{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0001","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[2,3],\"center\":0,\"x\":[1,2],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 3], 0, 1/2, 1, [2, 3, 0], [2, 3/2], 7/2, 7/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([2, 3], 0, 1/2, 1, [2, 3, 0], [2, 3/2], 7/2, 7/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163001,"parameters":{"coefficients":[2,3],"center":0,"x":[1,2],"order":1,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[2,1],[3,1]],\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0002","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[3,5,7],\"center\":0,\"x\":[1,2],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 7], 0, 1/2, 2, [3, 5, 14, 0], [3, 5/2, 7/4], 29/4, 29/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([3, 5, 7], 0, 1/2, 2, [3, 5, 14, 0], [3, 5/2, 7/4], 29/4, 29/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163002,"parameters":{"coefficients":[3,5,7],"center":0,"x":[1,2],"order":2,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[3,1],[5,1],[7,1]],\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0003","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[4,7,10,13],\"center\":0,\"x\":[1,2],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, 7, 10, 13], 0, 1/2, 3, [4, 7, 20, 78, 0], [4, 7/2, 5/2, 13/8], 93/8, 93/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([4, 7, 10, 13], 0, 1/2, 3, [4, 7, 20, 78, 0], [4, 7/2, 5/2, 13/8], 93/8, 93/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163003,"parameters":{"coefficients":[4,7,10,13],"center":0,"x":[1,2],"order":3,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[4,1],[7,1],[10,1],[13,1]],\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0004","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[5,9,13,17,4],\"center\":0,\"x\":[1,2],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5, 9, 13, 17, 4], 0, 1/2, 4, [5, 9, 26, 102, 96, 0], [5, 9/2, 13/4, 17/8, 1/4], 121/8, 121/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([5, 9, 13, 17, 4], 0, 1/2, 4, [5, 9, 26, 102, 96, 0], [5, 9/2, 13/4, 17/8, 1/4], 121/8, 121/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163004,"parameters":{"coefficients":[5,9,13,17,4],"center":0,"x":[1,2],"order":4,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[5,1],[9,1],[13,1],[17,1],[4,1]],\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0005","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[6,11,16,4,9,14],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([6, 11, 16, 4, 9, 14], 0, 1/2, 5, [6, 11, 32, 24, 216, 1680, 0], [6, 11/2, 4, 1/2, 9/16, 7/16], 17, 17, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([6, 11, 16, 4, 9, 14], 0, 1/2, 5, [6, 11, 32, 24, 216, 1680, 0], [6, 11/2, 4, 1/2, 9/16, 7/16], 17, 17, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163005,"parameters":{"coefficients":[6,11,16,4,9,14],"center":0,"x":[1,2],"order":5,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[6,1],[11,1],[16,1],[4,1],[9,1],[14,1]],\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0006","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[7,13],\"center\":0,\"x\":[1,2],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([7, 13], 0, 1/2, 1, [7, 13, 0], [7, 13/2], 27/2, 27/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([7, 13], 0, 1/2, 1, [7, 13, 0], [7, 13/2], 27/2, 27/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163006,"parameters":{"coefficients":[7,13],"center":0,"x":[1,2],"order":1,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[7,1],[13,1]],\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0007","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[8,15,5],\"center\":0,\"x\":[1,2],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8, 15, 5], 0, 1/2, 2, [8, 15, 10, 0], [8, 15/2, 5/4], 67/4, 67/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([8, 15, 5], 0, 1/2, 2, [8, 15, 10, 0], [8, 15/2, 5/4], 67/4, 67/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163007,"parameters":{"coefficients":[8,15,5],"center":0,"x":[1,2],"order":2,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[8,1],[15,1],[5,1]],\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0008","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[9,17,8,16],\"center\":0,\"x\":[1,2],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([9, 17, 8, 16], 0, 1/2, 3, [9, 17, 16, 96, 0], [9, 17/2, 2, 2], 43/2, 43/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([9, 17, 8, 16], 0, 1/2, 3, [9, 17, 16, 96, 0], [9, 17/2, 2, 2], 43/2, 43/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163008,"parameters":{"coefficients":[9,17,8,16],"center":0,"x":[1,2],"order":3,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[9,1],[17,1],[8,1],[16,1]],\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0009","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[10,2,11,3,12],\"center\":0,\"x\":[1,2],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([10, 2, 11, 3, 12], 0, 1/2, 4, [10, 2, 22, 18, 288, 0], [10, 1, 11/4, 3/8, 3/4], 119/8, 119/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([10, 2, 11, 3, 12], 0, 1/2, 4, [10, 2, 22, 18, 288, 0], [10, 1, 11/4, 3/8, 3/4], 119/8, 119/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163009,"parameters":{"coefficients":[10,2,11,3,12],"center":0,"x":[1,2],"order":4,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[10,1],[2,1],[11,1],[3,1],[12,1]],\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0010","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[11,4,14,7,17,10],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([11, 4, 14, 7, 17, 10], 0, 1/2, 5, [11, 4, 28, 42, 408, 1200, 0], [11, 2, 7/2, 7/8, 17/16, 5/16], 75/4, 75/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([11, 4, 14, 7, 17, 10], 0, 1/2, 5, [11, 4, 28, 42, 408, 1200, 0], [11, 2, 7/2, 7/8, 17/16, 5/16], 75/4, 75/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163010,"parameters":{"coefficients":[11,4,14,7,17,10],"center":0,"x":[1,2],"order":5,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[11,1],[4,1],[14,1],[7,1],[17,1],[10,1]],\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0011","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[12,6],\"center\":0,\"x\":[1,2],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([12, 6], 0, 1/2, 1, [12, 6, 0], [12, 3], 15, 15, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([12, 6], 0, 1/2, 1, [12, 6, 0], [12, 3], 15, 15, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163011,"parameters":{"coefficients":[12,6],"center":0,"x":[1,2],"order":1,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[12,1],[6,1]],\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0012","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[13,8,3],\"center\":0,\"x\":[1,2],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([13, 8, 3], 0, 1/2, 2, [13, 8, 6, 0], [13, 4, 3/4], 71/4, 71/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([13, 8, 3], 0, 1/2, 2, [13, 8, 6, 0], [13, 4, 3/4], 71/4, 71/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163012,"parameters":{"coefficients":[13,8,3],"center":0,"x":[1,2],"order":2,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[13,1],[8,1],[3,1]],\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0013","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[14,10,6,2],\"center\":0,\"x\":[1,2],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([14, 10, 6, 2], 0, 1/2, 3, [14, 10, 