{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r06-partial-differential-equation-0001","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[-12,-2,-3,1,-1,-2],\"operator\":[1,-2],\"source\":-12,\"x\":-2,\"y\":-3}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -2], [1, -2], -2, -3, -3, 11, -12, -13, 否)","explanation_zh":"依賴結構核對=([-12, -2, -3, 1, -1, -2], [1, -2], -2, -3, -3, 11, -12, -13, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246001,"parameters":{"coefficients":[-12,-2,-3,1,-1,-2],"operator":[1,-2],"source":-12,"x":-2,"y":-3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[-12,-2,-3,1,-1,-2],\"operator\":[1,-2],\"source\":-12,\"x\":-2,\"y\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0002","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[-11,-1,-2,2,0,-1],\"operator\":[2,-1],\"source\":-11,\"x\":-1,\"y\":-2}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-11, -1, -2, 2, 0, -1], [2, -1], -1, -2, -5, 2, -11, -1, 否)","explanation_zh":"依賴結構核對=([-11, -1, -2, 2, 0, -1], [2, -1], -1, -2, -5, 2, -11, -1, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246002,"parameters":{"coefficients":[-11,-1,-2,2,0,-1],"operator":[2,-1],"source":-11,"x":-1,"y":-2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[-11,-1,-2,2,0,-1],\"operator\":[2,-1],\"source\":-11,\"x\":-1,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0003","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[-10,0,-1,3,1,0],\"operator\":[3,0],\"source\":-10,\"x\":0,\"y\":-1}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0, -1, 3, 1, 0], [3, 0], 0, -1, -1, -1, -10, 7, 否)","explanation_zh":"依賴結構核對=([-10, 0, -1, 3, 1, 0], [3, 0], 0, -1, -1, -1, -10, 7, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246003,"parameters":{"coefficients":[-10,0,-1,3,1,0],"operator":[3,0],"source":-10,"x":0,"y":-1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[-10,0,-1,3,1,0],\"operator\":[3,0],\"source\":-10,\"x\":0,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0004","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[-9,1,0,4,-1,1],\"operator\":[1,1],\"source\":-9,\"x\":1,\"y\":0}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1, 0, 4, -1, 1], [1, 1], 1, 0, 9, -1, -9, 17, 否)","explanation_zh":"依賴結構核對=([-9, 1, 0, 4, -1, 1], [1, 1], 1, 0, 9, -1, -9, 17, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246004,"parameters":{"coefficients":[-9,1,0,4,-1,1],"operator":[1,1],"source":-9,"x":1,"y":0,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[-9,1,0,4,-1,1],\"operator\":[1,1],\"source\":-9,\"x\":1,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0005","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[-8,2,1,1,0,2],\"operator\":[2,2],\"source\":-8,\"x\":2,\"y\":1}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, 2], [2, 2], 2, 1, 6, 5, -8, 30, 否)","explanation_zh":"依賴結構核對=([-8, 2, 1, 1, 0, 2], [2, 2], 2, 1, 6, 5, -8, 30, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246005,"parameters":{"coefficients":[-8,2,1,1,0,2],"operator":[2,2],"source":-8,"x":2,"y":1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[-8,2,1,1,0,2],\"operator\":[2,2],\"source\":-8,\"x\":2,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0006","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[-7,-2,2,2,1,-2],\"operator\":[3,-2],\"source\":-7,\"x\":-2,\"y\":2}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-7, -2, 2, 2, 1, -2], [3, -2], -2, 2, -8, -8, -7, -1, 否)","explanation_zh":"依賴結構核對=([-7, -2, 2, 2, 1, -2], [3, -2], -2, 2, -8, -8, -7, -1, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246006,"parameters":{"coefficients":[-7,-2,2,2,1,-2],"operator":[3,-2],"source":-7,"x":-2,"y":2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[-7,-2,2,2,1,-2],\"operator\":[3,-2],\"source\":-7,\"x\":-2,\"y\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0007","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[-6,-1,3,3,-1,-1],\"operator\":[1,-1],\"source\":-6,\"x\":-1,\"y\":3}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-6, -1, 3, 3, -1, -1], [1, -1], -1, 3, -10, -2, -6, -2, 否)","explanation_zh":"依賴結構核對=([-6, -1, 3, 3, -1, -1], [1, -1], -1, 3, -10, -2, -6, -2, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246007,"parameters":{"coefficients":[-6,-1,3,3,-1,-1],"operator":[1,-1],"source":-6,"x":-1,"y":3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[-6,-1,3,3,-1,-1],\"operator\":[1,-1],\"source\":-6,\"x\":-1,\"y\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0008","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[-5,0,-3,4,0,0],\"operator\":[2,0],\"source\":-5,\"x\":0,\"y\":-3}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-5, 0, -3, 4, 0, 0], [2, 0], 0, -3, 0, -3, -5, 5, 否)","explanation_zh":"依賴結構核對=([-5, 0, -3, 4, 0, 0], [2, 0], 0, -3, 0, -3, -5, 5, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246008,"parameters":{"coefficients":[-5,0,-3,4,0,0],"operator":[2,0],"source":-5,"x":0,"y":-3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[-5,0,-3,4,0,0],\"operator\":[2,0],\"source\":-5,\"x\":0,\"y\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0009","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[-4,1,-2,1,1,1],\"operator\":[3,1],\"source\":-4,\"x\":1,\"y\":-2}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, 1], [3, 1], 1, -2, 1, -5, -4, 2, 否)","explanation_zh":"依賴結構核對=([-4, 1, -2, 1, 1, 1], [3, 1], 1, -2, 1, -5, -4, 2, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246009,"parameters":{"coefficients":[-4,1,-2,1,1,1],"operator":[3,1],"source":-4,"x":1,"y":-2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[-4,1,-2,1,1,1],\"operator\":[3,1],\"source\":-4,\"x\":1,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0010","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[-3,2,-1,2,-1,2],\"operator\":[1,2],\"source\":-3,\"x\":2,\"y\":-1}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2, -1, 2, -1, 2], [1, 2], 2, -1, 11, -7, -3, 0, 是)","explanation_zh":"依賴結構核對=([-3, 2, -1, 2, -1, 2], [1, 2], 2, -1, 11, -7, -3, 0, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246010,"parameters":{"coefficients":[-3,2,-1,2,-1,2],"operator":[1,2],"source":-3,"x":2,"y":-1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[-3,2,-1,2,-1,2],\"operator\":[1,2],\"source\":-3,\"x\":2,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0011","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[-2,-2,0,3,0,-2],\"operator\":[2,-2],\"source\":-2,\"x\":-2,\"y\":0}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, 0, 3, 0, -2], [2, -2], -2, 0, -14, 0, -2, -26, 否)","explanation_zh":"依賴結構核對=([-2, -2, 0, 3, 0, -2], [2, -2], -2, 0, -14, 0, -2, -26, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246011,"parameters":{"coefficients":[-2,-2,0,3,0,-2],"operator":[2,-2],"source":-2,"x":-2,"y":0,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[-2,-2,0,3,0,-2],\"operator\":[2,-2],\"source\":-2,\"x\":-2,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0012","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[-1,-1,1,4,1,-1],\"operator\":[3,-1],\"source\":-1,\"x\":-1,\"y\":1}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1, 