{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0001","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[1,2,3,4],\"order\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 是, f_zero_equals_f, 是)","explanation_zh":"Wave6第五層核對=(3, 0, 是, f_zero_equals_f, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112001,"parameters":{"coefficients":[1,2,3,4],"order":0,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[1,2,3,4],\"order\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0002","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[2,4,6,8,10],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(4, 1, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112002,"parameters":{"coefficients":[2,4,6,8,10],"order":1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[2,4,6,8,10],\"order\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0003","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[3,6,9,12,15,18],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 2, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(5, 2, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112003,"parameters":{"coefficients":[3,6,9,12,15,18],"order":2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[3,6,9,12,15,18],\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0004","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[4,8,12,16,20,24,28],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 3, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(6, 3, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112004,"parameters":{"coefficients":[4,8,12,16,20,24,28],"order":3,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[4,8,12,16,20,24,28],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0005","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[5,10,15,20],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(3, 4, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112005,"parameters":{"coefficients":[5,10,15,20],"order":4,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[5,10,15,20],\"order\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0006","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[6,12,18,24,30],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 5, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(4, 5, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112006,"parameters":{"coefficients":[6,12,18,24,30],"order":5,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[6,12,18,24,30],\"order\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0007","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[7,14,21,28,35,42],\"order\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 6, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(5, 6, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112007,"parameters":{"coefficients":[7,14,21,28,35,42],"order":6,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[7,14,21,28,35,42],\"order\":6}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0008","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[8,16,24,32,40,48,56],\"order\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 7, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(6, 7, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112008,"parameters":{"coefficients":[8,16,24,32,40,48,56],"order":7,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[8,16,24,32,40,48,56],\"order\":7}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0009","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[9,18,27,36],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(3, 3, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112009,"parameters":{"coefficients":[9,18,27,36],"order":3,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[9,18,27,36],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0010","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[10,20,30,40,50],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(4, 3, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112010,"parameters":{"coefficients":[10,20,30,40,50],"order":3,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[10,20,30,40,50],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0011","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[11,22,33,44,55,66],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 3, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(5, 3, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112011,"parameters":{"coefficients":[11,22,33,44,55,66],"order":3,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[11,22,33,44,55,66],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0012","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[12,24,36,48,60,72,84],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 3, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(6, 3, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112012,"parameters":{"coefficients":[12,24,36,48,60,72,84],"order":3,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[12,24,36,48,60,72,84],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0013","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[13,26,39,52],\"order\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(3, 2, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112013,"parameters":{"coefficients":[13,26,39,52],"order":2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[13,26,39,52],\"order\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0014","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[14,28,42,56,70],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(4, 1, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112014,"parameters":{"coefficients":[14,28,42,56,70],"order":1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[14,28,42,56,70],\"order\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0015","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[15,30,45,60,75,90],\"order\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 是, f_zero_equals_f, 是)","explanation_zh":"Wave6第五層核對=(5, 0, 是, f_zero_equals_f, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112015,"parameters":{"coefficients":[15,30,45,60,75,90],"order":0,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[15,30,45,60,75,90],\"order\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0016","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[16,32,48,64,80,96,112],\"order\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 7, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(6, 7, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112016,"parameters":{"coefficients":[16,32,48,64,80,96,112],"order":7,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[16,32,48,64,80,96,112],\"order\":7}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0017","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[17,34,51,68],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(3, 1, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112017,"parameters":{"coefficients":[17,34,51,68],"order":1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[17,34,51,68],\"order\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0018","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[18,36,54,72,90],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 5, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(4, 5, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112018,"parameters":{"coefficients":[18,36,54,72,90],"order":5,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[18,36,54,72,90],\"order\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0019","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[19,38,57,76,95,114],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(5, 4, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112019,"parameters":{"coefficients":[19,38,57,76,95,114],"order":4,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[19,38,57,76,95,114],\"order\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0020","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[20,40,60,80,100,120,140],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 3, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(6, 3, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112020,"parameters":{"coefficients":[20,40,60,80,100,120,140],"order":3,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[20,40,60,80,100,120,140],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0021","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[21,42,63,84],\"order\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 是, f_zero_equals_f, 是)","explanation_zh":"Wave6第五層核對=(3, 0, 是, f_zero_equals_f, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112021,"parameters":{"coefficients":[21,42,63,84],"order":0,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[21,42,63,84],\"order\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0022","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[22,44,66,88,110],\"order\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(4, 3, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112022,"parameters":{"coefficients":[22,44,66,88,110],"order":3,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[22,44,66,88,110],\"order\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0023","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[23,46,69,92,115,138],\"order\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 1, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(5, 1, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112023,"parameters":{"coefficients":[23,46,69,92,115,138],"order":1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[23,46,69,92,115,138],\"order\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0024","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[24,48,72,96,120,144,168],\"order\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 7, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(6, 7, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112024,"parameters":{"coefficients":[24,48,72,96,120,144,168],"order":7,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[24,48,72,96,120,144,168],\"order\":7}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0025","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={\"coefficients\":[25,50,75,100],\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 4, 是, recursive_derivative_chain_required, 是)","explanation_zh":"Wave6第五層核對=(3, 4, 是, recursive_derivative_chain_required, 是)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-order-qualification","seed":202611112025,"parameters":{"coefficients":[25,50,75,100],"order":4,"domain":"Exact_polynomial_higher_derivative","operation":"higher-order-qualification","semantic_template":"qf-mod42-higher-order-qualification/{\"coefficients\":[25,50,75,100],\"order\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0026","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[1,2,3,4],\"order\":0,\"point\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3, 4], 0, [1, 2, 3, 4], -2, -23, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([1, 2, 3, 4], 0, [1, 2, 3, 4], -2, -23, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112026,"parameters":{"coefficients":[1,2,3,4],"order":0,"point":-2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[1,2,3,4],\"order\":0,\"point\":-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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0027","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[2,4,6,8,10],\"order\":1,\"point\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 4, 6, 8, 10], 1, [4, 12, 24, 40], -1, -24, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([2, 4, 6, 8, 10], 1, [4, 12, 24, 40], -1, -24, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112027,"parameters":{"coefficients":[2,4,6,8,10],"order":1,"point":-1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[2,4,6,8,10],\"order\":1,\"point\":-1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0028","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[3,6,9,12,15,18],\"order\":2,\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 6, 9, 12, 15, 18], 2, [18, 72, 180, 360], 0, 18, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([3, 6, 9, 12, 15, 18], 2, [18, 72, 180, 360], 0, 18, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112028,"parameters":{"coefficients":[3,6,9,12,15,18],"order":2,"point":0,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[3,6,9,12,15,18],\"order\":2,\"point\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0029","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[4,8,12,16,20,24,28],\"order\":3,\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, 8, 12, 16, 20, 24, 28], 3, [96, 480, 1440, 3360], 1, 5376, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([4, 8, 12, 16, 20, 24, 28], 3, [96, 480, 1440, 3360], 1, 5376, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112029,"parameters":{"coefficients":[4,8,12,16,20,24,28],"order":3,"point":1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[4,8,12,16,20,24,28],\"order\":3,\"point\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0030","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[5,10,15,20],\"order\":4,\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5, 10, 15, 20], 4, [0], 2, 0, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([5, 10, 15, 20], 4, [0], 2, 0, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112030,"parameters":{"coefficients":[5,10,15,20],"order":4,"point":2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[5,10,15,20],\"order\":4,\"point\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0031","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[6,12,18,24,30],\"order\":5,\"point\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([6, 12, 18, 24, 30], 5, [0], -2, 0, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([6, 12, 18, 24, 30], 5, [0], -2, 0, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112031,"parameters":{"coefficients":[6,12,18,24,30],"order":5,"point":-2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[6,12,18,24,30],\"order\":5,\"point\":-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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0032","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[7,14,21,28,35,42],\"order\":6,\"point\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([7, 14, 21, 28, 35, 42], 6, [0], -1, 0, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([7, 14, 21, 28, 35, 42], 6, [0], -1, 0, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112032,"parameters":{"coefficients":[7,14,21,28,35,42],"order":6,"point":-1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[7,14,21,28,35,42],\"order\":6,\"point\":-1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0033","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[8,16,24,32,40,48,56],\"order\":7,\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8, 16, 24, 32, 40, 48, 56], 7, [0], 0, 0, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([8, 