12, 12, 0], [14, 5, 3/2, 1/4], 83/4, 83/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([14, 10, 6, 2], 0, 1/2, 3, [14, 10, 12, 12, 0], [14, 5, 3/2, 1/4], 83/4, 83/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163013,"parameters":{"coefficients":[14,10,6,2],"center":0,"x":[1,2],"order":3,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[14,1],[10,1],[6,1],[2,1]],\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0014","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[15,12,9,6,3],\"center\":0,\"x\":[1,2],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([15, 12, 9, 6, 3], 0, 1/2, 4, [15, 12, 18, 36, 72, 0], [15, 6, 9/4, 3/4, 3/16], 387/16, 387/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([15, 12, 9, 6, 3], 0, 1/2, 4, [15, 12, 18, 36, 72, 0], [15, 6, 9/4, 3/4, 3/16], 387/16, 387/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163014,"parameters":{"coefficients":[15,12,9,6,3],"center":0,"x":[1,2],"order":4,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[15,1],[12,1],[9,1],[6,1],[3,1]],\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0015","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[16,14,12,10,8,6],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([16, 14, 12, 10, 8, 6], 0, 1/2, 5, [16, 14, 24, 60, 192, 720, 0], [16, 7, 3, 5/4, 1/2, 3/16], 447/16, 447/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([16, 14, 12, 10, 8, 6], 0, 1/2, 5, [16, 14, 24, 60, 192, 720, 0], [16, 7, 3, 5/4, 1/2, 3/16], 447/16, 447/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163015,"parameters":{"coefficients":[16,14,12,10,8,6],"center":0,"x":[1,2],"order":5,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[16,1],[14,1],[12,1],[10,1],[8,1],[6,1]],\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0016","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[17,16],\"center\":0,\"x\":[1,2],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([17, 16], 0, 1/2, 1, [17, 16, 0], [17, 8], 25, 25, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([17, 16], 0, 1/2, 1, [17, 16, 0], [17, 8], 25, 25, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163016,"parameters":{"coefficients":[17,16],"center":0,"x":[1,2],"order":1,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[17,1],[16,1]],\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0017","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[1,1,1],\"center\":0,\"x\":[1,2],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1], 0, 1/2, 2, [1, 1, 2, 0], [1, 1/2, 1/4], 7/4, 7/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([1, 1, 1], 0, 1/2, 2, [1, 1, 2, 0], [1, 1/2, 1/4], 7/4, 7/4, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163017,"parameters":{"coefficients":[1,1,1],"center":0,"x":[1,2],"order":2,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[1,1],[1,1],[1,1]],\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0018","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[2,3,4,5],\"center\":0,\"x\":[1,2],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 3, 4, 5], 0, 1/2, 3, [2, 3, 8, 30, 0], [2, 3/2, 1, 5/8], 41/8, 41/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([2, 3, 4, 5], 0, 1/2, 3, [2, 3, 8, 30, 0], [2, 3/2, 1, 5/8], 41/8, 41/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163018,"parameters":{"coefficients":[2,3,4,5],"center":0,"x":[1,2],"order":3,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[2,1],[3,1],[4,1],[5,1]],\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0019","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[3,5,7,9,11],\"center\":0,\"x\":[1,2],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 7, 9, 11], 0, 1/2, 4, [3, 5, 14, 54, 264, 0], [3, 5/2, 7/4, 9/8, 11/16], 145/16, 145/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([3, 5, 7, 9, 11], 0, 1/2, 4, [3, 5, 14, 54, 264, 0], [3, 5/2, 7/4, 9/8, 11/16], 145/16, 145/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163019,"parameters":{"coefficients":[3,5,7,9,11],"center":0,"x":[1,2],"order":4,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[3,1],[5,1],[7,1],[9,1],[11,1]],\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0020","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[4,7,10,13,16,2],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, 7, 10, 13, 16, 2], 0, 1/2, 5, [4, 7, 20, 78, 384, 240, 0], [4, 7/2, 5/2, 13/8, 1, 1/16], 203/16, 203/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([4, 7, 10, 13, 16, 2], 0, 1/2, 5, [4, 7, 20, 78, 384, 240, 0], [4, 7/2, 5/2, 13/8, 1, 1/16], 203/16, 203/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163020,"parameters":{"coefficients":[4,7,10,13,16,2],"center":0,"x":[1,2],"order":5,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[4,1],[7,1],[10,1],[13,1],[16,1],[2,1]],\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0021","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[5,9],\"center\":0,\"x\":[1,2],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5, 9], 0, 1/2, 1, [5, 9, 0], [5, 9/2], 19/2, 19/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([5, 9], 0, 1/2, 1, [5, 9, 0], [5, 9/2], 19/2, 19/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163021,"parameters":{"coefficients":[5,9],"center":0,"x":[1,2],"order":1,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[5,1],[9,1]],\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0022","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[6,11,16],\"center\":0,\"x\":[1,2],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([6, 11, 16], 0, 1/2, 2, [6, 11, 32, 0], [6, 11/2, 4], 31/2, 31/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([6, 11, 16], 0, 1/2, 2, [6, 11, 32, 0], [6, 11/2, 4], 31/2, 31/2, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163022,"parameters":{"coefficients":[6,11,16],"center":0,"x":[1,2],"order":2,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[6,1],[11,1],[16,1]],\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0023","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[7,13,2,8],\"center\":0,\"x\":[1,2],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([7, 13, 2, 8], 0, 1/2, 3, [7, 13, 4, 48, 0], [7, 13/2, 1/2, 1], 15, 15, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([7, 13, 2, 8], 0, 1/2, 3, [7, 13, 4, 48, 0], [7, 13/2, 1/2, 1], 15, 15, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163023,"parameters":{"coefficients":[7,13,2,8],"center":0,"x":[1,2],"order":3,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[7,1],[13,1],[2,1],[8,1]],\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0024","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[8,15,5,12,2],\"center\":0,\"x\":[1,2],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8, 15, 5, 12, 2], 0, 1/2, 4, [8, 15, 10, 72, 48, 0], [8, 15/2, 5/4, 3/2, 1/8], 147/8, 147/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([8, 15, 5, 12, 2], 0, 1/2, 4, [8, 15, 10, 72, 48, 0], [8, 15/2, 5/4, 3/2, 1/8], 147/8, 147/8, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163024,"parameters":{"coefficients":[8,15,5,12,2],"center":0,"x":[1,2],"order":4,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[8,1],[15,1],[5,1],[12,1],[2,1]],\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0025","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[9,17,8,16,7,15],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([9, 17, 8, 16, 7, 15], 0, 1/2, 5, [9, 17, 16, 96, 168, 1800, 0], [9, 17/2, 2, 2, 7/16, 15/32], 717/32, 717/32, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)","explanation_zh":"Wave6第十層核對=([9, 