{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261024-r04-partial-derivative-0001","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[-12,-2,-3,1,-1,-1],\"x\":-2,\"y\":-3}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -1], -2, -3, -3)","explanation_zh":"依賴結構核對=([-12, -2, -3, 1, -1, -1], -2, -3, -3)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244001,"parameters":{"coefficients":[-12,-2,-3,1,-1,-1],"x":-2,"y":-3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-12,-2,-3,1,-1,-1],\"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-r04-partial-derivative-0002","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[-11,-1,-2,1,0,-1],\"x\":-1,\"y\":-2}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([-11, -1, -2, 1, 0, -1], -1, -2, -3)","explanation_zh":"依賴結構核對=([-11, -1, -2, 1, 0, -1], -1, -2, -3)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244002,"parameters":{"coefficients":[-11,-1,-2,1,0,-1],"x":-1,"y":-2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-11,-1,-2,1,0,-1],\"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-r04-partial-derivative-0003","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[-10,0,-1,1,1,-1],\"x\":0,\"y\":-1}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0, -1, 1, 1, -1], 0, -1, -1)","explanation_zh":"依賴結構核對=([-10, 0, -1, 1, 1, -1], 0, -1, -1)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244003,"parameters":{"coefficients":[-10,0,-1,1,1,-1],"x":0,"y":-1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-10,0,-1,1,1,-1],\"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-r04-partial-derivative-0004","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[-9,1,0,1,-1,-1],\"x\":1,\"y\":0}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1, 0, 1, -1, -1], 1, 0, 3)","explanation_zh":"依賴結構核對=([-9, 1, 0, 1, -1, -1], 1, 0, 3)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244004,"parameters":{"coefficients":[-9,1,0,1,-1,-1],"x":1,"y":0,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-9,1,0,1,-1,-1],\"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-r04-partial-derivative-0005","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[-8,2,1,1,0,-1],\"x\":2,\"y\":1}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, -1], 2, 1, 6)","explanation_zh":"依賴結構核對=([-8, 2, 1, 1, 0, -1], 2, 1, 6)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244005,"parameters":{"coefficients":[-8,2,1,1,0,-1],"x":2,"y":1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-8,2,1,1,0,-1],\"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-r04-partial-derivative-0006","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[-7,-2,2,1,1,-1],\"x\":-2,\"y\":2}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([-7, -2, 2, 1, 1, -1], -2, 2, -4)","explanation_zh":"依賴結構核對=([-7, -2, 2, 1, 1, -1], -2, 2, -4)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244006,"parameters":{"coefficients":[-7,-2,2,1,1,-1],"x":-2,"y":2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-7,-2,2,1,1,-1],\"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-r04-partial-derivative-0007","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[-6,-1,3,1,-1,-1],\"x\":-1,\"y\":3}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([-6, -1, 3, 1, -1, -1], -1, 3, -6)","explanation_zh":"依賴結構核對=([-6, -1, 3, 1, -1, -1], -1, 3, -6)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244007,"parameters":{"coefficients":[-6,-1,3,1,-1,-1],"x":-1,"y":3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-6,-1,3,1,-1,-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-r04-partial-derivative-0008","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[-5,0,-3,1,0,-1],\"x\":0,\"y\":-3}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([-5, 0, -3, 1, 0, -1], 0, -3, 0)","explanation_zh":"依賴結構核對=([-5, 0, -3, 1, 0, -1], 0, -3, 0)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244008,"parameters":{"coefficients":[-5,0,-3,1,0,-1],"x":0,"y":-3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-5,0,-3,1,0,-1],\"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-r04-partial-derivative-0009","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[-4,1,-2,1,1,-1],\"x\":1,\"y\":-2}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, -1], 1, -2, 1)","explanation_zh":"依賴結構核對=([-4, 1, -2, 1, 1, -1], 1, -2, 1)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244009,"parameters":{"coefficients":[-4,1,-2,1,1,-1],"x":1,"y":-2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-4,1,-2,1,1,-1],\"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-r04-partial-derivative-0010","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[-3,2,-1,1,-1,-1],\"x\":2,\"y\":-1}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2, -1, 1, -1, -1], 2, -1, 7)","explanation_zh":"依賴結構核對=([-3, 2, -1, 1, -1, -1], 2, -1, 