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":11,\"derivatives\":[24,48,72,96,120,144],\"order\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0025","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-jet-qualification；資料={\"derivatives\":[25,50,75],\"center\":12,\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 2, [25, 50, 75], 3, 是, complete_zero_through_n_derivative_jet)","explanation_zh":"Wave6第六層核對=(12, 2, [25, 50, 75], 3, 是, complete_zero_through_n_derivative_jet)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-jet-qualification","seed":202611122025,"parameters":{"derivatives":[25,50,75],"center":12,"order":2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-jet-qualification","semantic_template":"qf-mod43-taylor-jet-qualification/{\"center\":12,\"derivatives\":[25,50,75],\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0026","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[1,2,3],\"center\":-12,\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, 2, [1, 2, 3/2], [[0, 1], [1, 2], [2, 3/2]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(-12, 2, [1, 2, 3/2], [[0, 1], [1, 2], [2, 3/2]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122026,"parameters":{"derivatives":[1,2,3],"center":-12,"order":2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-12,\"derivatives\":[1,2,3],\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0027","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[2,4,6,8],\"center\":-11,\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, 3, [2, 4, 3, 4/3], [[0, 2], [1, 4], [2, 3], [3, 4/3]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(-11, 3, [2, 4, 3, 4/3], [[0, 2], [1, 4], [2, 3], [3, 4/3]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122027,"parameters":{"derivatives":[2,4,6,8],"center":-11,"order":3,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-11,\"derivatives\":[2,4,6,8],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0028","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[3,6,9,12,15],\"center\":-10,\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, 4, [3, 6, 9/2, 2, 5/8], [[0, 3], [1, 6], [2, 9/2], [3, 2], [4, 5/8]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(-10, 4, [3, 6, 9/2, 2, 5/8], [[0, 3], [1, 6], [2, 9/2], [3, 2], [4, 5/8]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122028,"parameters":{"derivatives":[3,6,9,12,15],"center":-10,"order":4,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-10,\"derivatives\":[3,6,9,12,15],\"order\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0029","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[4,8,12,16,20,24],\"center\":-9,\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, 5, [4, 8, 6, 8/3, 5/6, 1/5], [[0, 4], [1, 8], [2, 6], [3, 8/3], [4, 5/6], [5, 1/5]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(-9, 5, [4, 8, 6, 8/3, 5/6, 1/5], [[0, 4], [1, 8], [2, 6], [3, 8/3], [4, 5/6], [5, 1/5]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122029,"parameters":{"derivatives":[4,8,12,16,20,24],"center":-9,"order":5,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-9,\"derivatives\":[4,8,12,16,20,24],\"order\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0030","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[5,10,15],\"center\":-8,\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, 2, [5, 10, 15/2], [[0, 5], [1, 10], [2, 15/2]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(-8, 2, [5, 10, 15/2], [[0, 5], [1, 10], [2, 15/2]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122030,"parameters":{"derivatives":[5,10,15],"center":-8,"order":2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-8,\"derivatives\":[5,10,15],\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0031","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[6,12,18,24],\"center\":-7,\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, 3, [6, 12, 9, 4], [[0, 6], [1, 12], [2, 9], [3, 4]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(-7, 3, [6, 12, 9, 4], [[0, 6], [1, 12], [2, 9], [3, 4]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122031,"parameters":{"derivatives":[6,12,18,24],"center":-7,"order":3,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-7,\"derivatives\":[6,12,18,24],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0032","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[7,14,21,28,35],\"center\":-6,\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, 4, [7, 14, 21/2, 14/3, 35/24], [[0, 7], [1, 14], [2, 21/2], [3, 14/3], [4, 35/24]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(-6, 4, [7, 14, 21/2, 14/3, 35/24], [[0, 7], [1, 14], [2, 21/2], [3, 14/3], [4, 35/24]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122032,"parameters":{"derivatives":[7,14,21,28,35],"center":-6,"order":4,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-6,\"derivatives\":[7,14,21,28,35],\"order\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0033","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[8,16,24,32,40,48],\"center\":-5,\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, 5, [8, 16, 12, 16/3, 5/3, 2/5], [[0, 8], [1, 16], [2, 12], [3, 16/3], [4, 5/3], [5, 2/5]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(-5, 5, [8, 16, 12, 16/3, 5/3, 2/5], [[0, 8], [1, 16], [2, 12], [3, 16/3], [4, 5/3], [5, 2/5]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122033,"parameters":{"derivatives":[8,16,24,32,40,48],"center":-5,"order":5,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-5,\"derivatives\":[8,16,24,32,40,48],\"order\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0034","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[9,18,27],\"center\":-4,\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, 2, [9, 18, 27/2], [[0, 9], [1, 18], [2, 27/2]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(-4, 2, [9, 18, 27/2], [[0, 9], [1, 18], [2, 27/2]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122034,"parameters":{"derivatives":[9,18,27],"center":-4,"order":2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-4,\"derivatives\":[9,18,27],\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0035","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[10,20,30,40],\"center\":-3,\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 3, [10, 20, 15, 20/3], [[0, 10], [1, 20], [2, 15], [3, 