17, 8, 16, 7, 15], 0, 1/2, 5, [9, 17, 16, 96, 168, 1800, 0], [9, 17/2, 2, 2, 7/16, 15/32], 717/32, 717/32, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-qualification","seed":202611163025,"parameters":{"coefficients":[9,17,8,16,7,15],"center":0,"x":[1,2],"order":5,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-qualification","semantic_template":"qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[9,1],[17,1],[8,1],[16,1],[7,1],[15,1]],\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0026","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[2,3],\"center\":0,\"x\":[1,2],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [2, 3/2], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(1, 1, [2, 3/2], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163026,"parameters":{"coefficients":[2,3],"center":0,"x":[1,2],"order":1,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[2,1],[3,1]],\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0027","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[3,5,7],\"center\":0,\"x\":[1,2],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [3, 5/2, 7/4], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(2, 2, [3, 5/2, 7/4], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163027,"parameters":{"coefficients":[3,5,7],"center":0,"x":[1,2],"order":2,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[3,1],[5,1],[7,1]],\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0028","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[4,7,10,13],\"center\":0,\"x\":[1,2],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, [4, 7/2, 5/2, 13/8], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(3, 3, [4, 7/2, 5/2, 13/8], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163028,"parameters":{"coefficients":[4,7,10,13],"center":0,"x":[1,2],"order":3,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[4,1],[7,1],[10,1],[13,1]],\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0029","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[5,9,13,17,4],\"center\":0,\"x\":[1,2],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, [5, 9/2, 13/4, 17/8, 1/4], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(4, 4, [5, 9/2, 13/4, 17/8, 1/4], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163029,"parameters":{"coefficients":[5,9,13,17,4],"center":0,"x":[1,2],"order":4,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[5,1],[9,1],[13,1],[17,1],[4,1]],\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0030","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[6,11,16,4,9,14],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [6, 11/2, 4, 1/2, 9/16, 7/16], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(5, 5, [6, 11/2, 4, 1/2, 9/16, 7/16], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163030,"parameters":{"coefficients":[6,11,16,4,9,14],"center":0,"x":[1,2],"order":5,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[6,1],[11,1],[16,1],[4,1],[9,1],[14,1]],\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0031","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[7,13],\"center\":0,\"x\":[1,2],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [7, 13/2], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(1, 1, [7, 13/2], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163031,"parameters":{"coefficients":[7,13],"center":0,"x":[1,2],"order":1,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[7,1],[13,1]],\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0032","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[8,15,5],\"center\":0,\"x\":[1,2],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [8, 15/2, 5/4], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(2, 2, [8, 15/2, 5/4], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163032,"parameters":{"coefficients":[8,15,5],"center":0,"x":[1,2],"order":2,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[8,1],[15,1],[5,1]],\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0033","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[9,17,8,16],\"center\":0,\"x\":[1,2],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, [9, 17/2, 2, 2], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(3, 3, [9, 17/2, 2, 2], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163033,"parameters":{"coefficients":[9,17,8,16],"center":0,"x":[1,2],"order":3,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[9,1],[17,1],[8,1],[16,1]],\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0034","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[10,2,11,3,12],\"center\":0,\"x\":[1,2],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, [10, 1, 11/4, 3/8, 3/4], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(4, 4, [10, 1, 11/4, 3/8, 3/4], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163034,"parameters":{"coefficients":[10,2,11,3,12],"center":0,"x":[1,2],"order":4,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[10,1],[2,1],[11,1],[3,1],[12,1]],\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0035","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[11,4,14,7,17,10],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [11, 2, 7/2, 7/8, 17/16, 5/16], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(5, 5, [11, 2, 7/2, 7/8, 17/16, 5/16], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163035,"parameters":{"coefficients":[11,4,14,7,17,10],"center":0,"x":[1,2],"order":5,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[11,1],[4,1],[14,1],[7,1],[17,1],[10,1]],\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0036","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[12,6],\"center\":0,\"x\":[1,2],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [12, 3], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(1, 1, [12, 3], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163036,"parameters":{"coefficients":[12,6],"center":0,"x":[1,2],"order":1,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[12,1],[6,1]],\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0037","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[13,8,3],\"center\":0,\"x\":[1,2],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [13, 4, 3/4], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(2, 2, [13, 4, 3/4], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163037,"parameters":{"coefficients":[13,8,3],"center":0,"x":[1,2],"order":2,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[13,1],[8,1],[3,1]],\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0038","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[14,10,6,2],\"center\":0,\"x\":[1,2],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, [14, 5, 3/2, 1/4], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(3, 3, [14, 5, 3/2, 1/4], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163038,"parameters":{"coefficients":[14,10,6,2],"center":0,"x":[1,2],"order":3,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[14,1],[10,1],[6,1],[2,1]],\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0039","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[15,12,9,6,3],\"center\":0,\"x\":[1,2],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, [15, 6, 9/4, 3/4, 3/16], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(4, 4, [15, 6, 9/4, 3/4, 