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{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0014","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-order-classification；資料={\"derivative_orders\":[1,2,3]}。請依 [derivative_orders,order,includes_y] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3], 3, 否)","explanation_zh":"應用基礎核對=([1, 2, 3], 3, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-order-classification","seed":202610202014,"parameters":{"derivative_orders":[1,2,3],"domain":"Polynomial_candidate_ODE","operation":"ode-order-classification","semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0015","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-order-classification；資料={\"derivative_orders\":[0,1,2,3]}。請依 [derivative_orders,order,includes_y] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3], 3, 是)","explanation_zh":"應用基礎核對=([0, 1, 2, 3], 3, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-order-classification","seed":202610202015,"parameters":{"derivative_orders":[0,1,2,3],"domain":"Polynomial_candidate_ODE","operation":"ode-order-classification","semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0016","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-order-classification；資料={\"derivative_orders\":[4]}。請依 [derivative_orders,order,includes_y] 欄序回答。","answer":{"kind":"expression","value":"([4], 4, 否)","explanation_zh":"應用基礎核對=([4], 4, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-order-classification","seed":202610202016,"parameters":{"derivative_orders":[4],"domain":"Polynomial_candidate_ODE","operation":"ode-order-classification","semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0017","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-order-classification；資料={\"derivative_orders\":[0,4]}。請依 [derivative_orders,order,includes_y] 欄序回答。","answer":{"kind":"expression","value":"([0, 4], 4, 是)","explanation_zh":"應用基礎核對=([0, 4], 4, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-order-classification","seed":202610202017,"parameters":{"derivative_orders":[0,4],"domain":"Polynomial_candidate_ODE","operation":"ode-order-classification","semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0018","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-order-classification；資料={\"derivative_orders\":[1,4]}。請依 [derivative_orders,order,includes_y] 欄序回答。","answer":{"kind":"expression","value":"([1, 4], 4, 否)","explanation_zh":"應用基礎核對=([1, 4], 4, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-order-classification","seed":202610202018,"parameters":{"derivative_orders":[1,4],"domain":"Polynomial_candidate_ODE","operation":"ode-order-classification","semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0023","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-order-classification；資料={\"derivative_orders\":[0,1,2,4]}。請依 [derivative_orders,order,includes_y] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 4], 4, 是)","explanation_zh":"應用基礎核對=([0, 1, 2, 4], 4, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-order-classification","seed":202610202023,"parameters":{"derivative_orders":[0,1,2,4],"domain":"Polynomial_candidate_ODE","operation":"ode-order-classification","semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0,1,2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0024","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-order-classification；資料={\"derivative_orders\":[3,4]}。請依 [derivative_orders,order,includes_y] 欄序回答。","answer":{"kind":"expression","value":"([3, 4], 4, 否)","explanation_zh":"應用基礎核對=([3, 4], 4, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-order-classification","seed":202610202024,"parameters":{"derivative_orders":[3,4],"domain":"Polynomial_candidate_ODE","operation":"ode-order-classification","semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0025","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-order-classification；資料={\"derivative_orders\":[0,3,4]}。請依 [derivative_orders,order,includes_y] 欄序回答。","answer":{"kind":"expression","value":"([0, 3, 4], 4, 是)","explanation_zh":"應用基礎核對=([0, 3, 4], 4, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-order-classification","seed":202610202025,"parameters":{"derivative_orders":[0,3,4],"domain":"Polynomial_candidate_ODE","operation":"ode-order-classification","semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0026","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[-12,1,-3],\"p\":-3,\"q\":-20,\"x\":-3}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-12, 1, -3], -3, -42, 19, -3, -20, -87, 否)","explanation_zh":"應用基礎核對=([-12, 1, -3], -3, -42, 19, -3, -20, -87, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202026,"parameters":{"coefficients":[-12,1,-3],"p":-3,"q":-20,"x":-3,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-12,1,-3],\"p\":-3,\"q\":-20,\"x\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0027","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[-11,2,-1],\"p\":-2,\"q\":-18,\"x\":-2}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-11, 2, -1], -2, -19, 6, -2, -18, -14, 否)","explanation_zh":"應用基礎核對=([-11, 2, -1], -2, -19, 6, -2, -18, -14, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202027,"parameters":{"coefficients":[-11,2,-1],"p":-2,"q":-18,"x":-2,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-11,2,-1],\"p\":-2,\"q\":-18,\"x\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0028","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[-10,3,1],\"p\":-1,\"q\":-16,\"x\":-1}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-10, 3, 1], -1, -12, 1, -1, -16, 5, 否)","explanation_zh":"應用基礎核對=([-10, 3, 1], -1, -12, 1, -1, -16, 5, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202028,"parameters":{"coefficients":[-10,3,1],"p":-1,"q":-16,"x":-1,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-10,3,1],\"p\":-1,\"q\":-16,\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0029","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[-9,4,3],\"p\":0,\"q\":-14,\"x\":0}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-9, 4, 3], 0, -9, 4, 0, -14, 18, 否)","explanation_zh":"應用基礎核對=([-9, 4, 3], 0, -9, 4, 0, -14, 18, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202029,"parameters":{"coefficients":[-9,4,3],"p":0,"q":-14,"x":0,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-9,4,3],\"p\":0,\"q\":-14,\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0030","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[-8,5,-2],\"p\":1,\"q\":-12,\"x\":1}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-8, 5, -2], 1, -5, 1, 1, -12, 18, 否)","explanation_zh":"應用基礎核對=([-8, 5, -2], 1, -5, 1, 1, -12, 18, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202030,"parameters":{"coefficients":[-8,5,-2],"p":1,"q":-12,"x":1,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-8,5,-2],\"p\":1,\"q\":-12,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0031","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[-7,1,0],\"p\":2,\"q\":-10,\"x\":2}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-7, 1, 0], 2, -5, 1, 2, -10, 21, 否)","explanation_zh":"應用基礎核對=([-7, 1, 0], 2, -5, 1, 2, -10, 21, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202031,"parameters":{"coefficients":[-7,1,0],"p":2,"q":-10,"x":2,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-7,1,0],\"p\":2,\"q\":-10,\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0032","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[-6,2,2],\"p\":3,\"q\":-8,\"x\":3}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-6, 2, 2], 3, 18, 14, 3, -8, -32, 否)","explanation_zh":"應用基礎核對=([-6, 2, 2], 3, 18, 14, 3, -8, -32, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202032,"parameters":{"coefficients":[-6,2,2],"p":3,"q":-8,"x":3,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-6,2,2],\"p\":3,\"q\":-8,\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0033","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[-5,3,-3],\"p\":-3,\"q\":-6,\"x\":-3}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-5, 3, -3], -3, -41, 21, -3, -6, -96, 否)","explanation_zh":"應用基礎核對=([-5, 3, -3], -3, -41, 21, -3, -6, -96, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202033,"parameters":{"coefficients":[-5,3,-3],"p":-3,"q":-6,"x":-3,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-5,3,-3],\"p\":-3,\"q\":-6,\"x\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0034","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[-4,4,-1],\"p\":-2,\"q\":-4,\"x\":-2}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-4, 4, -1], -2, -16, 8, -2, -4, -20, 否)","explanation_zh":"應用基礎核對=([-4, 4, -1], -2, -16, 8, -2, -4, -20, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202034,"parameters":{"coefficients":[-4,4,-1],"p":-2,"q":-4,"x":-2,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-4,4,-1],\"p\":-2,\"q\":-4,\"x\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0035","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[-3,5,1],\"p\":-1,\"q\":-2,\"x\":-1}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-3, 5, 1], -1, -7, 3, -1, -2, -2, 否)","explanation_zh":"應用基礎核對=([-3, 5, 1], -1, -7, 3, -1, -2, -2, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202035,"parameters":{"coefficients":[-3,5,1],"p":-1,"q":-2,"x":-1,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-3,5,1],\"p\":-1,\"q\":-2,\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0036","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[-2,1,3],\"p\":0,\"q\":0,\"x\":0}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-2, 1, 3], 0, -2, 1, 0, 0, 1, 否)","explanation_zh":"應用基礎核對=([-2, 1, 3], 0, -2, 1, 0, 0, 1, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202036,"parameters":{"coefficients":[-2,1,3],"p":0,"q":0,"x":0,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-2,1,3],\"p\":0,\"q\":0,\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0037","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[-1,2,-2],\"p\":1,\"q\":2,\"x\":1}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-1, 2, -2], 1, -1, -2, 1, 2, -3, 否)","explanation_zh":"應用基礎核對=([-1, 2, -2], 1, -1, -2, 1, 2, -3, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202037,"parameters":{"coefficients":[-1,2,-2],"p":1,"q":2,"x":1,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-1,2,-2],\"p\":1,\"q\":2,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0038","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[0,3,0],\"p\":2,\"q\":4,\"x\":2}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([0, 3, 0], 2, 6, 3, 2, 4, -13, 否)","explanation_zh":"應用基礎核對=([0, 3, 0], 2, 6, 3, 2, 4, -13, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202038,"parameters":{"coefficients":[0,3,0],"p":2,"q":4,"x":2,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[0,3,0],\"p\":2,\"q\":4,\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0039","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[1,4,2],\"p\":3,\"q\":6,\"x\":3}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([1, 4, 2], 3, 31, 16, 3, 6, -83, 否)","explanation_zh":"應用基礎核對=([1, 4, 2], 3, 31, 16, 3, 6, -83, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202039,"parameters":{"coefficients":[1,4,2],"p":3,"q":6,"x":3,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[1,4,2],\"p\":3,\"q\":6,\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0040","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[2,5,-3],\"p\":-3,\"q\":8,\"x\":-3}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([2, 