7)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244010,"parameters":{"coefficients":[-3,2,-1,1,-1,-1],"x":2,"y":-1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-3,2,-1,1,-1,-1],\"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-r04-partial-derivative-0011","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[-2,-2,0,1,0,-1],\"x\":-2,\"y\":0}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, 0, 1, 0, -1], -2, 0, -6)","explanation_zh":"依賴結構核對=([-2, -2, 0, 1, 0, -1], -2, 0, -6)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244011,"parameters":{"coefficients":[-2,-2,0,1,0,-1],"x":-2,"y":0,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-2,-2,0,1,0,-1],\"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-r04-partial-derivative-0012","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[-1,-1,1,1,1,-1],\"x\":-1,\"y\":1}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1, 1, 1, 1, -1], -1, 1, -2)","explanation_zh":"依賴結構核對=([-1, -1, 1, 1, 1, -1], -1, 1, -2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244012,"parameters":{"coefficients":[-1,-1,1,1,1,-1],"x":-1,"y":1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[-1,-1,1,1,1,-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-r04-partial-derivative-0013","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[0,0,2,1,-1,-1],\"x\":0,\"y\":2}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, -1], 0, 2, -2)","explanation_zh":"依賴結構核對=([0, 0, 2, 1, -1, -1], 0, 2, -2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244013,"parameters":{"coefficients":[0,0,2,1,-1,-1],"x":0,"y":2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[0,0,2,1,-1,-1],\"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-r04-partial-derivative-0014","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[1,1,3,1,0,-1],\"x\":1,\"y\":3}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3, 1, 0, -1], 1, 3, 3)","explanation_zh":"依賴結構核對=([1, 1, 3, 1, 0, -1], 1, 3, 3)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244014,"parameters":{"coefficients":[1,1,3,1,0,-1],"x":1,"y":3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[1,1,3,1,0,-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-r04-partial-derivative-0015","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[2,2,-3,1,1,-1],\"x\":2,\"y\":-3}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, -3, 1, 1, -1], 2, -3, 3)","explanation_zh":"依賴結構核對=([2, 2, -3, 1, 1, -1], 2, -3, 3)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244015,"parameters":{"coefficients":[2,2,-3,1,1,-1],"x":2,"y":-3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[2,2,-3,1,1,-1],\"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-r04-partial-derivative-0016","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[3,-2,-2,1,-1,-1],\"x\":-2,\"y\":-2}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([3, -2, -2, 1, -1, -1], -2, -2, -4)","explanation_zh":"依賴結構核對=([3, -2, -2, 1, -1, -1], -2, -2, -4)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244016,"parameters":{"coefficients":[3,-2,-2,1,-1,-1],"x":-2,"y":-2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[3,-2,-2,1,-1,-1],\"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-r04-partial-derivative-0017","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[4,-1,-1,1,0,-1],\"x\":-1,\"y\":-1}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], -1, -1, -3)","explanation_zh":"依賴結構核對=([4, -1, -1, 1, 0, -1], -1, -1, -3)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244017,"parameters":{"coefficients":[4,-1,-1,1,0,-1],"x":-1,"y":-1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[4,-1,-1,1,0,-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-r04-partial-derivative-0018","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[5,0,0,1,1,-1],\"x\":0,\"y\":0}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([5, 0, 0, 1, 1, -1], 0, 0, 0)","explanation_zh":"依賴結構核對=([5, 0, 0, 1, 1, -1], 0, 0, 0)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244018,"parameters":{"coefficients":[5,0,0,1,1,-1],"x":0,"y":0,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[5,0,0,1,1,-1],\"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-r04-partial-derivative-0019","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[6,1,1,1,-1,-1],\"x\":1,\"y\":1}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([6, 1, 1, 1, -1, -1], 1, 1, 2)","explanation_zh":"依賴結構核對=([6, 1, 1, 1, -1, -1], 1, 1, 2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244019,"parameters":{"coefficients":[6,1,1,1,-1,-1],"x":1,"y":1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[6,1,1,1,-1,-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-r04-partial-derivative-0020","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[7,2,2,1,0,-1],\"x\":2,\"y\":2}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([7, 2, 2, 1, 0, -1], 2, 2, 6)","explanation_zh":"依賴結構核對=([7, 