20/3]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(-3, 3, [10, 20, 15, 20/3], [[0, 10], [1, 20], [2, 15], [3, 20/3]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122035,"parameters":{"derivatives":[10,20,30,40],"center":-3,"order":3,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-3,\"derivatives\":[10,20,30,40],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0036","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[11,22,33,44,55],\"center\":-2,\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 4, [11, 22, 33/2, 22/3, 55/24], [[0, 11], [1, 22], [2, 33/2], [3, 22/3], [4, 55/24]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(-2, 4, [11, 22, 33/2, 22/3, 55/24], [[0, 11], [1, 22], [2, 33/2], [3, 22/3], [4, 55/24]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122036,"parameters":{"derivatives":[11,22,33,44,55],"center":-2,"order":4,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-2,\"derivatives\":[11,22,33,44,55],\"order\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0037","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[12,24,36,48,60,72],\"center\":-1,\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, 5, [12, 24, 18, 8, 5/2, 3/5], [[0, 12], [1, 24], [2, 18], [3, 8], [4, 5/2], [5, 3/5]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(-1, 5, [12, 24, 18, 8, 5/2, 3/5], [[0, 12], [1, 24], [2, 18], [3, 8], [4, 5/2], [5, 3/5]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122037,"parameters":{"derivatives":[12,24,36,48,60,72],"center":-1,"order":5,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":-1,\"derivatives\":[12,24,36,48,60,72],\"order\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0038","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[13,26,39],\"center\":0,\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, [13, 26, 39/2], [[0, 13], [1, 26], [2, 39/2]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(0, 2, [13, 26, 39/2], [[0, 13], [1, 26], [2, 39/2]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122038,"parameters":{"derivatives":[13,26,39],"center":0,"order":2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":0,\"derivatives\":[13,26,39],\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0039","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[14,28,42,56],\"center\":1,\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 3, [14, 28, 21, 28/3], [[0, 14], [1, 28], [2, 21], [3, 28/3]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(1, 3, [14, 28, 21, 28/3], [[0, 14], [1, 28], [2, 21], [3, 28/3]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122039,"parameters":{"derivatives":[14,28,42,56],"center":1,"order":3,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":1,\"derivatives\":[14,28,42,56],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0040","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[15,30,45,60,75],\"center\":2,\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 4, [15, 30, 45/2, 10, 25/8], [[0, 15], [1, 30], [2, 45/2], [3, 10], [4, 25/8]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(2, 4, [15, 30, 45/2, 10, 25/8], [[0, 15], [1, 30], [2, 45/2], [3, 10], [4, 25/8]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122040,"parameters":{"derivatives":[15,30,45,60,75],"center":2,"order":4,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":2,\"derivatives\":[15,30,45,60,75],\"order\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0041","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[16,32,48,64,80,96],\"center\":3,\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, [16, 32, 24, 32/3, 10/3, 4/5], [[0, 16], [1, 32], [2, 24], [3, 32/3], [4, 10/3], [5, 4/5]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(3, 5, [16, 32, 24, 32/3, 10/3, 4/5], [[0, 16], [1, 32], [2, 24], [3, 32/3], [4, 10/3], [5, 4/5]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122041,"parameters":{"derivatives":[16,32,48,64,80,96],"center":3,"order":5,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":3,\"derivatives\":[16,32,48,64,80,96],\"order\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0042","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[17,34,51],\"center\":4,\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 2, [17, 34, 51/2], [[0, 17], [1, 34], [2, 51/2]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(4, 2, [17, 34, 51/2], [[0, 17], [1, 34], [2, 51/2]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122042,"parameters":{"derivatives":[17,34,51],"center":4,"order":2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":4,\"derivatives\":[17,34,51],\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0043","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[18,36,54,72],\"center\":5,\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 3, [18, 36, 27, 12], [[0, 18], [1, 36], [2, 27], [3, 12]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(5, 3, [18, 36, 27, 12], [[0, 18], [1, 36], [2, 27], [3, 12]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122043,"parameters":{"derivatives":[18,36,54,72],"center":5,"order":3,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":5,\"derivatives\":[18,36,54,72],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0044","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[19,38,57,76,95],\"center\":6,\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 4, [19, 38, 57/2, 38/3, 95/24], [[0, 19], [1, 38], [2, 57/2], [3, 38/3], [4, 95/24]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(6, 4, [19, 38, 57/2, 38/3, 95/24], [[0, 19], [1, 38], [2, 57/2], [3, 38/3], [4, 95/24]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122044,"parameters":{"derivatives":[19,38,57,76,95],"center":6,"order":4,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":6,\"derivatives\":[19,38,57,76,95],\"order\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0045","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[20,40,60,80,100,120],\"center\":7,\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 5, [20, 40, 30, 40/3, 25/6, 1], [[0, 20], [1, 40], [2, 30], [3, 40/3], [4, 25/6], [5, 1]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(7, 5, [20, 40, 30, 40/3, 25/6, 1], [[0, 20], [1, 40], [2, 30], [3, 40/3], [4, 25/6], [5, 