1, 4, 1, -1], [3, -1], -1, 1, -8, -2, -1, -21, 否)","explanation_zh":"依賴結構核對=([-1, -1, 1, 4, 1, -1], [3, -1], -1, 1, -8, -2, -1, -21, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246012,"parameters":{"coefficients":[-1,-1,1,4,1,-1],"operator":[3,-1],"source":-1,"x":-1,"y":1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[-1,-1,1,4,1,-1],\"operator\":[3,-1],\"source\":-1,\"x\":-1,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0013","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[0,0,2,1,-1,0],\"operator\":[1,0],\"source\":0,\"x\":0,\"y\":2}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, 0], [1, 0], 0, 2, -2, 2, 0, -2, 否)","explanation_zh":"依賴結構核對=([0, 0, 2, 1, -1, 0], [1, 0], 0, 2, -2, 2, 0, -2, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246013,"parameters":{"coefficients":[0,0,2,1,-1,0],"operator":[1,0],"source":0,"x":0,"y":2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[0,0,2,1,-1,0],\"operator\":[1,0],\"source\":0,\"x\":0,\"y\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0014","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[1,1,3,2,0,1],\"operator\":[2,1],\"source\":1,\"x\":1,\"y\":3}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3, 2, 0, 1], [2, 1], 1, 3, 5, 9, 1, 18, 否)","explanation_zh":"依賴結構核對=([1, 1, 3, 2, 0, 1], [2, 1], 1, 3, 5, 9, 1, 18, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246014,"parameters":{"coefficients":[1,1,3,2,0,1],"operator":[2,1],"source":1,"x":1,"y":3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[1,1,3,2,0,1],\"operator\":[2,1],\"source\":1,\"x\":1,\"y\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0015","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[2,2,-3,3,1,2],\"operator\":[3,2],\"source\":2,\"x\":2,\"y\":-3}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, -3, 3, 1, 2], [3, 2], 2, -3, 11, -13, 2, 5, 否)","explanation_zh":"依賴結構核對=([2, 2, -3, 3, 1, 2], [3, 2], 2, -3, 11, -13, 2, 5, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246015,"parameters":{"coefficients":[2,2,-3,3,1,2],"operator":[3,2],"source":2,"x":2,"y":-3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[2,2,-3,3,1,2],\"operator\":[3,2],\"source\":2,\"x\":2,\"y\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0016","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[3,-2,-2,4,-1,-2],\"operator\":[1,-2],\"source\":3,\"x\":-2,\"y\":-2}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([3, -2, -2, 4, -1, -2], [1, -2], -2, -2, -16, 8, 3, -35, 否)","explanation_zh":"依賴結構核對=([3, -2, -2, 4, -1, -2], [1, -2], -2, -2, -16, 8, 3, -35, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246016,"parameters":{"coefficients":[3,-2,-2,4,-1,-2],"operator":[1,-2],"source":3,"x":-2,"y":-2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[3,-2,-2,4,-1,-2],\"operator\":[1,-2],\"source\":3,\"x\":-2,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0017","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[4,-1,-1,1,0,-1],\"operator\":[2,-1],\"source\":4,\"x\":-1,\"y\":-1}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], [2, -1], -1, -1, -3, 1, 4, -11, 否)","explanation_zh":"依賴結構核對=([4, -1, -1, 1, 0, -1], [2, -1], -1, -1, -3, 1, 4, -11, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246017,"parameters":{"coefficients":[4,-1,-1,1,0,-1],"operator":[2,-1],"source":4,"x":-1,"y":-1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[4,-1,-1,1,0,-1],\"operator\":[2,-1],\"source\":4,\"x\":-1,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0018","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[5,0,0,2,1,0],\"operator\":[3,0],\"source\":5,\"x\":0,\"y\":0}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([5, 0, 0, 2, 1, 0], [3, 0], 0, 0, 0, 0, 5, -5, 否)","explanation_zh":"依賴結構核對=([5, 0, 0, 2, 1, 0], [3, 0], 0, 0, 0, 0, 5, -5, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246018,"parameters":{"coefficients":[5,0,0,2,1,0],"operator":[3,0],"source":5,"x":0,"y":0,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[5,0,0,2,1,0],\"operator\":[3,0],\"source\":5,\"x\":0,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0019","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[6,1,1,3,-1,1],\"operator\":[1,1],\"source\":6,\"x\":1,\"y\":1}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([6, 1, 1, 3, -1, 1], [1, 1], 1, 1, 6, 2, 6, 2, 否)","explanation_zh":"依賴結構核對=([6, 1, 1, 3, -1, 1], [1, 1], 1, 1, 6, 2, 6, 2, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246019,"parameters":{"coefficients":[6,1,1,3,-1,1],"operator":[1,1],"source":6,"x":1,"y":1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[6,1,1,3,-1,1],\"operator\":[1,1],\"source\":6,\"x\":1,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0020","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[7,2,2,4,0,2],\"operator\":[2,2],\"source\":7,\"x\":2,\"y\":2}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([7, 2, 2, 4, 0, 2], [2, 2], 2, 2, 18, 10, 7, 49, 否)","explanation_zh":"依賴結構核對=([7, 2, 2, 4, 0, 2], [2, 2], 2, 2, 18, 10, 7, 49, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246020,"parameters":{"coefficients":[7,2,2,4,0,2],"operator":[2,2],"source":7,"x":2,"y":2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[7,2,2,4,0,2],\"operator\":[2,2],\"source\":7,\"x\":2,\"y\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0021","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[8,-2,3,1,1,-2],\"operator\":[3,-2],\"source\":8,\"x\":-2,\"y\":3}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -2], [3, -2], -2, 3, -3, -11, 8, 5, 否)","explanation_zh":"依賴結構核對=([8, -2, 3, 1, 1, -2], [3, -2], -2, 3, -3, -11, 8, 5, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246021,"parameters":{"coefficients":[8,-2,3,1,1,-2],"operator":[3,-2],"source":8,"x":-2,"y":3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[8,-2,3,1,1,-2],\"operator\":[3,-2],\"source\":8,\"x\":-2,\"y\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0022","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[9,-1,-3,2,-1,-1],\"operator\":[1,-1],\"source\":9,\"x\":-1,\"y\":-3}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([9, -1, -3, 2, -1, -1], [1, -1], -1, -3, -2, 4, 9, -15, 否)","explanation_zh":"依賴結構核對=([9, -1, -3, 2, -1, -1], [1, -1], -1, -3, -2, 4, 9, -15, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246022,"parameters":{"coefficients":[9,-1,-3,2,-1,-1],"operator":[1,-1],"source":9,"x":-1,"y":-3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[9,-1,-3,2,-1,-1],\"operator\":[1,-1],\"source\":9,\"x\":-1,\"y\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0023","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[10,0,-2,3,0,0],\"operator\":[2,0],\"source\":10,\"x\":0,\"y\":-2}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([10, 0, -2, 3, 0, 0], [2, 0], 0, -2, 0, -2, 10, -10, 否)","explanation_zh":"依賴結構核對=([10, 0, -2, 3, 0, 0], [2, 0], 0, -2, 0, -2, 10, -10, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246023,"parameters":{"coefficients":[10,0,-2,3,0,0],"operator":[2,0],"source":10,"x":0,"y":-2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[10,0,-2,3,0,0],\"operator\":[2,0],\"source\":10,\"x\":0,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0024","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[11,1,-1,4,1,1],\"operator\":[3,1],\"source\":11,\"x\":1,\"y\":-1}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([11, 1, -1, 4, 1, 1], [3, 1], 1, -1, 8, -2, 11, 11, 否)","explanation_zh":"依賴結構核對=([11, 1, -1, 4, 1, 1], [3, 1], 1, -1, 8, -2, 11, 11, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246024,"parameters":{"coefficients":[11,1,-1,4,1,1],"operator":[3,1],"source":11,"x":1,"y":-1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[11,1,-1,4,1,1],\"operator\":[3,1],\"source\":11,\"x\":1,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0025","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-first-order-point-residual；資料={\"coefficients\":[12,2,0,1,-1,2],\"operator\":[1,2],\"source\":12,\"x\":2,\"y\":0}。