5, -3], -3, -40, 23, -3, 8, -105, 否)","explanation_zh":"應用基礎核對=([2, 5, -3], -3, -40, 23, -3, 8, -105, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202040,"parameters":{"coefficients":[2,5,-3],"p":-3,"q":8,"x":-3,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[2,5,-3],\"p\":-3,\"q\":8,\"x\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0041","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[3,1,-1],\"p\":-2,\"q\":10,\"x\":-2}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([3, 1, -1], -2, -3, 5, -2, 10, -11, 否)","explanation_zh":"應用基礎核對=([3, 1, -1], -2, -3, 5, -2, 10, -11, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202041,"parameters":{"coefficients":[3,1,-1],"p":-2,"q":10,"x":-2,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[3,1,-1],\"p\":-2,\"q\":10,\"x\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0042","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[4,2,1],\"p\":-1,\"q\":12,\"x\":-1}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([4, 2, 1], -1, 3, 0, -1, 12, -9, 否)","explanation_zh":"應用基礎核對=([4, 2, 1], -1, 3, 0, -1, 12, -9, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202042,"parameters":{"coefficients":[4,2,1],"p":-1,"q":12,"x":-1,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[4,2,1],\"p\":-1,\"q\":12,\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0043","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[5,3,3],\"p\":0,\"q\":14,\"x\":0}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([5, 3, 3], 0, 5, 3, 0, 14, -11, 否)","explanation_zh":"應用基礎核對=([5, 3, 3], 0, 5, 3, 0, 14, -11, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202043,"parameters":{"coefficients":[5,3,3],"p":0,"q":14,"x":0,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[5,3,3],\"p\":0,\"q\":14,\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0044","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[6,4,-2],\"p\":1,\"q\":16,\"x\":1}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([6, 4, -2], 1, 8, 0, 1, 16, -24, 否)","explanation_zh":"應用基礎核對=([6, 4, -2], 1, 8, 0, 1, 16, -24, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202044,"parameters":{"coefficients":[6,4,-2],"p":1,"q":16,"x":1,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[6,4,-2],\"p\":1,\"q\":16,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0045","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[7,5,0],\"p\":2,\"q\":18,\"x\":2}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([7, 5, 0], 2, 17, 5, 2, 18, -47, 否)","explanation_zh":"應用基礎核對=([7, 5, 0], 2, 17, 5, 2, 18, -47, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202045,"parameters":{"coefficients":[7,5,0],"p":2,"q":18,"x":2,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[7,5,0],\"p\":2,\"q\":18,\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0046","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[8,1,2],\"p\":3,\"q\":20,\"x\":3}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([8, 1, 2], 3, 29, 13, 3, 20, -94, 否)","explanation_zh":"應用基礎核對=([8, 1, 2], 3, 29, 13, 3, 20, -94, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202046,"parameters":{"coefficients":[8,1,2],"p":3,"q":20,"x":3,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[8,1,2],\"p\":3,\"q\":20,\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0047","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[9,2,-3],\"p\":-3,\"q\":22,\"x\":-3}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([9, 2, -3], -3, -24, 20, -3, 22, -74, 否)","explanation_zh":"應用基礎核對=([9, 2, -3], -3, -24, 20, -3, 22, -74, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202047,"parameters":{"coefficients":[9,2,-3],"p":-3,"q":22,"x":-3,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[9,2,-3],\"p\":-3,\"q\":22,\"x\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0048","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[10,3,-1],\"p\":-2,\"q\":24,\"x\":-2}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([10, 3, -1], -2, 0, 7, -2, 24, -17, 否)","explanation_zh":"應用基礎核對=([10, 3, -1], -2, 0, 7, -2, 24, -17, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202048,"parameters":{"coefficients":[10,3,-1],"p":-2,"q":24,"x":-2,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[10,3,-1],\"p\":-2,\"q\":24,\"x\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0049","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[11,4,1],\"p\":-1,\"q\":26,\"x\":-1}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([11, 4, 1], -1, 8, 2, -1, 26, -16, 否)","explanation_zh":"應用基礎核對=([11, 4, 1], -1, 8, 2, -1, 26, -16, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202049,"parameters":{"coefficients":[11,4,1],"p":-1,"q":26,"x":-1,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[11,4,1],\"p\":-1,\"q\":26,\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0050","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[12,5,3],\"p\":0,\"q\":28,\"x\":0}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([12, 5, 3], 0, 12, 5, 0, 28, -23, 否)","explanation_zh":"應用基礎核對=([12, 5, 3], 0, 12, 5, 0, 28, -23, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-first-order-residual","seed":202610202050,"parameters":{"coefficients":[12,5,3],"p":0,"q":28,"x":0,"domain":"Polynomial_candidate_ODE","operation":"ode-first-order-residual","semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[12,5,3],\"p\":0,\"q\":28,\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0051","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[-8,-2,-3,0],\"a\":1,\"b\":-3,\"c\":-4,\"rhs\":-20,\"x\":-2}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-8, -2, -3, 