16, 24, 32, 40, 48, 56], 7, [0], 0, 0, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112033,"parameters":{"coefficients":[8,16,24,32,40,48,56],"order":7,"point":0,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[8,16,24,32,40,48,56],\"order\":7,\"point\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0034","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[9,18,27,36],\"order\":3,\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([9, 18, 27, 36], 3, [216], 1, 216, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([9, 18, 27, 36], 3, [216], 1, 216, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112034,"parameters":{"coefficients":[9,18,27,36],"order":3,"point":1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[9,18,27,36],\"order\":3,\"point\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0035","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[10,20,30,40,50],\"order\":3,\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([10, 20, 30, 40, 50], 3, [240, 1200], 2, 2640, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([10, 20, 30, 40, 50], 3, [240, 1200], 2, 2640, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112035,"parameters":{"coefficients":[10,20,30,40,50],"order":3,"point":2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[10,20,30,40,50],\"order\":3,\"point\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0036","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[11,22,33,44,55,66],\"order\":3,\"point\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([11, 22, 33, 44, 55, 66], 3, [264, 1320, 3960], -2, 13464, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([11, 22, 33, 44, 55, 66], 3, [264, 1320, 3960], -2, 13464, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112036,"parameters":{"coefficients":[11,22,33,44,55,66],"order":3,"point":-2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[11,22,33,44,55,66],\"order\":3,\"point\":-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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0037","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[12,24,36,48,60,72,84],\"order\":3,\"point\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([12, 24, 36, 48, 60, 72, 84], 3, [288, 1440, 4320, 10080], -1, -6912, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([12, 24, 36, 48, 60, 72, 84], 3, [288, 1440, 4320, 10080], -1, -6912, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112037,"parameters":{"coefficients":[12,24,36,48,60,72,84],"order":3,"point":-1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[12,24,36,48,60,72,84],\"order\":3,\"point\":-1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0038","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[13,26,39,52],\"order\":2,\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([13, 26, 39, 52], 2, [78, 312], 0, 78, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([13, 26, 39, 52], 2, [78, 312], 0, 78, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112038,"parameters":{"coefficients":[13,26,39,52],"order":2,"point":0,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[13,26,39,52],\"order\":2,\"point\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0039","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[14,28,42,56,70],\"order\":1,\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([14, 28, 42, 56, 70], 1, [28, 84, 168, 280], 1, 560, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([14, 28, 42, 56, 70], 1, [28, 84, 168, 280], 1, 560, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112039,"parameters":{"coefficients":[14,28,42,56,70],"order":1,"point":1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[14,28,42,56,70],\"order\":1,\"point\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0040","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[15,30,45,60,75,90],\"order\":0,\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([15, 30, 45, 60, 75, 90], 0, [15, 30, 45, 60, 75, 90], 2, 4815, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([15, 30, 45, 60, 75, 90], 0, [15, 30, 45, 60, 75, 90], 2, 4815, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112040,"parameters":{"coefficients":[15,30,45,60,75,90],"order":0,"point":2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[15,30,45,60,75,90],\"order\":0,\"point\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0041","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[16,32,48,64,80,96,112],\"order\":7,\"point\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([16, 32, 48, 64, 80, 96, 112], 7, [0], -2, 0, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([16, 32, 48, 64, 80, 96, 112], 7, [0], -2, 0, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112041,"parameters":{"coefficients":[16,32,48,64,80,96,112],"order":7,"point":-2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[16,32,48,64,80,96,112],\"order\":7,\"point\":-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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0042","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[17,34,51,68],\"order\":1,\"point\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([17, 34, 51, 68], 1, [34, 102, 204], -1, 136, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([17, 34, 51, 68], 1, [34, 102, 204], -1, 136, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112042,"parameters":{"coefficients":[17,34,51,68],"order":1,"point":-1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[17,34,51,68],\"order\":1,\"point\":-1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0043","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[18,36,54,72,90],\"order\":5,\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([18, 36, 54, 72, 90], 5, [0], 0, 0, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([18, 36, 54, 72, 90], 5, [0], 0, 0, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112043,"parameters":{"coefficients":[18,36,54,72,90],"order":5,"point":0,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[18,36,54,72,90],\"order\":5,\"point\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0044","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[19,38,57,76,95,114],\"order\":4,\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([19, 38, 57, 76, 95, 114], 4, [2280, 13680], 1, 15960, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([19, 38, 57, 76, 95, 114], 4, [2280, 13680], 1, 15960, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112044,"parameters":{"coefficients":[19,38,57,76,95,114],"order":4,"point":1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[19,38,57,76,95,114],\"order\":4,\"point\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0045","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[20,40,60,80,100,120,140],\"order\":3,\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([20, 40, 60, 80, 100, 120, 140], 3, [480, 2400, 7200, 16800], 2, 168480, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([20, 40, 60, 80, 100, 120, 140], 3, [480, 2400, 7200, 16800], 2, 168480, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112045,"parameters":{"coefficients":[20,40,60,80,100,120,140],"order":3,"point":2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[20,40,60,80,100,120,140],\"order\":3,\"point\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0046","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[21,42,63,84],\"order\":0,\"point\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([21, 42, 63, 84], 0, [21, 42, 63, 