請依 [coefficients,operator,x,y,u_x,u_y,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, 2], [1, 2], 2, 0, 6, -2, 12, -10, 否)","explanation_zh":"依賴結構核對=([12, 2, 0, 1, -1, 2], [1, 2], 2, 0, 6, -2, 12, -10, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-first-order-point-residual","seed":202610246025,"parameters":{"coefficients":[12,2,0,1,-1,2],"operator":[1,2],"source":12,"x":2,"y":0,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-first-order-point-residual","semantic_template":"qf-mod24-pde-first-order-point-residual/{\"coefficients\":[12,2,0,1,-1,2],\"operator\":[1,2],\"source\":12,\"x\":2,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0026","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[-12,-2,-3,1,-1,-2],\"principal\":[1,-2,1],\"source\":-12,\"x\":-2,\"y\":-3}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -2], [1, -2, 1], -2, -3, 2, -1, -4, -12, 12, 否)","explanation_zh":"依賴結構核對=([-12, -2, -3, 1, -1, -2], [1, -2, 1], -2, -3, 2, -1, -4, -12, 12, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246026,"parameters":{"coefficients":[-12,-2,-3,1,-1,-2],"principal":[1,-2,1],"source":-12,"x":-2,"y":-3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[-12,-2,-3,1,-1,-2],\"principal\":[1,-2,1],\"source\":-12,\"x\":-2,\"y\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0027","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[-11,-1,-2,2,0,-1],\"principal\":[2,-1,2],\"source\":-11,\"x\":-1,\"y\":-2}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-11, -1, -2, 2, 0, -1], [2, -1, 2], -1, -2, 4, 0, -2, -11, 15, 否)","explanation_zh":"依賴結構核對=([-11, -1, -2, 2, 0, -1], [2, -1, 2], -1, -2, 4, 0, -2, -11, 15, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246027,"parameters":{"coefficients":[-11,-1,-2,2,0,-1],"principal":[2,-1,2],"source":-11,"x":-1,"y":-2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[-11,-1,-2,2,0,-1],\"principal\":[2,-1,2],\"source\":-11,\"x\":-1,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0028","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[-10,0,-1,3,1,0],\"principal\":[3,0,3],\"source\":-10,\"x\":0,\"y\":-1}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0, -1, 3, 1, 0], [3, 0, 3], 0, -1, 6, 1, 0, -10, 28, 否)","explanation_zh":"依賴結構核對=([-10, 0, -1, 3, 1, 0], [3, 0, 3], 0, -1, 6, 1, 0, -10, 28, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246028,"parameters":{"coefficients":[-10,0,-1,3,1,0],"principal":[3,0,3],"source":-10,"x":0,"y":-1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[-10,0,-1,3,1,0],\"principal\":[3,0,3],\"source\":-10,\"x\":0,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0029","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[-9,1,0,4,-1,1],\"principal\":[1,1,4],\"source\":-9,\"x\":1,\"y\":0}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1, 0, 4, -1, 1], [1, 1, 4], 1, 0, 8, -1, 2, -9, 24, 否)","explanation_zh":"依賴結構核對=([-9, 1, 0, 4, -1, 1], [1, 1, 4], 1, 0, 8, -1, 2, -9, 24, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246029,"parameters":{"coefficients":[-9,1,0,4,-1,1],"principal":[1,1,4],"source":-9,"x":1,"y":0,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[-9,1,0,4,-1,1],\"principal\":[1,1,4],\"source\":-9,\"x\":1,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0030","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[-8,2,1,1,0,2],\"principal\":[2,2,1],\"source\":-8,\"x\":2,\"y\":1}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, 2], [2, 2, 1], 2, 1, 2, 0, 4, -8, 16, 否)","explanation_zh":"依賴結構核對=([-8, 2, 1, 1, 0, 2], [2, 2, 1], 2, 1, 2, 0, 4, -8, 16, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246030,"parameters":{"coefficients":[-8,2,1,1,0,2],"principal":[2,2,1],"source":-8,"x":2,"y":1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[-8,2,1,1,0,2],\"principal\":[2,2,1],\"source\":-8,\"x\":2,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0031","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[-7,-2,2,2,1,-2],\"principal\":[3,-2,2],\"source\":-7,\"x\":-2,\"y\":2}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-7, -2, 2, 2, 1, -2], [3, -2, 2], -2, 2, 4, 1, -4, -7, 9, 否)","explanation_zh":"依賴結構核對=([-7, -2, 2, 2, 1, -2], [3, -2, 2], -2, 2, 4, 1, -4, -7, 9, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246031,"parameters":{"coefficients":[-7,-2,2,2,1,-2],"principal":[3,-2,2],"source":-7,"x":-2,"y":2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[-7,-2,2,2,1,-2],\"principal\":[3,-2,2],\"source\":-7,\"x\":-2,\"y\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0032","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[-6,-1,3,3,-1,-1],\"principal\":[1,-1,3],\"source\":-6,\"x\":-1,\"y\":3}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-6, -1, 3, 3, -1, -1], [1, -1, 3], -1, 3, 6, -1, -2, -6, 7, 否)","explanation_zh":"依賴結構核對=([-6, -1, 3, 3, -1, -1], [1, -1, 3], -1, 3, 6, -1, -2, -6, 7, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246032,"parameters":{"coefficients":[-6,-1,3,3,-1,-1],"principal":[1,-1,3],"source":-6,"x":-1,"y":3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[-6,-1,3,3,-1,-1],\"principal\":[1,-1,3],\"source\":-6,\"x\":-1,\"y\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0033","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[-5,0,-3,4,0,0],\"principal\":[2,0,4],\"source\":-5,\"x\":0,\"y\":-3}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-5, 0, -3, 4, 0, 0], [2, 0, 4], 0, -3, 8, 0, 0, -5, 21, 否)","explanation_zh":"依賴結構核對=([-5, 0, -3, 4, 0, 0], [2, 0, 4], 0, -3, 8, 0, 0, -5, 21, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246033,"parameters":{"coefficients":[-5,0,-3,4,0,0],"principal":[2,0,4],"source":-5,"x":0,"y":-3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[-5,0,-3,4,0,0],\"principal\":[2,0,4],\"source\":-5,\"x\":0,\"y\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0034","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[-4,1,-2,1,1,1],\"principal\":[3,1,1],\"source\":-4,\"x\":1,\"y\":-2}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, 