0], -2, -16, 10, -6, 1, -3, -4, -20, 48, 否)","explanation_zh":"應用基礎核對=([-8, -2, -3, 0], -2, -16, 10, -6, 1, -3, -4, -20, 48, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202051,"parameters":{"coefficients":[-8,-2,-3,0],"a":1,"b":-3,"c":-4,"rhs":-20,"x":-2,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":1,\"b\":-3,\"c\":-4,\"coefficients\":[-8,-2,-3,0],\"rhs\":-20,\"x\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0052","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[-7,-1,0,1],\"a\":2,\"b\":-2,\"c\":-3,\"rhs\":-17,\"x\":-1}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-7, -1, 0, 1], -1, -7, 2, -6, 2, -2, -3, -17, 22, 否)","explanation_zh":"應用基礎核對=([-7, -1, 0, 1], -1, -7, 2, -6, 2, -2, -3, -17, 22, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202052,"parameters":{"coefficients":[-7,-1,0,1],"a":2,"b":-2,"c":-3,"rhs":-17,"x":-1,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":2,\"b\":-2,\"c\":-3,\"coefficients\":[-7,-1,0,1],\"rhs\":-17,\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0053","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[-6,0,3,2],\"a\":3,\"b\":-1,\"c\":-2,\"rhs\":-14,\"x\":0}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-6, 0, 3, 2], 0, -6, 0, 6, 3, -1, -2, -14, 44, 否)","explanation_zh":"應用基礎核對=([-6, 0, 3, 2], 0, -6, 0, 6, 3, -1, -2, -14, 44, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202053,"parameters":{"coefficients":[-6,0,3,2],"a":3,"b":-1,"c":-2,"rhs":-14,"x":0,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":3,\"b\":-1,\"c\":-2,\"coefficients\":[-6,0,3,2],\"rhs\":-14,\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0054","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[-5,1,-1,0],\"a\":4,\"b\":0,\"c\":-1,\"rhs\":-11,\"x\":1}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-5, 1, -1, 0], 1, -5, -1, -2, 4, 0, -1, -11, 8, 否)","explanation_zh":"應用基礎核對=([-5, 1, -1, 0], 1, -5, -1, -2, 4, 0, -1, -11, 8, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202054,"parameters":{"coefficients":[-5,1,-1,0],"a":4,"b":0,"c":-1,"rhs":-11,"x":1,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":4,\"b\":0,\"c\":-1,\"coefficients\":[-5,1,-1,0],\"rhs\":-11,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0055","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[-4,2,2,1],\"a\":5,\"b\":1,\"c\":0,\"rhs\":-8,\"x\":2}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-4, 2, 2, 1], 2, 16, 22, 16, 5, 1, 0, -8, 110, 否)","explanation_zh":"應用基礎核對=([-4, 2, 2, 1], 2, 16, 22, 16, 5, 1, 0, -8, 110, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202055,"parameters":{"coefficients":[-4,2,2,1],"a":5,"b":1,"c":0,"rhs":-8,"x":2,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":5,\"b\":1,\"c\":0,\"coefficients\":[-4,2,2,1],\"rhs\":-8,\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0056","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[-3,-2,-2,2],\"a\":1,\"b\":2,\"c\":1,\"rhs\":-5,\"x\":-2}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-3, -2, -2, 2], -2, -23, 30, -28, 1, 2, 1, -5, 14, 否)","explanation_zh":"應用基礎核對=([-3, -2, -2, 2], -2, -23, 30, -28, 1, 2, 1, -5, 14, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202056,"parameters":{"coefficients":[-3,-2,-2,2],"a":1,"b":2,"c":1,"rhs":-5,"x":-2,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":1,\"b\":2,\"c\":1,\"coefficients\":[-3,-2,-2,2],\"rhs\":-5,\"x\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0057","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[-2,-1,1,0],\"a\":2,\"b\":3,\"c\":2,\"rhs\":-2,\"x\":-1}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-2, -1, 1, 0], -1, 0, -3, 2, 2, 3, 2, -2, -3, 否)","explanation_zh":"應用基礎核對=([-2, -1, 1, 0], -1, 0, -3, 2, 2, 3, 2, -2, -3, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202057,"parameters":{"coefficients":[-2,-1,1,0],"a":2,"b":3,"c":2,"rhs":-2,"x":-1,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":2,\"b\":3,\"c\":2,\"coefficients\":[-2,-1,1,0],\"rhs\":-2,\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0058","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[-1,0,-3,1],\"a\":3,\"b\":-3,\"c\":3,\"rhs\":1,\"x\":0}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([-1, 0, -3, 1], 0, -1, 0, -6, 3, -3, 3, 1, -22, 否)","explanation_zh":"應用基礎核對=([-1, 0, -3, 1], 0, -1, 0, -6, 3, -3, 3, 1, -22, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202058,"parameters":{"coefficients":[-1,0,-3,1],"a":3,"b":-3,"c":3,"rhs":1,"x":0,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":3,\"b\":-3,\"c\":3,\"coefficients\":[-1,0,-3,1],\"rhs\":1,\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0059","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[0,1,0,2],\"a\":4,\"b\":-2,\"c\":4,\"rhs\":4,\"x\":1}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 2], 1, 3, 7, 12, 4, -2, 4, 4, 42, 否)","explanation_zh":"應用基礎核對=([0, 1, 0, 2], 1, 3, 7, 12, 4, -2, 4, 4, 42, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202059,"parameters":{"coefficients":[0,1,0,2],"a":4,"b":-2,"c":4,"rhs":4,"x":1,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":4,\"b\":-2,\"c\":4,\"coefficients\":[0,1,0,2],\"rhs\":4,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0060","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[1,2,3,0],\"a\":5,\"b\":-1,\"c\":-4,\"rhs\":7,\"x\":2}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 3, 0], 2, 17, 14, 6, 5, -1, -4, 7, -59, 否)","explanation_zh":"應用基礎核對=([1, 2, 3, 0], 2, 17, 14, 6, 5, -1, -4, 