1]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122045,"parameters":{"derivatives":[20,40,60,80,100,120],"center":7,"order":5,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":7,\"derivatives\":[20,40,60,80,100,120],\"order\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0046","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[21,42,63],\"center\":8,\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 2, [21, 42, 63/2], [[0, 21], [1, 42], [2, 63/2]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(8, 2, [21, 42, 63/2], [[0, 21], [1, 42], [2, 63/2]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122046,"parameters":{"derivatives":[21,42,63],"center":8,"order":2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":8,\"derivatives\":[21,42,63],\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0047","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[22,44,66,88],\"center\":9,\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 3, [22, 44, 33, 44/3], [[0, 22], [1, 44], [2, 33], [3, 44/3]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(9, 3, [22, 44, 33, 44/3], [[0, 22], [1, 44], [2, 33], [3, 44/3]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122047,"parameters":{"derivatives":[22,44,66,88],"center":9,"order":3,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":9,\"derivatives\":[22,44,66,88],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0048","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[23,46,69,92,115],\"center\":10,\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 4, [23, 46, 69/2, 46/3, 115/24], [[0, 23], [1, 46], [2, 69/2], [3, 46/3], [4, 115/24]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(10, 4, [23, 46, 69/2, 46/3, 115/24], [[0, 23], [1, 46], [2, 69/2], [3, 46/3], [4, 115/24]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122048,"parameters":{"derivatives":[23,46,69,92,115],"center":10,"order":4,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":10,\"derivatives\":[23,46,69,92,115],\"order\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0049","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[24,48,72,96,120,144],\"center\":11,\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 5, [24, 48, 36, 16, 5, 6/5], [[0, 24], [1, 48], [2, 36], [3, 16], [4, 5], [5, 6/5]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(11, 5, [24, 48, 36, 16, 5, 6/5], [[0, 24], [1, 48], [2, 36], [3, 16], [4, 5], [5, 6/5]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122049,"parameters":{"derivatives":[24,48,72,96,120,144],"center":11,"order":5,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":11,\"derivatives\":[24,48,72,96,120,144],\"order\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0050","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[25,50,75],\"center\":12,\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 2, [25, 50, 75/2], [[0, 25], [1, 50], [2, 75/2]], coefficient_k_equals_derivative_k_over_k_factorial)","explanation_zh":"Wave6第六層核對=(12, 2, [25, 50, 75/2], [[0, 25], [1, 50], [2, 75/2]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-coefficient-construction","seed":202611122050,"parameters":{"derivatives":[25,50,75],"center":12,"order":2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-coefficient-construction","semantic_template":"qf-mod43-taylor-coefficient-construction/{\"center\":12,\"derivatives\":[25,50,75],\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0051","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[1,2,3],\"center\":-12,\"point\":-14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3], [1, 2, 6], [1, 2, 3], -14, 9, 9, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([1, 2, 3], [1, 2, 6], [1, 2, 3], -14, 9, 9, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122051,"parameters":{"shifted_coefficients":[1,2,3],"center":-12,"point":-14,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-12,\"point\":-14,\"shifted_coefficients\":[1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0052","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[2,4,6,8],\"center\":-11,\"point\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 4, 6, 8], [2, 4, 12, 48], [2, 4, 6, 8], -12, -4, -4, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([2, 4, 6, 8], [2, 4, 12, 48], [2, 4, 6, 8], -12, -4, -4, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122052,"parameters":{"shifted_coefficients":[2,4,6,8],"center":-11,"point":-12,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-11,\"point\":-12,\"shifted_coefficients\":[2,4,6,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0053","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[3,6,9,12,15],\"center\":-10,\"point\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 6, 9, 12, 15], [3, 6, 18, 72, 360], [3, 6, 9, 12, 15], -10, 3, 3, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([3, 6, 9, 12, 15], [3, 6, 18, 72, 360], [3, 6, 9, 12, 15], -10, 3, 3, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122053,"parameters":{"shifted_coefficients":[3,6,9,12,15],"center":-10,"point":-10,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-10,\"point\":-10,\"shifted_coefficients\":[3,6,9,12,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0054","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[4,8,12,16,20,24],\"center\":-9,\"point\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, 8, 12, 16, 20, 24], [4, 8, 24, 96, 480, 2880], [4, 8, 12, 16, 20, 24], -8, 84, 84, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([4, 8, 12, 16, 20, 24], [4, 8, 24, 96, 480, 2880], [4, 8, 12, 16, 20, 24], -8, 84, 84, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122054,"parameters":{"shifted_coefficients":[4,8,12,16,20,24],"center":-9,"point":-8,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-9,\"point\":-8,\"shifted_coefficients\":[4,8,12,16,20,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0055","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[5,10,15],\"center\":-8,\"point\":-6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5, 10, 15], [5, 10, 30], [5, 10, 15], -6, 85, 85, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([5, 10, 15], [5, 10, 30], [5, 10, 15], -6, 85, 85, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122055,"parameters":{"shifted_coefficients":[5,10,15],"center":-8,"point":-6,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-8,\"point\":-6,\"shifted_coefficients\":[5,10,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0056","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[6,12,18,24],\"center\":-7,\"point\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([6, 