3/16], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163039,"parameters":{"coefficients":[15,12,9,6,3],"center":0,"x":[1,2],"order":4,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[15,1],[12,1],[9,1],[6,1],[3,1]],\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0040","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[16,14,12,10,8,6],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [16, 7, 3, 5/4, 1/2, 3/16], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(5, 5, [16, 7, 3, 5/4, 1/2, 3/16], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163040,"parameters":{"coefficients":[16,14,12,10,8,6],"center":0,"x":[1,2],"order":5,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[16,1],[14,1],[12,1],[10,1],[8,1],[6,1]],\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0041","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[17,16],\"center\":0,\"x\":[1,2],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [17, 8], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(1, 1, [17, 8], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163041,"parameters":{"coefficients":[17,16],"center":0,"x":[1,2],"order":1,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[17,1],[16,1]],\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0042","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[1,1,1],\"center\":0,\"x\":[1,2],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [1, 1/2, 1/4], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(2, 2, [1, 1/2, 1/4], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163042,"parameters":{"coefficients":[1,1,1],"center":0,"x":[1,2],"order":2,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[1,1],[1,1],[1,1]],\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0043","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[2,3,4,5],\"center\":0,\"x\":[1,2],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, [2, 3/2, 1, 5/8], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(3, 3, [2, 3/2, 1, 5/8], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163043,"parameters":{"coefficients":[2,3,4,5],"center":0,"x":[1,2],"order":3,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[2,1],[3,1],[4,1],[5,1]],\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0044","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[3,5,7,9,11],\"center\":0,\"x\":[1,2],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, [3, 5/2, 7/4, 9/8, 11/16], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(4, 4, [3, 5/2, 7/4, 9/8, 11/16], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163044,"parameters":{"coefficients":[3,5,7,9,11],"center":0,"x":[1,2],"order":4,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[3,1],[5,1],[7,1],[9,1],[11,1]],\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0045","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[4,7,10,13,16,2],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [4, 7/2, 5/2, 13/8, 1, 1/16], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(5, 5, [4, 7/2, 5/2, 13/8, 1, 1/16], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163045,"parameters":{"coefficients":[4,7,10,13,16,2],"center":0,"x":[1,2],"order":5,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[4,1],[7,1],[10,1],[13,1],[16,1],[2,1]],\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0046","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[5,9],\"center\":0,\"x\":[1,2],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [5, 9/2], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(1, 1, [5, 9/2], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163046,"parameters":{"coefficients":[5,9],"center":0,"x":[1,2],"order":1,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[5,1],[9,1]],\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0047","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[6,11,16],\"center\":0,\"x\":[1,2],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [6, 11/2, 4], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(2, 2, [6, 11/2, 4], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163047,"parameters":{"coefficients":[6,11,16],"center":0,"x":[1,2],"order":2,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[6,1],[11,1],[16,1]],\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0048","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[7,13,2,8],\"center\":0,\"x\":[1,2],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, [7, 13/2, 1/2, 1], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(3, 3, [7, 13/2, 1/2, 1], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163048,"parameters":{"coefficients":[7,13,2,8],"center":0,"x":[1,2],"order":3,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[7,1],[13,1],[2,1],[8,1]],\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0049","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[8,15,5,12,2],\"center\":0,\"x\":[1,2],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, [8, 15/2, 5/4, 3/2, 1/8], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(4, 4, [8, 15/2, 5/4, 3/2, 1/8], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163049,"parameters":{"coefficients":[8,15,5,12,2],"center":0,"x":[1,2],"order":4,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[8,1],[15,1],[5,1],[12,1],[2,1]],\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0050","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-zero-remainder；資料={\"coefficients\":[9,17,8,16,7,15],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 5, [9, 17/2, 2, 2, 7/16, 15/32], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)","explanation_zh":"Wave6第十層核對=(5, 5, [9, 17/2, 2, 2, 7/16, 15/32], 0, 是, degree_at_most_n_polynomial_has_zero_remainder)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-zero-remainder","seed":202611163050,"parameters":{"coefficients":[9,17,8,16,7,15],"center":0,"x":[1,2],"order":5,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-zero-remainder","semantic_template":"qf-mod47-taylor-zero-remainder/{\"center\":[0,1],\"coefficients\":[[9,1],[17,1],[8,1],[16,1],[7,1],[15,1]],\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0051","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[2,3,2],\"center\":0,\"x\":[1,2],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 4, 1/4, 是, 1/2, 1/2, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(2, 1, 4, 1/4, 是, 1/2, 1/2, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163051,"parameters":{"coefficients":[2,3,2],"center":0,"x":[1,2],"order":1,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[2,1],[3,1],[2,1]],\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0052","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[3,5,7,3],\"center\":0,\"x\":[1,2],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 18, 1/4, 是, 3/8, 3/8, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(3, 2, 18, 1/4, 是, 3/8, 3/8, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163052,"parameters":{"coefficients":[3,5,7,3],"center":0,"x":[1,2],"order":2,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[3,1],[5,1],[7,1],[3,1]],\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0053","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[4,7,10,13,4],\"center\":0,\"x\":[1,2],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 96, 1/4, 是, 1/4, 1/4, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(4, 3, 96, 1/4, 是, 1/4, 1/4, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163053,"parameters":{"coefficients":[4,7,10,13,4],"center":0,"x":[1,2],"order":3,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[4,1],[7,1],[10,1],[13,1],[4,1]],\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0054","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[5,9,13,17,4,5],\"center\":0,\"x\":[1,2],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 