7, -59, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202060,"parameters":{"coefficients":[1,2,3,0],"a":5,"b":-1,"c":-4,"rhs":7,"x":2,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":5,\"b\":-1,\"c\":-4,\"coefficients\":[1,2,3,0],\"rhs\":7,\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0061","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[2,-2,-1,1],\"a\":1,\"b\":0,\"c\":-3,\"rhs\":10,\"x\":-2}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([2, -2, -1, 1], -2, -6, 14, -14, 1, 0, -3, 10, -6, 否)","explanation_zh":"應用基礎核對=([2, -2, -1, 1], -2, -6, 14, -14, 1, 0, -3, 10, -6, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202061,"parameters":{"coefficients":[2,-2,-1,1],"a":1,"b":0,"c":-3,"rhs":10,"x":-2,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":1,\"b\":0,\"c\":-3,\"coefficients\":[2,-2,-1,1],\"rhs\":10,\"x\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0062","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[3,-1,2,2],\"a\":2,\"b\":1,\"c\":-2,\"rhs\":13,\"x\":-1}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([3, -1, 2, 2], -1, 4, 1, -8, 2, 1, -2, 13, -36, 否)","explanation_zh":"應用基礎核對=([3, -1, 2, 2], -1, 4, 1, -8, 2, 1, -2, 13, -36, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202062,"parameters":{"coefficients":[3,-1,2,2],"a":2,"b":1,"c":-2,"rhs":13,"x":-1,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":2,\"b\":1,\"c\":-2,\"coefficients\":[3,-1,2,2],\"rhs\":13,\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0063","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[4,0,-2,0],\"a\":3,\"b\":2,\"c\":-1,\"rhs\":16,\"x\":0}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([4, 0, -2, 0], 0, 4, 0, -4, 3, 2, -1, 16, -32, 否)","explanation_zh":"應用基礎核對=([4, 0, -2, 0], 0, 4, 0, -4, 3, 2, -1, 16, -32, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202063,"parameters":{"coefficients":[4,0,-2,0],"a":3,"b":2,"c":-1,"rhs":16,"x":0,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":3,\"b\":2,\"c\":-1,\"coefficients\":[4,0,-2,0],\"rhs\":16,\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0064","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[5,1,1,1],\"a\":4,\"b\":3,\"c\":0,\"rhs\":19,\"x\":1}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([5, 1, 1, 1], 1, 8, 6, 8, 4, 3, 0, 19, 31, 否)","explanation_zh":"應用基礎核對=([5, 1, 1, 1], 1, 8, 6, 8, 4, 3, 0, 19, 31, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202064,"parameters":{"coefficients":[5,1,1,1],"a":4,"b":3,"c":0,"rhs":19,"x":1,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":4,\"b\":3,\"c\":0,\"coefficients\":[5,1,1,1],\"rhs\":19,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0065","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[6,2,-3,2],\"a\":5,\"b\":-3,\"c\":1,\"rhs\":22,\"x\":2}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([6, 2, -3, 2], 2, 14, 14, 18, 5, -3, 1, 22, 40, 否)","explanation_zh":"應用基礎核對=([6, 2, -3, 2], 2, 14, 14, 18, 5, -3, 1, 22, 40, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202065,"parameters":{"coefficients":[6,2,-3,2],"a":5,"b":-3,"c":1,"rhs":22,"x":2,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":5,\"b\":-3,\"c\":1,\"coefficients\":[6,2,-3,2],\"rhs\":22,\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0066","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[7,-2,0,0],\"a\":1,\"b\":-2,\"c\":2,\"rhs\":25,\"x\":-2}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([7, -2, 0, 0], -2, 11, -2, 0, 1, -2, 2, 25, 1, 否)","explanation_zh":"應用基礎核對=([7, -2, 0, 0], -2, 11, -2, 0, 1, -2, 2, 25, 1, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202066,"parameters":{"coefficients":[7,-2,0,0],"a":1,"b":-2,"c":2,"rhs":25,"x":-2,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":1,\"b\":-2,\"c\":2,\"coefficients\":[7,-2,0,0],\"rhs\":25,\"x\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0067","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[8,-1,3,1],\"a\":2,\"b\":-1,\"c\":3,\"rhs\":28,\"x\":-1}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([8, -1, 3, 1], -1, 11, -4, 0, 2, -1, 3, 28, 9, 否)","explanation_zh":"應用基礎核對=([8, -1, 3, 1], -1, 11, -4, 0, 2, -1, 3, 28, 9, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202067,"parameters":{"coefficients":[8,-1,3,1],"a":2,"b":-1,"c":3,"rhs":28,"x":-1,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":2,\"b\":-1,\"c\":3,\"coefficients\":[8,-1,3,1],\"rhs\":28,\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0068","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[9,0,-1,2],\"a\":3,\"b\":0,\"c\":4,\"rhs\":31,\"x\":0}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([9, 0, -1, 2], 0, 9, 0, -2, 3, 0, 4, 31, -1, 否)","explanation_zh":"應用基礎核對=([9, 0, -1, 2], 0, 9, 0, -2, 3, 0, 4, 31, -1, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202068,"parameters":{"coefficients":[9,0,-1,2],"a":3,"b":0,"c":4,"rhs":31,"x":0,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":3,\"b\":0,\"c\":4,\"coefficients\":[9,0,-1,2],\"rhs\":31,\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0069","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[10,1,2,0],\"a\":4,\"b\":1,\"c\":-4,\"rhs\":34,\"x\":1}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([10, 1, 2, 0], 1, 13, 5, 4, 4, 1, -4, 34, -65, 否)","explanation_zh":"應用基礎核對=([10, 1, 2, 0], 1, 13, 5, 4, 4, 1, -4, 34, -65, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202069,"parameters":{"coefficients":[10,1,2,0],"a":4,"b":1,"c":-4,"rhs":34,"x":1,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":4,\"b\":1,\"c\":-4,\"coefficients\":[10,1,2,0],\"rhs\":34,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0070","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[11,2,-2,1],\"a\":5,\"b\":2,\"c\":-3,\"rhs\":37,\"x\":2}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([11, 