1], [3, 1, 1], 1, -2, 2, 1, 2, -4, 13, 否)","explanation_zh":"依賴結構核對=([-4, 1, -2, 1, 1, 1], [3, 1, 1], 1, -2, 2, 1, 2, -4, 13, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246034,"parameters":{"coefficients":[-4,1,-2,1,1,1],"principal":[3,1,1],"source":-4,"x":1,"y":-2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[-4,1,-2,1,1,1],\"principal\":[3,1,1],\"source\":-4,\"x\":1,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0035","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[-3,2,-1,2,-1,2],\"principal\":[1,2,2],\"source\":-3,\"x\":2,\"y\":-1}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2, -1, 2, -1, 2], [1, 2, 2], 2, -1, 4, -1, 4, -3, 13, 否)","explanation_zh":"依賴結構核對=([-3, 2, -1, 2, -1, 2], [1, 2, 2], 2, -1, 4, -1, 4, -3, 13, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246035,"parameters":{"coefficients":[-3,2,-1,2,-1,2],"principal":[1,2,2],"source":-3,"x":2,"y":-1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[-3,2,-1,2,-1,2],\"principal\":[1,2,2],\"source\":-3,\"x\":2,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0036","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[-2,-2,0,3,0,-2],\"principal\":[2,-2,3],\"source\":-2,\"x\":-2,\"y\":0}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, 0, 3, 0, -2], [2, -2, 3], -2, 0, 6, 0, -4, -2, 2, 否)","explanation_zh":"依賴結構核對=([-2, -2, 0, 3, 0, -2], [2, -2, 3], -2, 0, 6, 0, -4, -2, 2, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246036,"parameters":{"coefficients":[-2,-2,0,3,0,-2],"principal":[2,-2,3],"source":-2,"x":-2,"y":0,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[-2,-2,0,3,0,-2],\"principal\":[2,-2,3],\"source\":-2,\"x\":-2,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0037","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[-1,-1,1,4,1,-1],\"principal\":[3,-1,4],\"source\":-1,\"x\":-1,\"y\":1}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1, 1, 4, 1, -1], [3, -1, 4], -1, 1, 8, 1, -2, -1, 16, 否)","explanation_zh":"依賴結構核對=([-1, -1, 1, 4, 1, -1], [3, -1, 4], -1, 1, 8, 1, -2, -1, 16, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246037,"parameters":{"coefficients":[-1,-1,1,4,1,-1],"principal":[3,-1,4],"source":-1,"x":-1,"y":1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[-1,-1,1,4,1,-1],\"principal\":[3,-1,4],\"source\":-1,\"x\":-1,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0038","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[0,0,2,1,-1,0],\"principal\":[1,0,1],\"source\":0,\"x\":0,\"y\":2}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, 0], [1, 0, 1], 0, 2, 2, -1, 0, 0, 2, 否)","explanation_zh":"依賴結構核對=([0, 0, 2, 1, -1, 0], [1, 0, 1], 0, 2, 2, -1, 0, 0, 2, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246038,"parameters":{"coefficients":[0,0,2,1,-1,0],"principal":[1,0,1],"source":0,"x":0,"y":2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[0,0,2,1,-1,0],\"principal\":[1,0,1],\"source\":0,\"x\":0,\"y\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0039","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[1,1,3,2,0,1],\"principal\":[2,1,2],\"source\":1,\"x\":1,\"y\":3}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3, 2, 0, 1], [2, 1, 2], 1, 3, 4, 0, 2, 1, 11, 否)","explanation_zh":"依賴結構核對=([1, 1, 3, 2, 0, 1], [2, 1, 2], 1, 3, 4, 0, 2, 1, 11, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246039,"parameters":{"coefficients":[1,1,3,2,0,1],"principal":[2,1,2],"source":1,"x":1,"y":3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[1,1,3,2,0,1],\"principal\":[2,1,2],\"source\":1,\"x\":1,\"y\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0040","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[2,2,-3,3,1,2],\"principal\":[3,2,3],\"source\":2,\"x\":2,\"y\":-3}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, -3, 3, 1, 2], [3, 2, 3], 2, -3, 6, 1, 4, 2, 30, 否)","explanation_zh":"依賴結構核對=([2, 2, -3, 3, 1, 2], [3, 2, 3], 2, -3, 6, 1, 4, 2, 30, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246040,"parameters":{"coefficients":[2,2,-3,3,1,2],"principal":[3,2,3],"source":2,"x":2,"y":-3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[2,2,-3,3,1,2],\"principal\":[3,2,3],\"source\":2,\"x\":2,\"y\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0041","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[3,-2,-2,4,-1,-2],\"principal\":[1,-2,4],\"source\":3,\"x\":-2,\"y\":-2}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([3, -2, -2, 4, -1, -2], [1, -2, 4], -2, -2, 8, -1, -4, 3, -9, 否)","explanation_zh":"依賴結構核對=([3, -2, -2, 4, -1, -2], [1, -2, 4], -2, -2, 8, -1, -4, 3, -9, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246041,"parameters":{"coefficients":[3,-2,-2,4,-1,-2],"principal":[1,-2,4],"source":3,"x":-2,"y":-2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[3,-2,-2,4,-1,-2],\"principal\":[1,-2,4],\"source\":3,\"x\":-2,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0042","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[4,-1,-1,1,0,-1],\"principal\":[2,-1,1],\"source\":4,\"x\":-1,\"y\":-1}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], [2, -1, 1], -1, -1, 2, 0, -2, 4, -2, 否)","explanation_zh":"依賴結構核對=([4, -1, -1, 1, 0, -1], [2, -1, 1], -1, -1, 2, 0, -2, 4, -2, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246042,"parameters":{"coefficients":[4,-1,-1,1,0,-1],"principal":[2,-1,1],"source":4,"x":-1,"y":-1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[4,-1,-1,1,0,-1],\"principal\":[2,-1,1],\"source\":4,\"x\":-1,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0043","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[5,0,0,2,1,0],\"principal\":[3,0,2],\"source\":5,\"x\":0,\"y\":0}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([5, 0, 0, 2, 1, 0], [3, 0, 2], 0, 0, 4, 1, 0, 5, 7, 否)","explanation_zh":"依賴結構核對=([5, 0, 0, 2, 1, 0], [3, 0, 2], 0, 0, 4, 1, 0, 5, 7, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246043,"parameters":{"coefficients":[5,0,0,2,1,0],"principal":[3,0,2],"source":5,"x":0,"y":0,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[5,0,0,2,1,0],\"principal\":[3,0,2],\"source\":5,\"x\":0,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0044","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[6,1,1,3,-1,1],\"principal\":[1,1,3],\"source\":6,\"x\":1,\"y\":1}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([6, 1, 1, 3, -1, 1], [1, 1, 3], 1, 1, 6, -1, 2, 6, 5, 否)","explanation_zh":"依賴結構核對=([6, 1, 1, 3, -1, 1], [1, 1, 3], 1, 1, 6, -1, 2, 6, 5, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246044,"parameters":{"coefficients":[6,1,1,3,-1,1],"principal":[1,1,3],"source":6,"x":1,"y":1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[6,1,1,3,-1,1],\"principal\":[1,1,3],\"source\":6,\"x\":1,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0045","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[7,2,2,4,0,2],\"principal\":[2,2,4],\"source\":7,\"x\":2,\"y\":2}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([7, 2, 2, 4, 0, 2], [2, 2, 4], 2, 2, 8, 0, 4, 7, 25, 否)","explanation_zh":"依賴結構核對=([7, 2, 2, 4, 0, 