600, 1/4, 是, 5/32, 5/32, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(5, 4, 600, 1/4, 是, 5/32, 5/32, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163054,"parameters":{"coefficients":[5,9,13,17,4,5],"center":0,"x":[1,2],"order":4,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[5,1],[9,1],[13,1],[17,1],[4,1],[5,1]],\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0055","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[6,11,16,4,9,14,6],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 5, 4320, 1/4, 是, 3/32, 3/32, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(6, 5, 4320, 1/4, 是, 3/32, 3/32, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163055,"parameters":{"coefficients":[6,11,16,4,9,14,6],"center":0,"x":[1,2],"order":5,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[6,1],[11,1],[16,1],[4,1],[9,1],[14,1],[6,1]],\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0056","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[7,13,7],\"center\":0,\"x\":[1,2],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 14, 1/4, 是, 7/4, 7/4, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(2, 1, 14, 1/4, 是, 7/4, 7/4, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163056,"parameters":{"coefficients":[7,13,7],"center":0,"x":[1,2],"order":1,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[7,1],[13,1],[7,1]],\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0057","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[8,15,5,8],\"center\":0,\"x\":[1,2],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 48, 1/4, 是, 1, 1, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(3, 2, 48, 1/4, 是, 1, 1, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163057,"parameters":{"coefficients":[8,15,5,8],"center":0,"x":[1,2],"order":2,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[8,1],[15,1],[5,1],[8,1]],\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0058","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[9,17,8,16,9],\"center\":0,\"x\":[1,2],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 216, 1/4, 是, 9/16, 9/16, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(4, 3, 216, 1/4, 是, 9/16, 9/16, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163058,"parameters":{"coefficients":[9,17,8,16,9],"center":0,"x":[1,2],"order":3,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[9,1],[17,1],[8,1],[16,1],[9,1]],\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0059","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[10,2,11,3,12,10],\"center\":0,\"x\":[1,2],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 1200, 1/4, 是, 5/16, 5/16, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(5, 4, 1200, 1/4, 是, 5/16, 5/16, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163059,"parameters":{"coefficients":[10,2,11,3,12,10],"center":0,"x":[1,2],"order":4,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[10,1],[2,1],[11,1],[3,1],[12,1],[10,1]],\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0060","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[11,4,14,7,17,10,11],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 5, 7920, 1/4, 是, 11/64, 11/64, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(6, 5, 7920, 1/4, 是, 11/64, 11/64, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163060,"parameters":{"coefficients":[11,4,14,7,17,10,11],"center":0,"x":[1,2],"order":5,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[11,1],[4,1],[14,1],[7,1],[17,1],[10,1],[11,1]],\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0061","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[12,6,12],\"center\":0,\"x\":[1,2],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 24, 1/4, 是, 3, 3, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(2, 1, 24, 1/4, 是, 3, 3, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163061,"parameters":{"coefficients":[12,6,12],"center":0,"x":[1,2],"order":1,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[12,1],[6,1],[12,1]],\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0062","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[13,8,3,13],\"center\":0,\"x\":[1,2],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 78, 1/4, 是, 13/8, 13/8, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(3, 2, 78, 1/4, 是, 13/8, 13/8, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163062,"parameters":{"coefficients":[13,8,3,13],"center":0,"x":[1,2],"order":2,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[13,1],[8,1],[3,1],[13,1]],\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0063","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[14,10,6,2,14],\"center\":0,\"x\":[1,2],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 336, 1/4, 是, 7/8, 7/8, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(4, 3, 336, 1/4, 是, 7/8, 7/8, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163063,"parameters":{"coefficients":[14,10,6,2,14],"center":0,"x":[1,2],"order":3,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[14,1],[10,1],[6,1],[2,1],[14,1]],\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0064","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[15,12,9,6,3,15],\"center\":0,\"x\":[1,2],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 1800, 1/4, 是, 15/32, 15/32, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(5, 4, 1800, 1/4, 是, 15/32, 15/32, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163064,"parameters":{"coefficients":[15,12,9,6,3,15],"center":0,"x":[1,2],"order":4,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[15,1],[12,1],[9,1],[6,1],[3,1],[15,1]],\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0065","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[16,14,12,10,8,6,16],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 5, 11520, 1/4, 是, 1/4, 1/4, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(6, 5, 11520, 1/4, 是, 1/4, 1/4, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163065,"parameters":{"coefficients":[16,14,12,10,8,6,16],"center":0,"x":[1,2],"order":5,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[16,1],[14,1],[12,1],[10,1],[8,1],[6,1],[16,1]],\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0066","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[17,16,17],\"center\":0,\"x\":[1,2],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 34, 1/4, 是, 17/4, 17/4, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(2, 1, 34, 1/4, 是, 17/4, 17/4, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163066,"parameters":{"coefficients":[17,16,17],"center":0,"x":[1,2],"order":1,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[17,1],[16,1],[17,1]],\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0067","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[1,1,1,18],\"center\":0,\"x\":[1,2],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 108, 1/4, 是, 9/4, 9/4, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(3, 2, 108, 1/4, 是, 9/4, 9/4, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163067,"parameters":{"coefficients":[1,1,1,18],"center":0,"x":[1,2],"order":2,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[1,1],[1,1],[1,1],[18,1]],\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0068","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[2,3,4,5,19],\"center\":0,\"x\":[1,2],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 