12, 18, 24], [6, 12, 36, 144], [6, 12, 18, 24], -9, -138, -138, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([6, 12, 18, 24], [6, 12, 36, 144], [6, 12, 18, 24], -9, -138, -138, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122056,"parameters":{"shifted_coefficients":[6,12,18,24],"center":-7,"point":-9,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-7,\"point\":-9,\"shifted_coefficients\":[6,12,18,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0057","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[7,14,21,28,35],\"center\":-6,\"point\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([7, 14, 21, 28, 35], [7, 14, 42, 168, 840], [7, 14, 21, 28, 35], -7, 21, 21, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([7, 14, 21, 28, 35], [7, 14, 42, 168, 840], [7, 14, 21, 28, 35], -7, 21, 21, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122057,"parameters":{"shifted_coefficients":[7,14,21,28,35],"center":-6,"point":-7,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-6,\"point\":-7,\"shifted_coefficients\":[7,14,21,28,35]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0058","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[8,16,24,32,40,48],\"center\":-5,\"point\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8, 16, 24, 32, 40, 48], [8, 16, 48, 192, 960, 5760], [8, 16, 24, 32, 40, 48], -5, 8, 8, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([8, 16, 24, 32, 40, 48], [8, 16, 48, 192, 960, 5760], [8, 16, 24, 32, 40, 48], -5, 8, 8, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122058,"parameters":{"shifted_coefficients":[8,16,24,32,40,48],"center":-5,"point":-5,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-5,\"point\":-5,\"shifted_coefficients\":[8,16,24,32,40,48]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0059","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[9,18,27],\"center\":-4,\"point\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([9, 18, 27], [9, 18, 54], [9, 18, 27], -3, 54, 54, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([9, 18, 27], [9, 18, 54], [9, 18, 27], -3, 54, 54, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122059,"parameters":{"shifted_coefficients":[9,18,27],"center":-4,"point":-3,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-4,\"point\":-3,\"shifted_coefficients\":[9,18,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0060","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[10,20,30,40],\"center\":-3,\"point\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([10, 20, 30, 40], [10, 20, 60, 240], [10, 20, 30, 40], -1, 490, 490, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([10, 20, 30, 40], [10, 20, 60, 240], [10, 20, 30, 40], -1, 490, 490, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122060,"parameters":{"shifted_coefficients":[10,20,30,40],"center":-3,"point":-1,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-3,\"point\":-1,\"shifted_coefficients\":[10,20,30,40]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0061","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[11,22,33,44,55],\"center\":-2,\"point\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([11, 22, 33, 44, 55], [11, 22, 66, 264, 1320], [11, 22, 33, 44, 55], -4, 627, 627, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([11, 22, 33, 44, 55], [11, 22, 66, 264, 1320], [11, 22, 33, 44, 55], -4, 627, 627, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122061,"parameters":{"shifted_coefficients":[11,22,33,44,55],"center":-2,"point":-4,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-2,\"point\":-4,\"shifted_coefficients\":[11,22,33,44,55]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0062","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[12,24,36,48,60,72],\"center\":-1,\"point\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([12, 24, 36, 48, 60, 72], [12, 24, 72, 288, 1440, 8640], [12, 24, 36, 48, 60, 72], -2, -36, -36, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([12, 24, 36, 48, 60, 72], [12, 24, 72, 288, 1440, 8640], [12, 24, 36, 48, 60, 72], -2, -36, -36, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122062,"parameters":{"shifted_coefficients":[12,24,36,48,60,72],"center":-1,"point":-2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-1,\"point\":-2,\"shifted_coefficients\":[12,24,36,48,60,72]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0063","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[13,26,39],\"center\":0,\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([13, 26, 39], [13, 26, 78], [13, 26, 39], 0, 13, 13, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([13, 26, 39], [13, 26, 78], [13, 26, 39], 0, 13, 13, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122063,"parameters":{"shifted_coefficients":[13,26,39],"center":0,"point":0,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":0,\"point\":0,\"shifted_coefficients\":[13,26,39]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0064","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[14,28,42,56],\"center\":1,\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([14, 28, 42, 56], [14, 28, 84, 336], [14, 28, 42, 56], 2, 140, 140, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([14, 28, 42, 56], [14, 28, 84, 336], [14, 28, 42, 56], 2, 140, 140, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122064,"parameters":{"shifted_coefficients":[14,28,42,56],"center":1,"point":2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":1,\"point\":2,\"shifted_coefficients\":[14,28,42,56]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0065","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[15,30,45,60,75],\"center\":2,\"point\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([15, 30, 45, 60, 75], [15, 30, 90, 360, 1800], [15, 30, 45, 60, 75], 4, 1935, 1935, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([15, 30, 45, 60, 75], [15, 30, 90, 360, 1800], [15, 30, 45, 60, 75], 4, 1935, 1935, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122065,"parameters":{"shifted_coefficients":[15,30,45,60,75],"center":2,"point":4,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":2,\"point\":4,\"shifted_coefficients\":[15,30,45,60,75]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0066","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[16,32,48,64,80,96],\"center\":3,\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([16, 