2, 2, 1, 0, -1], 2, 2, 6)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244020,"parameters":{"coefficients":[7,2,2,1,0,-1],"x":2,"y":2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[7,2,2,1,0,-1],\"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-r04-partial-derivative-0021","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[8,-2,3,1,1,-1],\"x\":-2,\"y\":3}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -1], -2, 3, -3)","explanation_zh":"依賴結構核對=([8, -2, 3, 1, 1, -1], -2, 3, -3)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244021,"parameters":{"coefficients":[8,-2,3,1,1,-1],"x":-2,"y":3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[8,-2,3,1,1,-1],\"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-r04-partial-derivative-0022","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[9,-1,-3,1,-1,-1],\"x\":-1,\"y\":-3}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([9, -1, -3, 1, -1, -1], -1, -3, 0)","explanation_zh":"依賴結構核對=([9, -1, -3, 1, -1, -1], -1, -3, 0)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244022,"parameters":{"coefficients":[9,-1,-3,1,-1,-1],"x":-1,"y":-3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[9,-1,-3,1,-1,-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-r04-partial-derivative-0023","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[10,0,-2,1,0,-1],\"x\":0,\"y\":-2}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([10, 0, -2, 1, 0, -1], 0, -2, 0)","explanation_zh":"依賴結構核對=([10, 0, -2, 1, 0, -1], 0, -2, 0)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244023,"parameters":{"coefficients":[10,0,-2,1,0,-1],"x":0,"y":-2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[10,0,-2,1,0,-1],\"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-r04-partial-derivative-0024","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[11,1,-1,1,1,-1],\"x\":1,\"y\":-1}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([11, 1, -1, 1, 1, -1], 1, -1, 2)","explanation_zh":"依賴結構核對=([11, 1, -1, 1, 1, -1], 1, -1, 2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244024,"parameters":{"coefficients":[11,1,-1,1,1,-1],"x":1,"y":-1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[11,1,-1,1,1,-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-r04-partial-derivative-0025","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-x；資料={\"coefficients\":[12,2,0,1,-1,-1],\"x\":2,\"y\":0}。請依 [coefficients,x,y,partial_x] 欄序回答。","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, -1], 2, 0, 6)","explanation_zh":"依賴結構核對=([12, 2, 0, 1, -1, -1], 2, 0, 6)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-x","seed":202610244025,"parameters":{"coefficients":[12,2,0,1,-1,-1],"x":2,"y":0,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-x","semantic_template":"qf-mod24-partial-partial-x/{\"coefficients\":[12,2,0,1,-1,-1],\"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-r04-partial-derivative-0026","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[-12,-2,-3,1,-1,-1],\"x\":-2,\"y\":-3}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -1], -2, -3, 5)","explanation_zh":"依賴結構核對=([-12, -2, -3, 1, -1, -1], -2, -3, 5)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244026,"parameters":{"coefficients":[-12,-2,-3,1,-1,-1],"x":-2,"y":-3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-12,-2,-3,1,-1,-1],\"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-r04-partial-derivative-0027","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[-11,-1,-2,1,0,-1],\"x\":-1,\"y\":-2}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([-11, -1, -2, 1, 0, -1], -1, -2, 2)","explanation_zh":"依賴結構核對=([-11, -1, -2, 1, 0, -1], -1, -2, 2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244027,"parameters":{"coefficients":[-11,-1,-2,1,0,-1],"x":-1,"y":-2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-11,-1,-2,1,0,-1],\"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-r04-partial-derivative-0028","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[-10,0,-1,1,1,-1],\"x\":0,\"y\":-1}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0, -1, 1, 1, -1], 0, -1, 1)","explanation_zh":"依賴結構核對=([-10, 0, -1, 1, 1, -1], 0, -1, 1)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244028,"parameters":{"coefficients":[-10,0,-1,1,1,-1],"x":0,"y":-1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-10,0,-1,1,1,-1],\"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-r04-partial-derivative-0029","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[-9,1,0,1,-1,-1],\"x\":1,\"y\":0}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1, 0, 1, -1, -1], 1, 0, -1)","explanation_zh":"依賴結構核對=([-9, 1, 0, 1, -1, -1], 1, 