2], [2, 2, 4], 2, 2, 8, 0, 4, 7, 25, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246045,"parameters":{"coefficients":[7,2,2,4,0,2],"principal":[2,2,4],"source":7,"x":2,"y":2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[7,2,2,4,0,2],\"principal\":[2,2,4],\"source\":7,\"x\":2,\"y\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0046","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[8,-2,3,1,1,-2],\"principal\":[3,-2,1],\"source\":8,\"x\":-2,\"y\":3}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -2], [3, -2, 1], -2, 3, 2, 1, -4, 8, -8, 否)","explanation_zh":"依賴結構核對=([8, -2, 3, 1, 1, -2], [3, -2, 1], -2, 3, 2, 1, -4, 8, -8, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246046,"parameters":{"coefficients":[8,-2,3,1,1,-2],"principal":[3,-2,1],"source":8,"x":-2,"y":3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[8,-2,3,1,1,-2],\"principal\":[3,-2,1],\"source\":8,\"x\":-2,\"y\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0047","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[9,-1,-3,2,-1,-1],\"principal\":[1,-1,2],\"source\":9,\"x\":-1,\"y\":-3}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([9, -1, -3, 2, -1, -1], [1, -1, 2], -1, -3, 4, -1, -2, 9, -8, 否)","explanation_zh":"依賴結構核對=([9, -1, -3, 2, -1, -1], [1, -1, 2], -1, -3, 4, -1, -2, 9, -8, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246047,"parameters":{"coefficients":[9,-1,-3,2,-1,-1],"principal":[1,-1,2],"source":9,"x":-1,"y":-3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[9,-1,-3,2,-1,-1],\"principal\":[1,-1,2],\"source\":9,\"x\":-1,\"y\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0048","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[10,0,-2,3,0,0],\"principal\":[2,0,3],\"source\":10,\"x\":0,\"y\":-2}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([10, 0, -2, 3, 0, 0], [2, 0, 3], 0, -2, 6, 0, 0, 10, 2, 否)","explanation_zh":"依賴結構核對=([10, 0, -2, 3, 0, 0], [2, 0, 3], 0, -2, 6, 0, 0, 10, 2, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246048,"parameters":{"coefficients":[10,0,-2,3,0,0],"principal":[2,0,3],"source":10,"x":0,"y":-2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[10,0,-2,3,0,0],\"principal\":[2,0,3],\"source\":10,\"x\":0,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0049","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[11,1,-1,4,1,1],\"principal\":[3,1,4],\"source\":11,\"x\":1,\"y\":-1}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([11, 1, -1, 4, 1, 1], [3, 1, 4], 1, -1, 8, 1, 2, 11, 22, 否)","explanation_zh":"依賴結構核對=([11, 1, -1, 4, 1, 1], [3, 1, 4], 1, -1, 8, 1, 2, 11, 22, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246049,"parameters":{"coefficients":[11,1,-1,4,1,1],"principal":[3,1,4],"source":11,"x":1,"y":-1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[11,1,-1,4,1,1],\"principal\":[3,1,4],\"source\":11,\"x\":1,\"y\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0050","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-second-order-point-residual；資料={\"coefficients\":[12,2,0,1,-1,2],\"principal\":[1,2,1],\"source\":12,\"x\":2,\"y\":0}。請依 [coefficients,principal,x,y,u_xx,u_xy,u_yy,source,residual,satisfies_at_point] 欄序回答。","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, 2], [1, 2, 1], 2, 0, 2, -1, 4, 12, -8, 否)","explanation_zh":"依賴結構核對=([12, 2, 0, 1, -1, 2], [1, 2, 1], 2, 0, 2, -1, 4, 12, -8, 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-second-order-point-residual","seed":202610246050,"parameters":{"coefficients":[12,2,0,1,-1,2],"principal":[1,2,1],"source":12,"x":2,"y":0,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-second-order-point-residual","semantic_template":"qf-mod24-pde-second-order-point-residual/{\"coefficients\":[12,2,0,1,-1,2],\"principal\":[1,2,1],\"source\":12,\"x\":2,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0051","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[1,-12,1]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, -12, 1], a*u_xx+b*u_xy+c*u_yy, 140, hyperbolic)","explanation_zh":"依賴結構核對=([1, -12, 1], a*u_xx+b*u_xy+c*u_yy, 140, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246051,"parameters":{"principal":[1,-12,1],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[1,-12,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0052","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[2,-11,2]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([2, -11, 2], a*u_xx+b*u_xy+c*u_yy, 105, hyperbolic)","explanation_zh":"依賴結構核對=([2, -11, 2], a*u_xx+b*u_xy+c*u_yy, 105, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246052,"parameters":{"principal":[2,-11,2],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[2,-11,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0053","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[3,-10,3]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([3, -10, 3], a*u_xx+b*u_xy+c*u_yy, 64, hyperbolic)","explanation_zh":"依賴結構核對=([3, -10, 3], a*u_xx+b*u_xy+c*u_yy, 64, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246053,"parameters":{"principal":[3,-10,3],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[3,-10,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0054","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[1,-9,4]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, -9, 4], a*u_xx+b*u_xy+c*u_yy, 65, hyperbolic)","explanation_zh":"依賴結構核對=([1, -9, 4], a*u_xx+b*u_xy+c*u_yy, 65, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246054,"parameters":{"principal":[1,-9,4],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[1,-9,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0055","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[2,-8,1]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([2, -8, 1], a*u_xx+b*u_xy+c*u_yy, 56, hyperbolic)","explanation_zh":"依賴結構核對=([2, -8, 1], a*u_xx+b*u_xy+c*u_yy, 56, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246055,"parameters":{"principal":[2,-8,1],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[2,-8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0056","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[3,-7,2]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([3, -7, 2], a*u_xx+b*u_xy+c*u_yy, 25, hyperbolic)","explanation_zh":"依賴結構核對=([3, -7, 2], a*u_xx+b*u_xy+c*u_yy, 25, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246056,"parameters":{"principal":[3,-7,2],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[3,-7,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0057","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[1,-6,3]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, -6, 3], a*u_xx+b*u_xy+c*u_yy, 24, hyperbolic)","explanation_zh":"依賴結構核對=([1, -6, 3], a*u_xx+b*u_xy+c*u_yy, 24, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246057,"parameters":{"principal":[1,-6,3],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[1,-6,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0058","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[2,-5,4]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([2, -5, 