32, 48, 64, 80, 96], [16, 32, 96, 384, 1920, 11520], [16, 32, 48, 64, 80, 96], 1, -2160, -2160, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([16, 32, 48, 64, 80, 96], [16, 32, 96, 384, 1920, 11520], [16, 32, 48, 64, 80, 96], 1, -2160, -2160, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122066,"parameters":{"shifted_coefficients":[16,32,48,64,80,96],"center":3,"point":1,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":3,\"point\":1,\"shifted_coefficients\":[16,32,48,64,80,96]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0067","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[17,34,51],\"center\":4,\"point\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([17, 34, 51], [17, 34, 102], [17, 34, 51], 3, 34, 34, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([17, 34, 51], [17, 34, 102], [17, 34, 51], 3, 34, 34, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122067,"parameters":{"shifted_coefficients":[17,34,51],"center":4,"point":3,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":4,\"point\":3,\"shifted_coefficients\":[17,34,51]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0068","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[18,36,54,72],\"center\":5,\"point\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([18, 36, 54, 72], [18, 36, 108, 432], [18, 36, 54, 72], 5, 18, 18, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([18, 36, 54, 72], [18, 36, 108, 432], [18, 36, 54, 72], 5, 18, 18, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122068,"parameters":{"shifted_coefficients":[18,36,54,72],"center":5,"point":5,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":5,\"point\":5,\"shifted_coefficients\":[18,36,54,72]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0069","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[19,38,57,76,95],\"center\":6,\"point\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([19, 38, 57, 76, 95], [19, 38, 114, 456, 2280], [19, 38, 57, 76, 95], 7, 285, 285, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([19, 38, 57, 76, 95], [19, 38, 114, 456, 2280], [19, 38, 57, 76, 95], 7, 285, 285, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122069,"parameters":{"shifted_coefficients":[19,38,57,76,95],"center":6,"point":7,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":6,\"point\":7,\"shifted_coefficients\":[19,38,57,76,95]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0070","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[20,40,60,80,100,120],\"center\":7,\"point\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([20, 40, 60, 80, 100, 120], [20, 40, 120, 480, 2400, 14400], [20, 40, 60, 80, 100, 120], 9, 6420, 6420, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([20, 40, 60, 80, 100, 120], [20, 40, 120, 480, 2400, 14400], [20, 40, 60, 80, 100, 120], 9, 6420, 6420, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122070,"parameters":{"shifted_coefficients":[20,40,60,80,100,120],"center":7,"point":9,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":7,\"point\":9,\"shifted_coefficients\":[20,40,60,80,100,120]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0071","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[21,42,63],\"center\":8,\"point\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([21, 42, 63], [21, 42, 126], [21, 42, 63], 6, 189, 189, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([21, 42, 63], [21, 42, 126], [21, 42, 63], 6, 189, 189, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122071,"parameters":{"shifted_coefficients":[21,42,63],"center":8,"point":6,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":8,\"point\":6,\"shifted_coefficients\":[21,42,63]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0072","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[22,44,66,88],\"center\":9,\"point\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([22, 44, 66, 88], [22, 44, 132, 528], [22, 44, 66, 88], 8, -44, -44, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([22, 44, 66, 88], [22, 44, 132, 528], [22, 44, 66, 88], 8, -44, -44, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122072,"parameters":{"shifted_coefficients":[22,44,66,88],"center":9,"point":8,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":9,\"point\":8,\"shifted_coefficients\":[22,44,66,88]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0073","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[23,46,69,92,115],\"center\":10,\"point\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([23, 46, 69, 92, 115], [23, 46, 138, 552, 2760], [23, 46, 69, 92, 115], 10, 23, 23, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([23, 46, 69, 92, 115], [23, 46, 138, 552, 2760], [23, 46, 69, 92, 115], 10, 23, 23, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122073,"parameters":{"shifted_coefficients":[23,46,69,92,115],"center":10,"point":10,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":10,\"point\":10,\"shifted_coefficients\":[23,46,69,92,115]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0074","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[24,48,72,96,120,144],\"center\":11,\"point\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([24, 48, 72, 96, 120, 144], [24, 48, 144, 576, 2880, 17280], [24, 48, 72, 96, 120, 144], 12, 504, 504, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([24, 48, 72, 96, 120, 144], [24, 48, 144, 576, 2880, 17280], [24, 48, 72, 96, 120, 144], 12, 504, 504, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122074,"parameters":{"shifted_coefficients":[24,48,72,96,120,144],"center":11,"point":12,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":11,\"point\":12,\"shifted_coefficients\":[24,48,72,96,120,144]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0075","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[25,50,75],\"center\":12,\"point\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([25, 50, 75], [25, 50, 150], [25, 50, 75], 14, 425, 425, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)","explanation_zh":"Wave6第六層核對=([25, 50, 75], [25, 50, 150], [25, 50, 75], 14, 425, 425, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-exact-polynomial-reconstruction","seed":202611122075,"parameters":{"shifted_coefficients":[25,50,75],"center":12,"point":14,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-exact-polynomial-reconstruction","semantic_template":"qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":12,\"point\":14,\"shifted_coefficients\":[25,50,75]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0076","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"missing_derivative\",\"derivatives\":[1,2],\"center\":-12,\"order\":2,\"alternate_center\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(missing_derivative, 2, [1, 2], -12, -11, 否, all_derivatives_zero_through_n_are_required)","explanation_zh":"Wave6第六層核對=(missing_derivative, 2, [1, 2], -12, -11, 否, all_derivatives_zero_through_n_are_required)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122076,"parameters":{"mode":"missing_derivative","derivatives":[1,2],"center":-12,"order":2,"alternate_center":-11,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-11,\"center\":-12,\"derivatives\":[1,2],\"mode\":\"missing_derivative\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0077","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"wrong_center\",\"derivatives\":[2,4,6],\"center\":-11,\"order\":2,\"alternate_center\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(wrong_center, 2, [2, 4, 6], -11, -10, 否, a_jet_is_bound_to_its_declared_center)","explanation_zh":"Wave6第六層核對=(wrong_center, 2, [2, 4, 6], -11, -10, 否, a_jet_is_bound_to_its_declared_center)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122077,"parameters":{"mode":"wrong_center","derivatives":[2,4,6],"center":-11,"order":2,"alternate_center":-10,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-10,\"center\":-11,\"derivatives\":[2,4,6],\"mode\":\"wrong_center\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0078","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"missing_factorial\",\"derivatives\":[3,6,9],\"center\":-10,\"order\":2,\"alternate_center\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(missing_factorial, 2, [3, 6, 9], -10, -9, 否, coefficients_require_division_by_k_factorial)","explanation_zh":"Wave6第六層核對=(missing_factorial, 2, [3, 6, 9], -10, -9, 否, coefficients_require_division_by_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122078,"parameters":{"mode":"missing_factorial","derivatives":[3,6,9],"center":-10,"order":2,"alternate_center":-9,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-9,\"center\":-10,\"derivatives\":[3,6,9],\"mode\":\"missing_factorial\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0079","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"remainder_claim\",\"derivatives\":[4,8,12],\"center\":-9,\"order\":2,\"alternate_center\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(remainder_claim, 2, [4, 8, 12], -9, -8, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)","explanation_zh":"Wave6第六層核對=(remainder_claim, 2, [4, 8, 12], -9, -8, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122079,"parameters":{"mode":"remainder_claim","derivatives":[4,8,12],"center":-9,"order":2,"alternate_center":-8,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-8,\"center\":-9,\"derivatives\":[4,8,12],\"mode\":\"remainder_claim\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0080","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"missing_derivative\",\"derivatives\":[5,10],\"center\":-8,\"order\":2,\"alternate_center\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(missing_derivative, 2, [5, 10], -8, -7, 否, all_derivatives_zero_through_n_are_required)","explanation_zh":"Wave6第六層核對=(missing_derivative, 2, [5, 10], -8, -7, 否, all_derivatives_zero_through_n_are_required)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122080,"parameters":{"mode":"missing_derivative","derivatives":[5,10],"center":-8,"order":2,"alternate_center":-7,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-7,\"center\":-8,\"derivatives\":[5,10],\"mode\":\"missing_derivative\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0081","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"wrong_center\",\"derivatives\":[6,12,18],\"center\":-7,\"order\":2,\"alternate_center\":-6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(wrong_center, 2, [6, 12, 18], -7, -6, 否, a_jet_is_bound_to_its_declared_center)","explanation_zh":"Wave6第六層核對=(wrong_center, 2, [6, 12, 18], -7, -6, 否, a_jet_is_bound_to_its_declared_center)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122081,"parameters":{"mode":"wrong_center","derivatives":[6,12,18],"center":-7,"order":2,"alternate_center":-6,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-6,\"center\":-7,\"derivatives\":[6,12,18],\"mode\":\"wrong_center\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0082","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"missing_factorial\",\"derivatives\":[7,14,21],\"center\":-6,\"order\":2,\"alternate_center\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(missing_factorial, 2, [7, 14, 21], -6, -5, 否, coefficients_require_division_by_k_factorial)","explanation_zh":"Wave6第六層核對=(missing_factorial, 2, [7, 14, 21], -6, -5, 否, coefficients_require_division_by_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122082,"parameters":{"mode":"missing_factorial","derivatives":[7,14,21],"center":-6,"order":2,"alternate_center":-5,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-5,\"center\":-6,\"derivatives\":[7,14,21],\"mode\":\"missing_factorial\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0083","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"remainder_claim\",\"derivatives\":[8,16,24],\"center\":-5,\"order\":2,\"alternate_center\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(remainder_claim, 2, [8, 16, 24], -5, -4, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)","explanation_zh":"Wave6第六層核對=(remainder_claim, 2, [8, 16, 24], -5, -4, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122083,"parameters":{"mode":"remainder_claim","derivatives":[8,16,24],"center":-5,"order":2,"alternate_center":-4,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-4,\"center\":-5,\"derivatives\":[8,16,24],\"mode\":\"remainder_claim\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0084","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"missing_derivative\",\"derivatives\":[9,18],\"center\":-4,\"order\":2,\"alternate_center\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(missing_derivative, 