84], -2, -483, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([21, 42, 63, 84], 0, [21, 42, 63, 84], -2, -483, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112046,"parameters":{"coefficients":[21,42,63,84],"order":0,"point":-2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[21,42,63,84],\"order\":0,\"point\":-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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0047","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[22,44,66,88,110],\"order\":3,\"point\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([22, 44, 66, 88, 110], 3, [528, 2640], -1, -2112, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([22, 44, 66, 88, 110], 3, [528, 2640], -1, -2112, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112047,"parameters":{"coefficients":[22,44,66,88,110],"order":3,"point":-1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[22,44,66,88,110],\"order\":3,\"point\":-1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0048","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[23,46,69,92,115,138],\"order\":1,\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([23, 46, 69, 92, 115, 138], 1, [46, 138, 276, 460, 690], 0, 46, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([23, 46, 69, 92, 115, 138], 1, [46, 138, 276, 460, 690], 0, 46, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112048,"parameters":{"coefficients":[23,46,69,92,115,138],"order":1,"point":0,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[23,46,69,92,115,138],\"order\":1,\"point\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0049","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[24,48,72,96,120,144,168],\"order\":7,\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([24, 48, 72, 96, 120, 144, 168], 7, [0], 1, 0, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([24, 48, 72, 96, 120, 144, 168], 7, [0], 1, 0, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112049,"parameters":{"coefficients":[24,48,72,96,120,144,168],"order":7,"point":1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[24,48,72,96,120,144,168],\"order\":7,\"point\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0050","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-polynomial-nth-derivative；資料={\"coefficients\":[25,50,75,100],\"order\":4,\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([25, 50, 75, 100], 4, [0], 2, 0, falling_factorial_coefficients_exact)","explanation_zh":"Wave6第五層核對=([25, 50, 75, 100], 4, [0], 2, 0, falling_factorial_coefficients_exact)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-polynomial-nth-derivative","seed":202611112050,"parameters":{"coefficients":[25,50,75,100],"order":4,"point":2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-polynomial-nth-derivative","semantic_template":"qf-mod42-higher-polynomial-nth-derivative/{\"coefficients\":[25,50,75,100],\"order\":4,\"point\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0051","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[1,2,3,4],\"order\":4,\"point\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 4, [0], -2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(3, 4, [0], -2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112051,"parameters":{"coefficients":[1,2,3,4],"order":4,"point":-2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[1,2,3,4],\"order\":4,\"point\":-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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0052","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[2,4,6,8,10],\"order\":6,\"point\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 6, [0], -1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(4, 6, [0], -1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112052,"parameters":{"coefficients":[2,4,6,8,10],"order":6,"point":-1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[2,4,6,8,10],\"order\":6,\"point\":-1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0053","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[3,6,9,12,15,18],\"order\":8,\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 8, [0], 0, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(5, 8, [0], 0, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112053,"parameters":{"coefficients":[3,6,9,12,15,18],"order":8,"point":0,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[3,6,9,12,15,18],\"order\":8,\"point\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0054","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[4,8,12,16,20,24,28],\"order\":7,\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 7, [0], 1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(6, 7, [0], 1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112054,"parameters":{"coefficients":[4,8,12,16,20,24,28],"order":7,"point":1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[4,8,12,16,20,24,28],\"order\":7,\"point\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0055","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[5,10,15,20],\"order\":5,\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, [0], 2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(3, 5, [0], 2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112055,"parameters":{"coefficients":[5,10,15,20],"order":5,"point":2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[5,10,15,20],\"order\":5,\"point\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0056","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[6,12,18,24,30],\"order\":7,\"point\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 7, [0], -2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(4, 7, [0], -2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112056,"parameters":{"coefficients":[6,12,18,24,30],"order":7,"point":-2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[6,12,18,24,30],\"order\":7,\"point\":-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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0057","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[7,14,21,28,35,42],\"order\":6,\"point\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 6, [0], -1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(5, 6, [0], -1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112057,"parameters":{"coefficients":[7,14,21,28,35,42],"order":6,"point":-1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[7,14,21,28,35,42],\"order\":6,\"point\":-1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0058","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[8,16,24,32,40,48,56],\"order\":8,\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 8, [0], 0, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(6, 8, [0], 0, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112058,"parameters":{"coefficients":[8,16,24,32,40,48,56],"order":8,"point":0,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[8,16,24,32,40,48,56],\"order\":8,\"point\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0059","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[9,18,27,36],\"order\":6,\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 6, [0], 1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(3, 6, [0], 1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112059,"parameters":{"coefficients":[9,18,27,36],"order":6,"point":1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[9,18,27,36],\"order\":6,\"point\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0060","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[10,20,30,40,50],\"order\":5,\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 