0, -1)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244029,"parameters":{"coefficients":[-9,1,0,1,-1,-1],"x":1,"y":0,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-9,1,0,1,-1,-1],\"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-r04-partial-derivative-0030","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[-8,2,1,1,0,-1],\"x\":2,\"y\":1}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, -1], 2, 1, -1)","explanation_zh":"依賴結構核對=([-8, 2, 1, 1, 0, -1], 2, 1, -1)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244030,"parameters":{"coefficients":[-8,2,1,1,0,-1],"x":2,"y":1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-8,2,1,1,0,-1],\"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-r04-partial-derivative-0031","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[-7,-2,2,1,1,-1],\"x\":-2,\"y\":2}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([-7, -2, 2, 1, 1, -1], -2, 2, -4)","explanation_zh":"依賴結構核對=([-7, -2, 2, 1, 1, -1], -2, 2, -4)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244031,"parameters":{"coefficients":[-7,-2,2,1,1,-1],"x":-2,"y":2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-7,-2,2,1,1,-1],\"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-r04-partial-derivative-0032","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[-6,-1,3,1,-1,-1],\"x\":-1,\"y\":3}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([-6, -1, 3, 1, -1, -1], -1, 3, -2)","explanation_zh":"依賴結構核對=([-6, -1, 3, 1, -1, -1], -1, 3, -2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244032,"parameters":{"coefficients":[-6,-1,3,1,-1,-1],"x":-1,"y":3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-6,-1,3,1,-1,-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-r04-partial-derivative-0033","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[-5,0,-3,1,0,-1],\"x\":0,\"y\":-3}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([-5, 0, -3, 1, 0, -1], 0, -3, 3)","explanation_zh":"依賴結構核對=([-5, 0, -3, 1, 0, -1], 0, -3, 3)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244033,"parameters":{"coefficients":[-5,0,-3,1,0,-1],"x":0,"y":-3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-5,0,-3,1,0,-1],\"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-r04-partial-derivative-0034","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[-4,1,-2,1,1,-1],\"x\":1,\"y\":-2}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, -1], 1, -2, 3)","explanation_zh":"依賴結構核對=([-4, 1, -2, 1, 1, -1], 1, -2, 3)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244034,"parameters":{"coefficients":[-4,1,-2,1,1,-1],"x":1,"y":-2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-4,1,-2,1,1,-1],\"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-r04-partial-derivative-0035","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[-3,2,-1,1,-1,-1],\"x\":2,\"y\":-1}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2, -1, 1, -1, -1], 2, -1, -1)","explanation_zh":"依賴結構核對=([-3, 2, -1, 1, -1, -1], 2, -1, -1)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244035,"parameters":{"coefficients":[-3,2,-1,1,-1,-1],"x":2,"y":-1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-3,2,-1,1,-1,-1],\"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-r04-partial-derivative-0036","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[-2,-2,0,1,0,-1],\"x\":-2,\"y\":0}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, 0, 1, 0, -1], -2, 0, 0)","explanation_zh":"依賴結構核對=([-2, -2, 0, 1, 0, -1], -2, 0, 0)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244036,"parameters":{"coefficients":[-2,-2,0,1,0,-1],"x":-2,"y":0,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-2,-2,0,1,0,-1],\"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-r04-partial-derivative-0037","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[-1,-1,1,1,1,-1],\"x\":-1,\"y\":1}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1, 1, 1, 1, -1], -1, 1, -2)","explanation_zh":"依賴結構核對=([-1, -1, 1, 1, 1, -1], -1, 1, -2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244037,"parameters":{"coefficients":[-1,-1,1,1,1,-1],"x":-1,"y":1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[-1,-1,1,1,1,-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-r04-partial-derivative-0038","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[0,0,2,1,-1,-1],\"x\":0,\"y\":2}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, -1], 0, 2, -2)","explanation_zh":"依賴結構核對=([0, 0, 2, 1, -1, -1], 0, 2, -2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244038,"parameters":{"coefficients":[0,0,2,1,-1,-1],"x":0,"y":2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[0,0,2,1,-1,-1],\"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-r04-partial-derivative-0039","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[1,1,3,1,0,-1],\"x\":1,\"y\":3}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3, 1, 0, -1], 1, 3, -3)","explanation_zh":"依賴結構核對=([1, 1, 3, 1, 0, -1], 1, 