2, [9, 18], -4, -3, 否, all_derivatives_zero_through_n_are_required)","explanation_zh":"Wave6第六層核對=(missing_derivative, 2, [9, 18], -4, -3, 否, all_derivatives_zero_through_n_are_required)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122084,"parameters":{"mode":"missing_derivative","derivatives":[9,18],"center":-4,"order":2,"alternate_center":-3,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-3,\"center\":-4,\"derivatives\":[9,18],\"mode\":\"missing_derivative\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0085","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"wrong_center\",\"derivatives\":[10,20,30],\"center\":-3,\"order\":2,\"alternate_center\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(wrong_center, 2, [10, 20, 30], -3, -2, 否, a_jet_is_bound_to_its_declared_center)","explanation_zh":"Wave6第六層核對=(wrong_center, 2, [10, 20, 30], -3, -2, 否, a_jet_is_bound_to_its_declared_center)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122085,"parameters":{"mode":"wrong_center","derivatives":[10,20,30],"center":-3,"order":2,"alternate_center":-2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-2,\"center\":-3,\"derivatives\":[10,20,30],\"mode\":\"wrong_center\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0086","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"missing_factorial\",\"derivatives\":[11,22,33],\"center\":-2,\"order\":2,\"alternate_center\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(missing_factorial, 2, [11, 22, 33], -2, -1, 否, coefficients_require_division_by_k_factorial)","explanation_zh":"Wave6第六層核對=(missing_factorial, 2, [11, 22, 33], -2, -1, 否, coefficients_require_division_by_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122086,"parameters":{"mode":"missing_factorial","derivatives":[11,22,33],"center":-2,"order":2,"alternate_center":-1,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":-1,\"center\":-2,\"derivatives\":[11,22,33],\"mode\":\"missing_factorial\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0087","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"remainder_claim\",\"derivatives\":[12,24,36],\"center\":-1,\"order\":2,\"alternate_center\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(remainder_claim, 2, [12, 24, 36], -1, 0, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)","explanation_zh":"Wave6第六層核對=(remainder_claim, 2, [12, 24, 36], -1, 0, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122087,"parameters":{"mode":"remainder_claim","derivatives":[12,24,36],"center":-1,"order":2,"alternate_center":0,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":0,\"center\":-1,\"derivatives\":[12,24,36],\"mode\":\"remainder_claim\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0088","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"missing_derivative\",\"derivatives\":[13,26],\"center\":0,\"order\":2,\"alternate_center\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(missing_derivative, 2, [13, 26], 0, 1, 否, all_derivatives_zero_through_n_are_required)","explanation_zh":"Wave6第六層核對=(missing_derivative, 2, [13, 26], 0, 1, 否, all_derivatives_zero_through_n_are_required)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122088,"parameters":{"mode":"missing_derivative","derivatives":[13,26],"center":0,"order":2,"alternate_center":1,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":1,\"center\":0,\"derivatives\":[13,26],\"mode\":\"missing_derivative\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0089","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"wrong_center\",\"derivatives\":[14,28,42],\"center\":1,\"order\":2,\"alternate_center\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(wrong_center, 2, [14, 28, 42], 1, 2, 否, a_jet_is_bound_to_its_declared_center)","explanation_zh":"Wave6第六層核對=(wrong_center, 2, [14, 28, 42], 1, 2, 否, a_jet_is_bound_to_its_declared_center)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122089,"parameters":{"mode":"wrong_center","derivatives":[14,28,42],"center":1,"order":2,"alternate_center":2,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":2,\"center\":1,\"derivatives\":[14,28,42],\"mode\":\"wrong_center\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0090","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"missing_factorial\",\"derivatives\":[15,30,45],\"center\":2,\"order\":2,\"alternate_center\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(missing_factorial, 2, [15, 30, 45], 2, 3, 否, coefficients_require_division_by_k_factorial)","explanation_zh":"Wave6第六層核對=(missing_factorial, 2, [15, 30, 45], 2, 3, 否, coefficients_require_division_by_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122090,"parameters":{"mode":"missing_factorial","derivatives":[15,30,45],"center":2,"order":2,"alternate_center":3,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":3,\"center\":2,\"derivatives\":[15,30,45],\"mode\":\"missing_factorial\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0091","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"remainder_claim\",\"derivatives\":[16,32,48],\"center\":3,\"order\":2,\"alternate_center\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(remainder_claim, 2, [16, 32, 48], 3, 4, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)","explanation_zh":"Wave6第六層核對=(remainder_claim, 2, [16, 32, 48], 3, 4, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122091,"parameters":{"mode":"remainder_claim","derivatives":[16,32,48],"center":3,"order":2,"alternate_center":4,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":4,\"center\":3,\"derivatives\":[16,32,48],\"mode\":\"remainder_claim\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0092","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"missing_derivative\",\"derivatives\":[17,34],\"center\":4,\"order\":2,\"alternate_center\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(missing_derivative, 2, [17, 34], 4, 5, 否, all_derivatives_zero_through_n_are_required)","explanation_zh":"Wave6第六層核對=(missing_derivative, 2, [17, 34], 4, 5, 否, all_derivatives_zero_through_n_are_required)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122092,"parameters":{"mode":"missing_derivative","derivatives":[17,34],"center":4,"order":2,"alternate_center":5,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":5,\"center\":4,\"derivatives\":[17,34],\"mode\":\"missing_derivative\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0093","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"wrong_center\",\"derivatives\":[18,36,54],\"center\":5,\"order\":2,\"alternate_center\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(wrong_center, 