5, [0], 2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(4, 5, [0], 2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112060,"parameters":{"coefficients":[10,20,30,40,50],"order":5,"point":2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[10,20,30,40,50],\"order\":5,\"point\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0061","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[11,22,33,44,55,66],\"order\":7,\"point\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 7, [0], -2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(5, 7, [0], -2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112061,"parameters":{"coefficients":[11,22,33,44,55,66],"order":7,"point":-2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[11,22,33,44,55,66],\"order\":7,\"point\":-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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0062","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[12,24,36,48,60,72,84],\"order\":9,\"point\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 9, [0], -1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(6, 9, [0], -1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112062,"parameters":{"coefficients":[12,24,36,48,60,72,84],"order":9,"point":-1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[12,24,36,48,60,72,84],\"order\":9,\"point\":-1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0063","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[13,26,39,52],\"order\":4,\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 4, [0], 0, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(3, 4, [0], 0, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112063,"parameters":{"coefficients":[13,26,39,52],"order":4,"point":0,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[13,26,39,52],\"order\":4,\"point\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0064","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[14,28,42,56,70],\"order\":6,\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 6, [0], 1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(4, 6, [0], 1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112064,"parameters":{"coefficients":[14,28,42,56,70],"order":6,"point":1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[14,28,42,56,70],\"order\":6,\"point\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0065","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[15,30,45,60,75,90],\"order\":8,\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 8, [0], 2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(5, 8, [0], 2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112065,"parameters":{"coefficients":[15,30,45,60,75,90],"order":8,"point":2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[15,30,45,60,75,90],\"order\":8,\"point\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0066","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[16,32,48,64,80,96,112],\"order\":7,\"point\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 7, [0], -2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(6, 7, [0], -2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112066,"parameters":{"coefficients":[16,32,48,64,80,96,112],"order":7,"point":-2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[16,32,48,64,80,96,112],\"order\":7,\"point\":-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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0067","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[17,34,51,68],\"order\":5,\"point\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, [0], -1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(3, 5, [0], -1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112067,"parameters":{"coefficients":[17,34,51,68],"order":5,"point":-1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[17,34,51,68],\"order\":5,\"point\":-1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0068","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[18,36,54,72,90],\"order\":7,\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 7, [0], 0, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(4, 7, [0], 0, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112068,"parameters":{"coefficients":[18,36,54,72,90],"order":7,"point":0,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[18,36,54,72,90],\"order\":7,\"point\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0069","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[19,38,57,76,95,114],\"order\":6,\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 6, [0], 1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(5, 6, [0], 1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112069,"parameters":{"coefficients":[19,38,57,76,95,114],"order":6,"point":1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[19,38,57,76,95,114],\"order\":6,\"point\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0070","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[20,40,60,80,100,120,140],\"order\":8,\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 8, [0], 2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(6, 8, [0], 2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112070,"parameters":{"coefficients":[20,40,60,80,100,120,140],"order":8,"point":2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[20,40,60,80,100,120,140],\"order\":8,\"point\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0071","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[21,42,63,84],\"order\":6,\"point\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 6, [0], -2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(3, 6, [0], -2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112071,"parameters":{"coefficients":[21,42,63,84],"order":6,"point":-2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[21,42,63,84],\"order\":6,\"point\":-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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0072","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[22,44,66,88,110],\"order\":5,\"point\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 5, [0], -1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(4, 5, [0], -1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112072,"parameters":{"coefficients":[22,44,66,88,110],"order":5,"point":-1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[22,44,66,88,110],\"order\":5,\"point\":-1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0073","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[23,46,69,92,115,138],\"order\":7,\"point\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 7, [0], 0, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(5, 7, [0], 0, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112073,"parameters":{"coefficients":[23,46,69,92,115,138],"order":7,"point":0,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[23,46,69,92,115,138],\"order\":7,\"point\":0}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0074","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[24,48,72,96,120,144,168],\"order\":9,\"point\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 9, [0], 1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(6, 9, [0], 1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112074,"parameters":{"coefficients":[24,48,72,96,120,144,168],"order":9,"point":1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[24,48,72,96,120,144,168],\"order\":9,\"point\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0075","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-zero-beyond-degree；資料={\"coefficients\":[25,50,75,100],\"order\":4,\"point\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 