2, -2, 1], 2, 15, 6, 8, 5, 2, -3, 37, -30, 否)","explanation_zh":"應用基礎核對=([11, 2, -2, 1], 2, 15, 6, 8, 5, 2, -3, 37, -30, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202070,"parameters":{"coefficients":[11,2,-2,1],"a":5,"b":2,"c":-3,"rhs":37,"x":2,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":5,\"b\":2,\"c\":-3,\"coefficients\":[11,2,-2,1],\"rhs\":37,\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0071","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[12,-2,1,2],\"a\":1,\"b\":3,\"c\":-2,\"rhs\":40,\"x\":-2}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([12, -2, 1, 2], -2, 4, 18, -22, 1, 3, -2, 40, -16, 否)","explanation_zh":"應用基礎核對=([12, -2, 1, 2], -2, 4, 18, -22, 1, 3, -2, 40, -16, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202071,"parameters":{"coefficients":[12,-2,1,2],"a":1,"b":3,"c":-2,"rhs":40,"x":-2,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":1,\"b\":3,\"c\":-2,\"coefficients\":[12,-2,1,2],\"rhs\":40,\"x\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0072","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[13,-1,-3,0],\"a\":2,\"b\":-3,\"c\":-1,\"rhs\":43,\"x\":-1}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([13, -1, -3, 0], -1, 11, 5, -6, 2, -3, -1, 43, -81, 否)","explanation_zh":"應用基礎核對=([13, -1, -3, 0], -1, 11, 5, -6, 2, -3, -1, 43, -81, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202072,"parameters":{"coefficients":[13,-1,-3,0],"a":2,"b":-3,"c":-1,"rhs":43,"x":-1,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":2,\"b\":-3,\"c\":-1,\"coefficients\":[13,-1,-3,0],\"rhs\":43,\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0073","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[14,0,0,1],\"a\":3,\"b\":-2,\"c\":0,\"rhs\":46,\"x\":0}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([14, 0, 0, 1], 0, 14, 0, 0, 3, -2, 0, 46, -46, 否)","explanation_zh":"應用基礎核對=([14, 0, 0, 1], 0, 14, 0, 0, 3, -2, 0, 46, -46, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202073,"parameters":{"coefficients":[14,0,0,1],"a":3,"b":-2,"c":0,"rhs":46,"x":0,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":3,\"b\":-2,\"c\":0,\"coefficients\":[14,0,0,1],\"rhs\":46,\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0074","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[15,1,3,2],\"a\":4,\"b\":-1,\"c\":1,\"rhs\":49,\"x\":1}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([15, 1, 3, 2], 1, 21, 13, 18, 4, -1, 1, 49, 31, 否)","explanation_zh":"應用基礎核對=([15, 1, 3, 2], 1, 21, 13, 18, 4, -1, 1, 49, 31, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202074,"parameters":{"coefficients":[15,1,3,2],"a":4,"b":-1,"c":1,"rhs":49,"x":1,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":4,\"b\":-1,\"c\":1,\"coefficients\":[15,1,3,2],\"rhs\":49,\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0075","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-second-order-residual；資料={\"coefficients\":[16,2,-1,0],\"a\":5,\"b\":0,\"c\":2,\"rhs\":52,\"x\":2}。請依 [coefficients,x,y,y_prime,y_second,a,b,c,rhs,residual,is_solution_at_x] 欄序回答。","answer":{"kind":"expression","value":"([16, 2, -1, 0], 2, 16, -2, -2, 5, 0, 2, 52, -30, 否)","explanation_zh":"應用基礎核對=([16, 2, -1, 0], 2, 16, -2, -2, 5, 0, 2, 52, -30, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-second-order-residual","seed":202610202075,"parameters":{"coefficients":[16,2,-1,0],"a":5,"b":0,"c":2,"rhs":52,"x":2,"domain":"Polynomial_candidate_ODE","operation":"ode-second-order-residual","semantic_template":"qf-app20-ode-second-order-residual/{\"a\":5,\"b\":0,\"c\":2,\"coefficients\":[16,2,-1,0],\"rhs\":52,\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0076","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[-9,-3,-2],\"x0\":-2,\"y0\":-10,\"dy0\":4}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([-9, -3, -2], -2, -11, 5, -10, 4, 否, 否, 否)","explanation_zh":"應用基礎核對=([-9, -3, -2], -2, -11, 5, -10, 4, 否, 否, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202076,"parameters":{"coefficients":[-9,-3,-2],"x0":-2,"y0":-10,"dy0":4,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-9,-3,-2],\"dy0\":4,\"x0\":-2,\"y0\":-10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0077","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[-8,-2,0],\"x0\":-1,\"y0\":-6,\"dy0\":-2}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([-8, -2, 0], -1, -6, -2, -6, -2, 是, 是, 是)","explanation_zh":"應用基礎核對=([-8, -2, 0], -1, -6, -2, -6, -2, 是, 是, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202077,"parameters":{"coefficients":[-8,-2,0],"x0":-1,"y0":-6,"dy0":-2,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-8,-2,0],\"dy0\":-2,\"x0\":-1,\"y0\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0078","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[-7,-1,2],\"x0\":0,\"y0\":-7,\"dy0\":-1}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([-7, -1, 2], 0, -7, -1, -7, -1, 是, 是, 是)","explanation_zh":"應用基礎核對=([-7, -1, 2], 0, -7, -1, -7, -1, 是, 是, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202078,"parameters":{"coefficients":[-7,-1,2],"x0":0,"y0":-7,"dy0":-1,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-7,-1,2],\"dy0\":-1,\"x0\":0,\"y0\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0079","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[-6,0,-1],\"x0\":1,\"y0\":-6,\"dy0\":-2}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([-6, 0, -1], 1, -7, -2, -6, -2, 否, 是, 否)","explanation_zh":"應用基礎核對=([-6, 0, -1], 1, -7, -2, -6, -2, 否, 是, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202079,"parameters":{"coefficients":[-6,0,-1],"x0":1,"y0":-6,"dy0":-2,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-6,0,-1],\"dy0\":-2,\"x0\":1,\"y0\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0080","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[-5,1,1],\"x0\":2,\"y0\":1,\"dy0\":4}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([-5, 1, 1], 2, 1, 5, 1, 4, 是, 否, 否)","explanation_zh":"應用基礎核對=([-5, 1, 1], 2, 1, 5, 1, 4, 是, 否, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202080,"parameters":{"coefficients":[-5,1,1],"x0":2,"y0":1,"dy0":4,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-5,1,1],\"dy0\":4,\"x0\":2,\"y0\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0081","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[-4,2,-2],\"x0\":-2,\"y0\":-16,\"dy0\":10}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([-4, 2, -2], -2, -16, 10, -16, 10, 是, 是, 是)","explanation_zh":"應用基礎核對=([-4, 2, -2], -2, -16, 10, -16, 10, 是, 是, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202081,"parameters":{"coefficients":[-4,2,-2],"x0":-2,"y0":-16,"dy0":10,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-4,2,-2],\"dy0\":10,\"x0\":-2,\"y0\":-16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0082","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[-3,3,0],\"x0\":-1,\"y0\":-5,\"dy0\":3}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([-3, 3, 0], -1, -6, 3, -5, 3, 否, 是, 否)","explanation_zh":"應用基礎核對=([-3, 3, 0], -1, -6, 3, -5, 3, 否, 是, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202082,"parameters":{"coefficients":[-3,3,0],"x0":-1,"y0":-5,"dy0":3,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-3,3,0],\"dy0\":3,\"x0\":-1,\"y0\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0083","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[-2,-3,2],\"x0\":0,\"y0\":-2,\"dy0\":-3}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([-2, -3, 2], 0, -2, -3, -2, -3, 是, 是, 是)","explanation_zh":"應用基礎核對=([-2, -3, 2], 0, -2, -3, -2, -3, 是, 是, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202083,"parameters":{"coefficients":[-2,-3,2],"x0":0,"y0":-2,"dy0":-3,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-2,-3,2],\"dy0\":-3,\"x0\":0,\"y0\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0084","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[-1,-2,-1],\"x0\":1,\"y0\":-4,\"dy0\":-5}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([-1, -2, -1], 1, -4, -4, -4, -5, 是, 否, 否)","explanation_zh":"應用基礎核對=([-1, -2, -1], 1, -4, -4, -4, -5, 是, 否, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202084,"parameters":{"coefficients":[-1,-2,-1],"x0":1,"y0":-4,"dy0":-5,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-1,-2,-1],\"dy0\":-5,\"x0\":1,\"y0\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0085","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[0,-1,1],\"x0\":2,\"y0\":3,\"dy0\":3}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([0, -1, 1], 2, 2, 3, 3, 3, 否, 是, 否)","explanation_zh":"應用基礎核對=([0, -1, 1], 2, 2, 3, 3, 3, 否, 是, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202085,"parameters":{"coefficients":[0,-1,1],"x0":2,"y0":3,"dy0":3,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[0,-1,1],\"dy0\":3,\"x0\":2,\"y0\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0086","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[1,0,-2],\"x0\":-2,\"y0\":-7,\"dy0\":8}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, -2], -2, -7, 8, -7, 8, 是, 是, 是)","explanation_zh":"應用基礎核對=([1, 0, -2], -2, -7, 8, -7, 8, 是, 是, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202086,"parameters":{"coefficients":[1,0,-2],"x0":-2,"y0":-7,"dy0":8,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[1,0,-2],\"dy0\":8,\"x0\":-2,\"y0\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0087","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[2,1,0],\"x0\":-1,\"y0\":1,\"dy0\":1}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 0], -1, 1, 1, 1, 1, 是, 是, 是)","explanation_zh":"應用基礎核對=([2, 1, 0], -1, 1, 1, 1, 1, 是, 是, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202087,"parameters":{"coefficients":[2,1,0],"x0":-1,"y0":1,"dy0":1,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[2,1,0],\"dy0\":1,\"x0\":-1,\"y0\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0088","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[3,2,2],\"x0\":0,\"y0\":4,\"dy0\":1}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([3, 2, 2], 0, 3, 2, 4, 1, 否, 否, 否)","explanation_zh":"應用基礎核對=([3, 2, 2], 0, 3, 2, 4, 1, 否, 否, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202088,"parameters":{"coefficients":[3,2,2],"x0":0,"y0":4,"dy0":1,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[3,2,2],\"dy0\":1,\"x0\":0,\"y0\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0089","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[4,3,-1],\"x0\":1,\"y0\":6,\"dy0\":1}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([4, 3, -1], 1, 6, 1, 6, 1, 是, 是, 是)","explanation_zh":"應用基礎核對=([4, 3, -1], 1, 6, 1, 6, 1, 是, 是, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202089,"parameters":{"coefficients":[4,3,-1],"x0":1,"y0":6,"dy0":1,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[4,3,-1],\"dy0\":1,\"x0\":1,\"y0\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0090","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[5,-3,1],\"x0\":2,\"y0\":3,\"dy0\":1}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([5, -3, 1], 2, 3, 1, 3, 1, 是, 是, 是)","explanation_zh":"應用基礎核對=([5, -3, 1], 2, 3, 1, 3, 1, 是, 是, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202090,"parameters":{"coefficients":[5,-3,1],"x0":2,"y0":3,"dy0":1,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[5,-3,1],\"dy0\":1,\"x0\":2,\"y0\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0091","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[6,-2,-2],\"x0\":-2,\"y0\":3,\"dy0\":6}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([6, -2, -2], -2, 2, 6, 3, 6, 否, 是, 否)","explanation_zh":"應用基礎核對=([6, -2, -2], -2, 2, 6, 3, 6, 否, 是, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202091,"parameters":{"coefficients":[6,-2,-2],"x0":-2,"y0":3,"dy0":6,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[6,-2,-2],\"dy0\":6,\"x0\":-2,\"y0\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0092","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[7,-1,0],\"x0\":-1,\"y0\":8,\"dy0\":-2}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([7, -1, 0], -1, 8, -1, 8, -2, 是, 否, 否)","explanation_zh":"應用基礎核對=([7, -1, 0], -1, 8, -1, 8, -2, 是, 否, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202092,"parameters":{"coefficients":[7,-1,0],"x0":-1,"y0":8,"dy0":-2,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[7,-1,0],\"dy0\":-2,\"x0\":-1,\"y0\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0093","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[8,0,2],\"x0\":0,\"y0\":8,\"dy0\":0}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([8, 0, 2], 0, 8, 0, 8, 0, 是, 是, 是)","explanation_zh":"應用基礎核對=([8, 0, 2], 0, 8, 0, 8, 0, 是, 是, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202093,"parameters":{"coefficients":[8,0,2],"x0":0,"y0":8,"dy0":0,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[8,0,2],\"dy0\":0,\"x0\":0,\"y0\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0094","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[9,1,-1],\"x0\":1,\"y0\":10,\"dy0\":-1}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([9, 1, -1], 1, 9, -1, 10, -1, 否, 是, 否)","explanation_zh":"應用基礎核對=([9, 1, -1], 1, 9, -1, 10, -1, 否, 是, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202094,"parameters":{"coefficients":[9,1,-1],"x0":1,"y0":10,"dy0":-1,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[9,1,-1],\"dy0\":-1,\"x0\":1,\"y0\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0095","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[10,2,1],\"x0\":2,\"y0\":18,\"dy0\":6}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([10, 2, 1], 2, 18, 6, 18, 6, 是, 是, 是)","explanation_zh":"應用基礎核對=([10, 2, 1], 2, 18, 6, 18, 6, 是, 是, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202095,"parameters":{"coefficients":[10,2,1],"x0":2,"y0":18,"dy0":6,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[10,2,1],\"dy0\":6,\"x0\":2,\"y0\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0096","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[11,3,-2],\"x0\":-2,\"y0\":-3,\"dy0\":10}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([11, 3, -2], -2, -3, 11, -3, 10, 是, 否, 否)","explanation_zh":"應用基礎核對=([11, 3, -2], -2, -3, 11, -3, 10, 是, 否, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202096,"parameters":{"coefficients":[11,3,-2],"x0":-2,"y0":-3,"dy0":10,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[11,3,-2],\"dy0\":10,\"x0\":-2,\"y0\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0097","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[12,-3,0],\"x0\":-1,\"y0\":16,\"dy0\":-3}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([12, -3, 0], -1, 15, -3, 16, -3, 否, 是, 否)","explanation_zh":"應用基礎核對=([12, -3, 0], -1, 15, -3, 16, -3, 否, 是, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202097,"parameters":{"coefficients":[12,-3,0],"x0":-1,"y0":16,"dy0":-3,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[12,-3,0],\"dy0\":-3,\"x0\":-1,\"y0\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0098","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[13,-2,2],\"x0\":0,\"y0\":13,\"dy0\":-2}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([13, -2, 2], 0, 13, -2, 13, -2, 是, 是, 是)","explanation_zh":"應用基礎核對=([13, -2, 2], 0, 13, -2, 13, -2, 是, 是, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202098,"parameters":{"coefficients":[13,-2,2],"x0":0,"y0":13,"dy0":-2,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[13,-2,2],\"dy0\":-2,\"x0\":0,\"y0\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0099","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[14,-1,-1],\"x0\":1,\"y0\":12,\"dy0\":-3}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([14, -1, -1], 1, 12, -3, 12, -3, 是, 是, 是)","explanation_zh":"應用基礎核對=([14, -1, -1], 1, 12, -3, 12, -3, 是, 是, 是)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202099,"parameters":{"coefficients":[14,-1,-1],"x0":1,"y0":12,"dy0":-3,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[14,-1,-1],\"dy0\":-3,\"x0\":1,\"y0\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0100","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-ordinary-differential-equation"],"title_zh":"多項式 ODE 候選核對","stem_zh":"整係數多項式候選。操作=ode-initial-condition-audit；資料={\"coefficients\":[15,0,1],\"x0\":2,\"y0\":20,\"dy0\":3}。請依 [coefficients,x0,y,y_prime,expected_y,expected_y_prime,y_match,y_prime_match,both_match] 欄序回答。","answer":{"kind":"expression","value":"([15, 0, 1], 2, 19, 4, 20, 3, 否, 否, 否)","explanation_zh":"應用基礎核對=([15, 0, 1], 2, 19, 4, 20, 3, 否, 否, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式 ODE 候選核對"]},"variant":{"family_id":"qf-app20-ode-initial-condition-audit","seed":202610202100,"parameters":{"coefficients":[15,0,1],"x0":2,"y0":20,"dy0":3,"domain":"Polynomial_candidate_ODE","operation":"ode-initial-condition-audit","semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[15,0,1],\"dy0\":3,\"x0\":2,\"y0\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}