4], a*u_xx+b*u_xy+c*u_yy, -7, elliptic)","explanation_zh":"依賴結構核對=([2, -5, 4], a*u_xx+b*u_xy+c*u_yy, -7, elliptic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246058,"parameters":{"principal":[2,-5,4],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[2,-5,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0059","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[3,-4,1]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([3, -4, 1], a*u_xx+b*u_xy+c*u_yy, 4, hyperbolic)","explanation_zh":"依賴結構核對=([3, -4, 1], a*u_xx+b*u_xy+c*u_yy, 4, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246059,"parameters":{"principal":[3,-4,1],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[3,-4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0060","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[1,-3,2]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, -3, 2], a*u_xx+b*u_xy+c*u_yy, 1, hyperbolic)","explanation_zh":"依賴結構核對=([1, -3, 2], a*u_xx+b*u_xy+c*u_yy, 1, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246060,"parameters":{"principal":[1,-3,2],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[1,-3,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0061","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[2,-2,3]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([2, -2, 3], a*u_xx+b*u_xy+c*u_yy, -20, elliptic)","explanation_zh":"依賴結構核對=([2, -2, 3], a*u_xx+b*u_xy+c*u_yy, -20, elliptic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246061,"parameters":{"principal":[2,-2,3],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[2,-2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0062","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[3,-1,4]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([3, -1, 4], a*u_xx+b*u_xy+c*u_yy, -47, elliptic)","explanation_zh":"依賴結構核對=([3, -1, 4], a*u_xx+b*u_xy+c*u_yy, -47, elliptic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246062,"parameters":{"principal":[3,-1,4],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[3,-1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0063","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[1,0,1]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1], a*u_xx+b*u_xy+c*u_yy, -4, elliptic)","explanation_zh":"依賴結構核對=([1, 0, 1], a*u_xx+b*u_xy+c*u_yy, -4, elliptic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246063,"parameters":{"principal":[1,0,1],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[1,0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0064","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[2,1,2]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([2, 1, 2], a*u_xx+b*u_xy+c*u_yy, -15, elliptic)","explanation_zh":"依賴結構核對=([2, 1, 2], a*u_xx+b*u_xy+c*u_yy, -15, elliptic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246064,"parameters":{"principal":[2,1,2],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[2,1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0065","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[3,2,3]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([3, 2, 3], a*u_xx+b*u_xy+c*u_yy, -32, elliptic)","explanation_zh":"依賴結構核對=([3, 2, 3], a*u_xx+b*u_xy+c*u_yy, -32, elliptic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246065,"parameters":{"principal":[3,2,3],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[3,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0066","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[1,3,4]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 4], a*u_xx+b*u_xy+c*u_yy, -7, elliptic)","explanation_zh":"依賴結構核對=([1, 3, 4], a*u_xx+b*u_xy+c*u_yy, -7, elliptic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246066,"parameters":{"principal":[1,3,4],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[1,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0067","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[2,4,1]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([2, 4, 1], a*u_xx+b*u_xy+c*u_yy, 8, hyperbolic)","explanation_zh":"依賴結構核對=([2, 4, 1], a*u_xx+b*u_xy+c*u_yy, 8, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246067,"parameters":{"principal":[2,4,1],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[2,4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0068","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[3,5,2]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([3, 5, 2], a*u_xx+b*u_xy+c*u_yy, 1, hyperbolic)","explanation_zh":"依賴結構核對=([3, 5, 2], a*u_xx+b*u_xy+c*u_yy, 1, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246068,"parameters":{"principal":[3,5,2],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[3,5,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0069","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[1,6,3]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 6, 3], a*u_xx+b*u_xy+c*u_yy, 24, hyperbolic)","explanation_zh":"依賴結構核對=([1, 6, 3], a*u_xx+b*u_xy+c*u_yy, 24, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246069,"parameters":{"principal":[1,6,3],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[1,6,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0070","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[2,7,4]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([2, 7, 4], a*u_xx+b*u_xy+c*u_yy, 17, hyperbolic)","explanation_zh":"依賴結構核對=([2, 7, 4], a*u_xx+b*u_xy+c*u_yy, 17, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246070,"parameters":{"principal":[2,7,4],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[2,7,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0071","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[3,8,1]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([3, 8, 1], a*u_xx+b*u_xy+c*u_yy, 52, hyperbolic)","explanation_zh":"依賴結構核對=([3, 8, 1], a*u_xx+b*u_xy+c*u_yy, 52, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246071,"parameters":{"principal":[3,8,1],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[3,8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0072","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[1,9,2]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 9, 2], a*u_xx+b*u_xy+c*u_yy, 73, hyperbolic)","explanation_zh":"依賴結構核對=([1, 9, 2], a*u_xx+b*u_xy+c*u_yy, 73, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246072,"parameters":{"principal":[1,9,2],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[1,9,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0073","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[2,10,3]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([2, 10, 3], a*u_xx+b*u_xy+c*u_yy, 76, hyperbolic)","explanation_zh":"依賴結構核對=([2, 10, 3], a*u_xx+b*u_xy+c*u_yy, 76, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246073,"parameters":{"principal":[2,10,3],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[2,10,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0074","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[3,11,4]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([3, 11, 4], a*u_xx+b*u_xy+c*u_yy, 73, hyperbolic)","explanation_zh":"依賴結構核對=([3, 11, 4], a*u_xx+b*u_xy+c*u_yy, 73, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246074,"parameters":{"principal":[3,11,4],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[3,11,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0075","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-principal-classification；資料={\"principal\":[1,12,1]}。