456, 1/4, 是, 19/16, 19/16, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(4, 3, 456, 1/4, 是, 19/16, 19/16, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163068,"parameters":{"coefficients":[2,3,4,5,19],"center":0,"x":[1,2],"order":3,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[2,1],[3,1],[4,1],[5,1],[19,1]],\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0069","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[3,5,7,9,11,20],\"center\":0,\"x\":[1,2],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 2400, 1/4, 是, 5/8, 5/8, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(5, 4, 2400, 1/4, 是, 5/8, 5/8, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163069,"parameters":{"coefficients":[3,5,7,9,11,20],"center":0,"x":[1,2],"order":4,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[3,1],[5,1],[7,1],[9,1],[11,1],[20,1]],\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0070","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[4,7,10,13,16,2,21],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 5, 15120, 1/4, 是, 21/64, 21/64, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(6, 5, 15120, 1/4, 是, 21/64, 21/64, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163070,"parameters":{"coefficients":[4,7,10,13,16,2,21],"center":0,"x":[1,2],"order":5,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[4,1],[7,1],[10,1],[13,1],[16,1],[2,1],[21,1]],\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0071","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[5,9,22],\"center\":0,\"x\":[1,2],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 44, 1/4, 是, 11/2, 11/2, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(2, 1, 44, 1/4, 是, 11/2, 11/2, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163071,"parameters":{"coefficients":[5,9,22],"center":0,"x":[1,2],"order":1,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[5,1],[9,1],[22,1]],\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0072","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[6,11,16,23],\"center\":0,\"x\":[1,2],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 138, 1/4, 是, 23/8, 23/8, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(3, 2, 138, 1/4, 是, 23/8, 23/8, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163072,"parameters":{"coefficients":[6,11,16,23],"center":0,"x":[1,2],"order":2,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[6,1],[11,1],[16,1],[23,1]],\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0073","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[7,13,2,8,24],\"center\":0,\"x\":[1,2],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 576, 1/4, 是, 3/2, 3/2, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(4, 3, 576, 1/4, 是, 3/2, 3/2, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163073,"parameters":{"coefficients":[7,13,2,8,24],"center":0,"x":[1,2],"order":3,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[7,1],[13,1],[2,1],[8,1],[24,1]],\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0074","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[8,15,5,12,2,25],\"center\":0,\"x\":[1,2],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 3000, 1/4, 是, 25/32, 25/32, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(5, 4, 3000, 1/4, 是, 25/32, 25/32, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163074,"parameters":{"coefficients":[8,15,5,12,2,25],"center":0,"x":[1,2],"order":4,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[8,1],[15,1],[5,1],[12,1],[2,1],[25,1]],\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0075","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-lagrange-witness；資料={\"coefficients\":[9,17,8,16,7,15,26],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 5, 18720, 1/4, 是, 13/32, 13/32, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)","explanation_zh":"Wave6第十層核對=(6, 5, 18720, 1/4, 是, 13/32, 13/32, 是, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-lagrange-witness","seed":202611163075,"parameters":{"coefficients":[9,17,8,16,7,15,26],"center":0,"x":[1,2],"order":5,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-lagrange-witness","semantic_template":"qf-mod47-taylor-lagrange-witness/{\"center\":[0,1],\"coefficients\":[[9,1],[17,1],[8,1],[16,1],[7,1],[15,1],[26,1]],\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0076","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"valid_bound\",\"coefficients\":[2,3,2],\"center\":0,\"x\":[1,2],\"order\":1,\"M\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(valid_bound, 1/2, 4, 4, 1/2, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(valid_bound, 1/2, 4, 4, 1/2, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163076,"parameters":{"coefficients":[2,3,2],"center":0,"x":[1,2],"order":1,"mode":"valid_bound","M":4,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[4,1],\"center\":[0,1],\"coefficients\":[[2,1],[3,1],[2,1]],\"mode\":\"valid_bound\",\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0077","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"undersized_M\",\"coefficients\":[3,5,7,3],\"center\":0,\"x\":[1,2],\"order\":2,\"M\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(undersized_M, 3/8, 18, 17, 17/48, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(undersized_M, 3/8, 18, 17, 17/48, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163077,"parameters":{"coefficients":[3,5,7,3],"center":0,"x":[1,2],"order":2,"mode":"undersized_M","M":17,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[17,1],\"center\":[0,1],\"coefficients\":[[3,1],[5,1],[7,1],[3,1]],\"mode\":\"undersized_M\",\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0078","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"valid_bound\",\"coefficients\":[4,7,10,13,4],\"center\":0,\"x\":[1,2],\"order\":3,\"M\":96}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(valid_bound, 1/4, 96, 96, 1/4, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(valid_bound, 1/4, 96, 96, 1/4, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163078,"parameters":{"coefficients":[4,7,10,13,4],"center":0,"x":[1,2],"order":3,"mode":"valid_bound","M":96,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[96,1],\"center\":[0,1],\"coefficients\":[[4,1],[7,1],[10,1],[13,1],[4,1]],\"mode\":\"valid_bound\",\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0079","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"undersized_M\",\"coefficients\":[5,9,13,17,4,5],\"center\":0,\"x\":[1,2],\"order\":4,\"M\":599}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(undersized_M, 5/32, 600, 599, 599/3840, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(undersized_M, 5/32, 600, 599, 599/3840, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163079,"parameters":{"coefficients":[5,9,13,17,4,5],"center":0,"x":[1,2],"order":4,"mode":"undersized_M","M":599,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[599,1],\"center\":[0,1],\"coefficients\":[[5,1],[9,1],[13,1],[17,1],[4,1],[5,1]],\"mode\":\"undersized_M\",\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0080","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"valid_bound\",\"coefficients\":[6,11,16,4,9,14,6],\"center\":0,\"x\":[1,2],\"order\":5,\"M\":4320}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(valid_bound, 3/32, 4320, 4320, 3/32, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(valid_bound, 3/32, 4320, 4320, 