4, [0], 2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)","explanation_zh":"Wave6第五層核對=(3, 4, [0], 2, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-zero-beyond-degree","seed":202611112075,"parameters":{"coefficients":[25,50,75,100],"order":4,"point":2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-zero-beyond-degree","semantic_template":"qf-mod42-higher-zero-beyond-degree/{\"coefficients\":[25,50,75,100],\"order\":4,\"point\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0076","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(1, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112076,"parameters":{"scale":1,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":1}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0077","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(2, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112077,"parameters":{"scale":2,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0078","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(3, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112078,"parameters":{"scale":3,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0079","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(4, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112079,"parameters":{"scale":4,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0080","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(5, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112080,"parameters":{"scale":5,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0081","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(6, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112081,"parameters":{"scale":6,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":6}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0082","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(7, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112082,"parameters":{"scale":7,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":7}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0083","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(8, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112083,"parameters":{"scale":8,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0084","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(9, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112084,"parameters":{"scale":9,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":9}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0085","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(10, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112085,"parameters":{"scale":10,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":10}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0086","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(11, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112086,"parameters":{"scale":11,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":11}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0087","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(12, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112087,"parameters":{"scale":12,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":12}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0088","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(13, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112088,"parameters":{"scale":13,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":13}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0089","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(14, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112089,"parameters":{"scale":14,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":14}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0090","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(15, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112090,"parameters":{"scale":15,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0091","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(16, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112091,"parameters":{"scale":16,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":16}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0092","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(17, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112092,"parameters":{"scale":17,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":17}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0093","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(18, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112093,"parameters":{"scale":18,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":18}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0094","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(19, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112094,"parameters":{"scale":19,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":19}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0095","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(20, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112095,"parameters":{"scale":20,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":20}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0096","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(21, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112096,"parameters":{"scale":21,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":21}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0097","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(22, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112097,"parameters":{"scale":22,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":22}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0098","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(23, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112098,"parameters":{"scale":23,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":23}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0099","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(24, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112099,"parameters":{"scale":24,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r02-higher-order-derivative-0100","batch_id":"batch-20261014-advance-20261111-r02-higher-order-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-higher-order-derivative"],"title_zh":"高階導數","stem_zh":"f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-missing-next-order-boundary；資料={\"scale\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)","explanation_zh":"Wave6第五層核對=(25, f(x)=scale*x*abs(x), 0, 是, 0, 否, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)；依完整明示資料獨立重算；polynomial nth derivative使用falling-factorial coefficients且order超過degree後為zero；存在第n階不推出第n+1階，也不推出analytic或Taylor series convergence。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["高階導數"]},"variant":{"family_id":"qf-mod42-higher-missing-next-order-boundary","seed":202611112100,"parameters":{"scale":25,"domain":"Exact_polynomial_higher_derivative","operation":"higher-missing-next-order-boundary","semantic_template":"qf-mod42-higher-missing-next-order-boundary/{\"scale\":25}"}},"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}