請依 [principal,convention,discriminant,classification] 欄序回答。","answer":{"kind":"expression","value":"([1, 12, 1], a*u_xx+b*u_xy+c*u_yy, 140, hyperbolic)","explanation_zh":"依賴結構核對=([1, 12, 1], a*u_xx+b*u_xy+c*u_yy, 140, hyperbolic)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-principal-classification","seed":202610246075,"parameters":{"principal":[1,12,1],"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-principal-classification","semantic_template":"qf-mod24-pde-principal-classification/{\"principal\":[1,12,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0076","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[-12,-2,2,1,1,0],\"diffusivity\":1}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([-12, -2, 2, 1, 1, 0], 1, [0, 1, 0], 否)","explanation_zh":"依賴結構核對=([-12, -2, 2, 1, 1, 0], 1, [0, 1, 0], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246076,"parameters":{"coefficients":[-12,-2,2,1,1,0],"diffusivity":1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[-12,-2,2,1,1,0],\"diffusivity\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0077","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[-11,-1,8,2,0,0],\"diffusivity\":2}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([-11, -1, 8, 2, 0, 0], 2, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([-11, -1, 8, 2, 0, 0], 2, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246077,"parameters":{"coefficients":[-11,-1,8,2,0,0],"diffusivity":2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[-11,-1,8,2,0,0],\"diffusivity\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0078","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[-10,0,18,3,0,0],\"diffusivity\":3}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0, 18, 3, 0, 0], 3, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([-10, 0, 18, 3, 0, 0], 3, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246078,"parameters":{"coefficients":[-10,0,18,3,0,0],"diffusivity":3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[-10,0,18,3,0,0],\"diffusivity\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0079","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[-9,1,8,1,0,0],\"diffusivity\":4}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1, 8, 1, 0, 0], 4, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([-9, 1, 8, 1, 0, 0], 4, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246079,"parameters":{"coefficients":[-9,1,8,1,0,0],"diffusivity":4,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[-9,1,8,1,0,0],\"diffusivity\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0080","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[-8,2,4,2,0,0],\"diffusivity\":1}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([-8, 2, 4, 2, 0, 0], 1, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([-8, 2, 4, 2, 0, 0], 1, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246080,"parameters":{"coefficients":[-8,2,4,2,0,0],"diffusivity":1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[-8,2,4,2,0,0],\"diffusivity\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0081","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[-7,-2,12,3,0,0],\"diffusivity\":2}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([-7, -2, 12, 3, 0, 0], 2, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([-7, -2, 12, 3, 0, 0], 2, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246081,"parameters":{"coefficients":[-7,-2,12,3,0,0],"diffusivity":2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[-7,-2,12,3,0,0],\"diffusivity\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0082","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[-6,-1,6,1,1,0],\"diffusivity\":3}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([-6, -1, 6, 1, 1, 0], 3, [0, 1, 0], 否)","explanation_zh":"依賴結構核對=([-6, -1, 6, 1, 1, 0], 3, [0, 1, 0], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246082,"parameters":{"coefficients":[-6,-1,6,1,1,0],"diffusivity":3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[-6,-1,6,1,1,0],\"diffusivity\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0083","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[-5,0,16,2,0,0],\"diffusivity\":4}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([-5, 0, 16, 2, 0, 0], 4, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([-5, 0, 16, 2, 0, 0], 4, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246083,"parameters":{"coefficients":[-5,0,16,2,0,0],"diffusivity":4,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[-5,0,16,2,0,0],\"diffusivity\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0084","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[-4,1,6,3,0,0],\"diffusivity\":1}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1, 6, 3, 0, 0], 1, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([-4, 1, 6, 3, 0, 0], 1, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246084,"parameters":{"coefficients":[-4,1,6,3,0,0],"diffusivity":1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[-4,1,6,3,0,0],\"diffusivity\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0085","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[-3,2,4,1,0,0],\"diffusivity\":2}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2, 4, 1, 0, 0], 2, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([-3, 2, 4, 1, 0, 0], 2, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246085,"parameters":{"coefficients":[-3,2,4,1,0,0],"diffusivity":2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[-3,2,4,1,0,0],\"diffusivity\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0086","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[-2,-2,12,2,0,0],\"diffusivity\":3}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, 12, 2, 0, 0], 3, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([-2, -2, 12, 2, 0, 0], 3, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246086,"parameters":{"coefficients":[-2,-2,12,2,0,0],"diffusivity":3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[-2,-2,12,2,0,0],\"diffusivity\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0087","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[-1,-1,24,3,0,0],\"diffusivity\":4}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1, 24, 3, 0, 0], 4, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([-1, -1, 24, 3, 0, 0], 4, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246087,"parameters":{"coefficients":[-1,-1,24,3,0,0],"diffusivity":4,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[-1,-1,24,3,0,0],\"diffusivity\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0088","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[0,0,2,1,1,0],\"diffusivity\":1}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 