3/32, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163080,"parameters":{"coefficients":[6,11,16,4,9,14,6],"center":0,"x":[1,2],"order":5,"mode":"valid_bound","M":4320,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[4320,1],\"center\":[0,1],\"coefficients\":[[6,1],[11,1],[16,1],[4,1],[9,1],[14,1],[6,1]],\"mode\":\"valid_bound\",\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0081","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"undersized_M\",\"coefficients\":[7,13,7],\"center\":0,\"x\":[1,2],\"order\":1,\"M\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(undersized_M, 7/4, 14, 13, 13/8, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(undersized_M, 7/4, 14, 13, 13/8, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163081,"parameters":{"coefficients":[7,13,7],"center":0,"x":[1,2],"order":1,"mode":"undersized_M","M":13,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[13,1],\"center\":[0,1],\"coefficients\":[[7,1],[13,1],[7,1]],\"mode\":\"undersized_M\",\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0082","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"valid_bound\",\"coefficients\":[8,15,5,8],\"center\":0,\"x\":[1,2],\"order\":2,\"M\":48}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(valid_bound, 1, 48, 48, 1, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(valid_bound, 1, 48, 48, 1, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163082,"parameters":{"coefficients":[8,15,5,8],"center":0,"x":[1,2],"order":2,"mode":"valid_bound","M":48,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[48,1],\"center\":[0,1],\"coefficients\":[[8,1],[15,1],[5,1],[8,1]],\"mode\":\"valid_bound\",\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0083","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"undersized_M\",\"coefficients\":[9,17,8,16,9],\"center\":0,\"x\":[1,2],\"order\":3,\"M\":215}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(undersized_M, 9/16, 216, 215, 215/384, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(undersized_M, 9/16, 216, 215, 215/384, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163083,"parameters":{"coefficients":[9,17,8,16,9],"center":0,"x":[1,2],"order":3,"mode":"undersized_M","M":215,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[215,1],\"center\":[0,1],\"coefficients\":[[9,1],[17,1],[8,1],[16,1],[9,1]],\"mode\":\"undersized_M\",\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0084","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"valid_bound\",\"coefficients\":[10,2,11,3,12,10],\"center\":0,\"x\":[1,2],\"order\":4,\"M\":1200}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(valid_bound, 5/16, 1200, 1200, 5/16, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(valid_bound, 5/16, 1200, 1200, 5/16, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163084,"parameters":{"coefficients":[10,2,11,3,12,10],"center":0,"x":[1,2],"order":4,"mode":"valid_bound","M":1200,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[1200,1],\"center\":[0,1],\"coefficients\":[[10,1],[2,1],[11,1],[3,1],[12,1],[10,1]],\"mode\":\"valid_bound\",\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0085","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"undersized_M\",\"coefficients\":[11,4,14,7,17,10,11],\"center\":0,\"x\":[1,2],\"order\":5,\"M\":7919}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(undersized_M, 11/64, 7920, 7919, 7919/46080, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(undersized_M, 11/64, 7920, 7919, 7919/46080, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163085,"parameters":{"coefficients":[11,4,14,7,17,10,11],"center":0,"x":[1,2],"order":5,"mode":"undersized_M","M":7919,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[7919,1],\"center\":[0,1],\"coefficients\":[[11,1],[4,1],[14,1],[7,1],[17,1],[10,1],[11,1]],\"mode\":\"undersized_M\",\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0086","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"valid_bound\",\"coefficients\":[12,6,12],\"center\":0,\"x\":[1,2],\"order\":1,\"M\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(valid_bound, 3, 24, 24, 3, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(valid_bound, 3, 24, 24, 3, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163086,"parameters":{"coefficients":[12,6,12],"center":0,"x":[1,2],"order":1,"mode":"valid_bound","M":24,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[24,1],\"center\":[0,1],\"coefficients\":[[12,1],[6,1],[12,1]],\"mode\":\"valid_bound\",\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0087","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"undersized_M\",\"coefficients\":[13,8,3,13],\"center\":0,\"x\":[1,2],\"order\":2,\"M\":77}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(undersized_M, 13/8, 78, 77, 77/48, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(undersized_M, 13/8, 78, 77, 77/48, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163087,"parameters":{"coefficients":[13,8,3,13],"center":0,"x":[1,2],"order":2,"mode":"undersized_M","M":77,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[77,1],\"center\":[0,1],\"coefficients\":[[13,1],[8,1],[3,1],[13,1]],\"mode\":\"undersized_M\",\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0088","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"valid_bound\",\"coefficients\":[14,10,6,2,14],\"center\":0,\"x\":[1,2],\"order\":3,\"M\":336}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(valid_bound, 7/8, 336, 336, 7/8, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(valid_bound, 7/8, 336, 336, 7/8, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163088,"parameters":{"coefficients":[14,10,6,2,14],"center":0,"x":[1,2],"order":3,"mode":"valid_bound","M":336,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[336,1],\"center\":[0,1],\"coefficients\":[[14,1],[10,1],[6,1],[2,1],[14,1]],\"mode\":\"valid_bound\",\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0089","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"undersized_M\",\"coefficients\":[15,12,9,6,3,15],\"center\":0,\"x\":[1,2],\"order\":4,\"M\":1799}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(undersized_M, 15/32, 1800, 1799, 1799/3840, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(undersized_M, 15/32, 1800, 1799, 1799/3840, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163089,"parameters":{"coefficients":[15,12,9,6,3,15],"center":0,"x":[1,2],"order":4,"mode":"undersized_M","M":1799,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[1799,1],\"center\":[0,1],\"coefficients\":[[15,1],[12,1],[9,1],[6,1],[3,1],[15,1]],\"mode\":\"undersized_M\",\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0090","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"valid_bound\",\"coefficients\":[16,14,12,10,8,6,16],\"center\":0,\"x\":[1,2],\"order\":5,\"M\":11520}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(valid_bound, 1/4, 11520, 11520, 1/4, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(valid_bound, 1/4, 11520, 11520, 1/4, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163090,"parameters":{"coefficients":[16,14,12,10,8,6,16],"center":0,"x":[1,2],"order":5,"mode":"valid_bound","M":11520,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[11520,1],\"center\":[0,1],\"coefficients\":[[16,1],[14,1],[12,1],[10,1],[8,1],[6,1],[16,1]],\"mode\":\"valid_bound\",\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0091","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"undersized_M\",\"coefficients\":[17,16,17],\"center\":0,\"x\":[1,2],\"order\":1,\"M\":33}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(undersized_M, 