2, [18, 36, 54], 5, 6, 否, a_jet_is_bound_to_its_declared_center)","explanation_zh":"Wave6第六層核對=(wrong_center, 2, [18, 36, 54], 5, 6, 否, a_jet_is_bound_to_its_declared_center)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122093,"parameters":{"mode":"wrong_center","derivatives":[18,36,54],"center":5,"order":2,"alternate_center":6,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":6,\"center\":5,\"derivatives\":[18,36,54],\"mode\":\"wrong_center\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0094","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"missing_factorial\",\"derivatives\":[19,38,57],\"center\":6,\"order\":2,\"alternate_center\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(missing_factorial, 2, [19, 38, 57], 6, 7, 否, coefficients_require_division_by_k_factorial)","explanation_zh":"Wave6第六層核對=(missing_factorial, 2, [19, 38, 57], 6, 7, 否, coefficients_require_division_by_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122094,"parameters":{"mode":"missing_factorial","derivatives":[19,38,57],"center":6,"order":2,"alternate_center":7,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":7,\"center\":6,\"derivatives\":[19,38,57],\"mode\":\"missing_factorial\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0095","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"remainder_claim\",\"derivatives\":[20,40,60],\"center\":7,\"order\":2,\"alternate_center\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(remainder_claim, 2, [20, 40, 60], 7, 8, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)","explanation_zh":"Wave6第六層核對=(remainder_claim, 2, [20, 40, 60], 7, 8, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122095,"parameters":{"mode":"remainder_claim","derivatives":[20,40,60],"center":7,"order":2,"alternate_center":8,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":8,\"center\":7,\"derivatives\":[20,40,60],\"mode\":\"remainder_claim\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0096","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"missing_derivative\",\"derivatives\":[21,42],\"center\":8,\"order\":2,\"alternate_center\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(missing_derivative, 2, [21, 42], 8, 9, 否, all_derivatives_zero_through_n_are_required)","explanation_zh":"Wave6第六層核對=(missing_derivative, 2, [21, 42], 8, 9, 否, all_derivatives_zero_through_n_are_required)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122096,"parameters":{"mode":"missing_derivative","derivatives":[21,42],"center":8,"order":2,"alternate_center":9,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":9,\"center\":8,\"derivatives\":[21,42],\"mode\":\"missing_derivative\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0097","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"wrong_center\",\"derivatives\":[22,44,66],\"center\":9,\"order\":2,\"alternate_center\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(wrong_center, 2, [22, 44, 66], 9, 10, 否, a_jet_is_bound_to_its_declared_center)","explanation_zh":"Wave6第六層核對=(wrong_center, 2, [22, 44, 66], 9, 10, 否, a_jet_is_bound_to_its_declared_center)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122097,"parameters":{"mode":"wrong_center","derivatives":[22,44,66],"center":9,"order":2,"alternate_center":10,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":10,\"center\":9,\"derivatives\":[22,44,66],\"mode\":\"wrong_center\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0098","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"missing_factorial\",\"derivatives\":[23,46,69],\"center\":10,\"order\":2,\"alternate_center\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(missing_factorial, 2, [23, 46, 69], 10, 11, 否, coefficients_require_division_by_k_factorial)","explanation_zh":"Wave6第六層核對=(missing_factorial, 2, [23, 46, 69], 10, 11, 否, coefficients_require_division_by_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122098,"parameters":{"mode":"missing_factorial","derivatives":[23,46,69],"center":10,"order":2,"alternate_center":11,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":11,\"center\":10,\"derivatives\":[23,46,69],\"mode\":\"missing_factorial\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0099","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"remainder_claim\",\"derivatives\":[24,48,72],\"center\":11,\"order\":2,\"alternate_center\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(remainder_claim, 2, [24, 48, 72], 11, 12, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)","explanation_zh":"Wave6第六層核對=(remainder_claim, 2, [24, 48, 72], 11, 12, 否, a_Taylor_polynomial_alone_gives_no_remainder_or_series_equality)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122099,"parameters":{"mode":"remainder_claim","derivatives":[24,48,72],"center":11,"order":2,"alternate_center":12,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":12,\"center\":11,\"derivatives\":[24,48,72],\"mode\":\"remainder_claim\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r02-taylor-polynomial-0100","batch_id":"batch-20261014-advance-20261112-r02-taylor-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-polynomial"],"title_zh":"Taylor多項式","stem_zh":"固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"missing_derivative\",\"derivatives\":[25,50],\"center\":12,\"order\":2,\"alternate_center\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(missing_derivative, 2, [25, 50], 12, 13, 否, all_derivatives_zero_through_n_are_required)","explanation_zh":"Wave6第六層核對=(missing_derivative, 2, [25, 50], 12, 13, 否, all_derivatives_zero_through_n_are_required)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor多項式"]},"variant":{"family_id":"qf-mod43-taylor-definition-boundary","seed":202611122100,"parameters":{"mode":"missing_derivative","derivatives":[25,50],"center":12,"order":2,"alternate_center":13,"domain":"Exact_finite_Taylor_polynomial","operation":"taylor-definition-boundary","semantic_template":"qf-mod43-taylor-definition-boundary/{\"alternate_center\":13,\"center\":12,\"derivatives\":[25,50],\"mode\":\"missing_derivative\",\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}