2, 1, 1, 0], 1, [0, 1, 0], 否)","explanation_zh":"依賴結構核對=([0, 0, 2, 1, 1, 0], 1, [0, 1, 0], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246088,"parameters":{"coefficients":[0,0,2,1,1,0],"diffusivity":1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[0,0,2,1,1,0],\"diffusivity\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0089","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[1,1,8,2,0,0],\"diffusivity\":2}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 8, 2, 0, 0], 2, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([1, 1, 8, 2, 0, 0], 2, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246089,"parameters":{"coefficients":[1,1,8,2,0,0],"diffusivity":2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[1,1,8,2,0,0],\"diffusivity\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0090","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[2,2,18,3,0,0],\"diffusivity\":3}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 18, 3, 0, 0], 3, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([2, 2, 18, 3, 0, 0], 3, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246090,"parameters":{"coefficients":[2,2,18,3,0,0],"diffusivity":3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[2,2,18,3,0,0],\"diffusivity\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0091","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[3,-2,8,1,0,0],\"diffusivity\":4}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([3, -2, 8, 1, 0, 0], 4, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([3, -2, 8, 1, 0, 0], 4, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246091,"parameters":{"coefficients":[3,-2,8,1,0,0],"diffusivity":4,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[3,-2,8,1,0,0],\"diffusivity\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0092","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[4,-1,4,2,0,0],\"diffusivity\":1}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([4, -1, 4, 2, 0, 0], 1, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([4, -1, 4, 2, 0, 0], 1, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246092,"parameters":{"coefficients":[4,-1,4,2,0,0],"diffusivity":1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[4,-1,4,2,0,0],\"diffusivity\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0093","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[5,0,12,3,0,0],\"diffusivity\":2}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([5, 0, 12, 3, 0, 0], 2, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([5, 0, 12, 3, 0, 0], 2, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246093,"parameters":{"coefficients":[5,0,12,3,0,0],"diffusivity":2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[5,0,12,3,0,0],\"diffusivity\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0094","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[6,1,6,1,1,0],\"diffusivity\":3}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([6, 1, 6, 1, 1, 0], 3, [0, 1, 0], 否)","explanation_zh":"依賴結構核對=([6, 1, 6, 1, 1, 0], 3, [0, 1, 0], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246094,"parameters":{"coefficients":[6,1,6,1,1,0],"diffusivity":3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[6,1,6,1,1,0],\"diffusivity\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0095","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[7,2,16,2,0,0],\"diffusivity\":4}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([7, 2, 16, 2, 0, 0], 4, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([7, 2, 16, 2, 0, 0], 4, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246095,"parameters":{"coefficients":[7,2,16,2,0,0],"diffusivity":4,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[7,2,16,2,0,0],\"diffusivity\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0096","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[8,-2,6,3,0,0],\"diffusivity\":1}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([8, -2, 6, 3, 0, 0], 1, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([8, -2, 6, 3, 0, 0], 1, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246096,"parameters":{"coefficients":[8,-2,6,3,0,0],"diffusivity":1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[8,-2,6,3,0,0],\"diffusivity\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0097","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[9,-1,4,1,0,0],\"diffusivity\":2}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([9, -1, 4, 1, 0, 0], 2, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([9, -1, 4, 1, 0, 0], 2, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246097,"parameters":{"coefficients":[9,-1,4,1,0,0],"diffusivity":2,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[9,-1,4,1,0,0],\"diffusivity\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0098","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[10,0,12,2,0,0],\"diffusivity\":3}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([10, 0, 12, 2, 0, 0], 3, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([10, 0, 12, 2, 0, 0], 3, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246098,"parameters":{"coefficients":[10,0,12,2,0,0],"diffusivity":3,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[10,0,12,2,0,0],\"diffusivity\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0099","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[11,1,24,3,0,0],\"diffusivity\":4}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([11, 1, 24, 3, 0, 0], 4, [0, 0, 0], 是)","explanation_zh":"依賴結構核對=([11, 1, 24, 3, 0, 0], 4, [0, 0, 0], 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246099,"parameters":{"coefficients":[11,1,24,3,0,0],"diffusivity":4,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[11,1,24,3,0,0],\"diffusivity\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0100","batch_id":"batch-20261013-advance-20261024-r06-partial-differential-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-differential-equation"],"title_zh":"線性PDE切片","stem_zh":"明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[12,2,2,1,1,0],\"diffusivity\":1}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。","answer":{"kind":"expression","value":"([12, 2, 2, 1, 1, 0], 1, [0, 1, 0], 否)","explanation_zh":"依賴結構核對=([12, 2, 2, 1, 1, 0], 1, [0, 1, 0], 否)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；點殘差不證全域解；不宣稱一般解存在唯一或適定性。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性PDE切片"]},"variant":{"family_id":"qf-mod24-pde-heat-polynomial-identity","seed":202610246100,"parameters":{"coefficients":[12,2,2,1,1,0],"diffusivity":1,"domain":"Polynomial_linear_PDE_operator_slice","operation":"pde-heat-polynomial-identity","semantic_template":"qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[12,2,2,1,1,0],\"diffusivity\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov長期性、PDE適定性、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"}}