17/4, 34, 33, 33/8, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(undersized_M, 17/4, 34, 33, 33/8, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163091,"parameters":{"coefficients":[17,16,17],"center":0,"x":[1,2],"order":1,"mode":"undersized_M","M":33,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[33,1],\"center\":[0,1],\"coefficients\":[[17,1],[16,1],[17,1]],\"mode\":\"undersized_M\",\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0092","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"valid_bound\",\"coefficients\":[1,1,1,18],\"center\":0,\"x\":[1,2],\"order\":2,\"M\":108}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(valid_bound, 9/4, 108, 108, 9/4, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(valid_bound, 9/4, 108, 108, 9/4, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163092,"parameters":{"coefficients":[1,1,1,18],"center":0,"x":[1,2],"order":2,"mode":"valid_bound","M":108,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[108,1],\"center\":[0,1],\"coefficients\":[[1,1],[1,1],[1,1],[18,1]],\"mode\":\"valid_bound\",\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0093","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"undersized_M\",\"coefficients\":[2,3,4,5,19],\"center\":0,\"x\":[1,2],\"order\":3,\"M\":455}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(undersized_M, 19/16, 456, 455, 455/384, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(undersized_M, 19/16, 456, 455, 455/384, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163093,"parameters":{"coefficients":[2,3,4,5,19],"center":0,"x":[1,2],"order":3,"mode":"undersized_M","M":455,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[455,1],\"center\":[0,1],\"coefficients\":[[2,1],[3,1],[4,1],[5,1],[19,1]],\"mode\":\"undersized_M\",\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0094","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"valid_bound\",\"coefficients\":[3,5,7,9,11,20],\"center\":0,\"x\":[1,2],\"order\":4,\"M\":2400}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(valid_bound, 5/8, 2400, 2400, 5/8, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(valid_bound, 5/8, 2400, 2400, 5/8, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163094,"parameters":{"coefficients":[3,5,7,9,11,20],"center":0,"x":[1,2],"order":4,"mode":"valid_bound","M":2400,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[2400,1],\"center\":[0,1],\"coefficients\":[[3,1],[5,1],[7,1],[9,1],[11,1],[20,1]],\"mode\":\"valid_bound\",\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0095","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"undersized_M\",\"coefficients\":[4,7,10,13,16,2,21],\"center\":0,\"x\":[1,2],\"order\":5,\"M\":15119}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(undersized_M, 21/64, 15120, 15119, 15119/46080, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(undersized_M, 21/64, 15120, 15119, 15119/46080, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163095,"parameters":{"coefficients":[4,7,10,13,16,2,21],"center":0,"x":[1,2],"order":5,"mode":"undersized_M","M":15119,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[15119,1],\"center\":[0,1],\"coefficients\":[[4,1],[7,1],[10,1],[13,1],[16,1],[2,1],[21,1]],\"mode\":\"undersized_M\",\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0096","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"valid_bound\",\"coefficients\":[5,9,22],\"center\":0,\"x\":[1,2],\"order\":1,\"M\":44}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(valid_bound, 11/2, 44, 44, 11/2, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(valid_bound, 11/2, 44, 44, 11/2, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163096,"parameters":{"coefficients":[5,9,22],"center":0,"x":[1,2],"order":1,"mode":"valid_bound","M":44,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[44,1],\"center\":[0,1],\"coefficients\":[[5,1],[9,1],[22,1]],\"mode\":\"valid_bound\",\"order\":1,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0097","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"undersized_M\",\"coefficients\":[6,11,16,23],\"center\":0,\"x\":[1,2],\"order\":2,\"M\":137}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(undersized_M, 23/8, 138, 137, 137/48, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(undersized_M, 23/8, 138, 137, 137/48, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163097,"parameters":{"coefficients":[6,11,16,23],"center":0,"x":[1,2],"order":2,"mode":"undersized_M","M":137,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[137,1],\"center\":[0,1],\"coefficients\":[[6,1],[11,1],[16,1],[23,1]],\"mode\":\"undersized_M\",\"order\":2,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0098","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"valid_bound\",\"coefficients\":[7,13,2,8,24],\"center\":0,\"x\":[1,2],\"order\":3,\"M\":576}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(valid_bound, 3/2, 576, 576, 3/2, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(valid_bound, 3/2, 576, 576, 3/2, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163098,"parameters":{"coefficients":[7,13,2,8,24],"center":0,"x":[1,2],"order":3,"mode":"valid_bound","M":576,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[576,1],\"center\":[0,1],\"coefficients\":[[7,1],[13,1],[2,1],[8,1],[24,1]],\"mode\":\"valid_bound\",\"order\":3,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0099","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"undersized_M\",\"coefficients\":[8,15,5,12,2,25],\"center\":0,\"x\":[1,2],\"order\":4,\"M\":2999}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(undersized_M, 25/32, 3000, 2999, 2999/3840, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(undersized_M, 25/32, 3000, 2999, 2999/3840, 否, 否, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163099,"parameters":{"coefficients":[8,15,5,12,2,25],"center":0,"x":[1,2],"order":4,"mode":"undersized_M","M":2999,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[2999,1],\"center\":[0,1],\"coefficients\":[[8,1],[15,1],[5,1],[12,1],[2,1],[25,1]],\"mode\":\"undersized_M\",\"order\":4,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r03-taylor-remainder-theorem-0100","batch_id":"batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-taylor-remainder-theorem"],"title_zh":"Taylor餘項定理","stem_zh":"center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-bound-boundary；資料={\"mode\":\"valid_bound\",\"coefficients\":[9,17,8,16,7,15,26],\"center\":0,\"x\":[1,2],\"order\":5,\"M\":18720}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(valid_bound, 13/32, 18720, 18720, 13/32, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)","explanation_zh":"Wave6第十層核對=(valid_bound, 13/32, 18720, 18720, 13/32, 是, 是, 否, exact_constant_n_plus_one_derivative_controls_the_bound_finite_jet_does_not_prove_general_theorem_or_analyticity)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Taylor餘項定理"]},"variant":{"family_id":"qf-mod47-taylor-bound-boundary","seed":202611163100,"parameters":{"coefficients":[9,17,8,16,7,15,26],"center":0,"x":[1,2],"order":5,"mode":"valid_bound","M":18720,"domain":"Exact_polynomial_taylor_remainder","operation":"taylor-bound-boundary","semantic_template":"qf-mod47-taylor-bound-boundary/{\"M\":[18720,1],\"center\":[0,1],\"coefficients\":[[9,1],[17,1],[8,1],[16,1],[7,1],[15,1],[26,1]],\"mode\":\"valid_bound\",\"order\":5,\"x\":[1,2]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}