3, -3)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244039,"parameters":{"coefficients":[1,1,3,1,0,-1],"x":1,"y":3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[1,1,3,1,0,-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-r04-partial-derivative-0040","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[2,2,-3,1,1,-1],\"x\":2,\"y\":-3}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, -3, 1, 1, -1], 2, -3, 5)","explanation_zh":"依賴結構核對=([2, 2, -3, 1, 1, -1], 2, -3, 5)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244040,"parameters":{"coefficients":[2,2,-3,1,1,-1],"x":2,"y":-3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[2,2,-3,1,1,-1],\"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-r04-partial-derivative-0041","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[3,-2,-2,1,-1,-1],\"x\":-2,\"y\":-2}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([3, -2, -2, 1, -1, -1], -2, -2, 4)","explanation_zh":"依賴結構核對=([3, -2, -2, 1, -1, -1], -2, -2, 4)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244041,"parameters":{"coefficients":[3,-2,-2,1,-1,-1],"x":-2,"y":-2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[3,-2,-2,1,-1,-1],\"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-r04-partial-derivative-0042","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[4,-1,-1,1,0,-1],\"x\":-1,\"y\":-1}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], -1, -1, 1)","explanation_zh":"依賴結構核對=([4, -1, -1, 1, 0, -1], -1, -1, 1)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244042,"parameters":{"coefficients":[4,-1,-1,1,0,-1],"x":-1,"y":-1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[4,-1,-1,1,0,-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-r04-partial-derivative-0043","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[5,0,0,1,1,-1],\"x\":0,\"y\":0}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([5, 0, 0, 1, 1, -1], 0, 0, 0)","explanation_zh":"依賴結構核對=([5, 0, 0, 1, 1, -1], 0, 0, 0)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244043,"parameters":{"coefficients":[5,0,0,1,1,-1],"x":0,"y":0,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[5,0,0,1,1,-1],\"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-r04-partial-derivative-0044","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[6,1,1,1,-1,-1],\"x\":1,\"y\":1}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([6, 1, 1, 1, -1, -1], 1, 1, -2)","explanation_zh":"依賴結構核對=([6, 1, 1, 1, -1, -1], 1, 1, -2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244044,"parameters":{"coefficients":[6,1,1,1,-1,-1],"x":1,"y":1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[6,1,1,1,-1,-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-r04-partial-derivative-0045","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[7,2,2,1,0,-1],\"x\":2,\"y\":2}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([7, 2, 2, 1, 0, -1], 2, 2, -2)","explanation_zh":"依賴結構核對=([7, 2, 2, 1, 0, -1], 2, 2, -2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244045,"parameters":{"coefficients":[7,2,2,1,0,-1],"x":2,"y":2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[7,2,2,1,0,-1],\"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-r04-partial-derivative-0046","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[8,-2,3,1,1,-1],\"x\":-2,\"y\":3}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -1], -2, 3, -5)","explanation_zh":"依賴結構核對=([8, -2, 3, 1, 1, -1], -2, 3, -5)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244046,"parameters":{"coefficients":[8,-2,3,1,1,-1],"x":-2,"y":3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[8,-2,3,1,1,-1],\"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-r04-partial-derivative-0047","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[9,-1,-3,1,-1,-1],\"x\":-1,\"y\":-3}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([9, -1, -3, 1, -1, -1], -1, -3, 4)","explanation_zh":"依賴結構核對=([9, -1, -3, 1, -1, -1], -1, -3, 4)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244047,"parameters":{"coefficients":[9,-1,-3,1,-1,-1],"x":-1,"y":-3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[9,-1,-3,1,-1,-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-r04-partial-derivative-0048","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[10,0,-2,1,0,-1],\"x\":0,\"y\":-2}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([10, 0, -2, 1, 0, -1], 0, -2, 2)","explanation_zh":"依賴結構核對=([10, 0, -2, 1, 0, -1], 0, -2, 2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244048,"parameters":{"coefficients":[10,0,-2,1,0,-1],"x":0,"y":-2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[10,0,-2,1,0,-1],\"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-r04-partial-derivative-0049","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[11,1,-1,1,1,-1],\"x\":1,\"y\":-1}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([11, 1, -1, 1, 1, -1], 1, -1, 2)","explanation_zh":"依賴結構核對=([11, 1, -1, 1, 1, -1], 1, -1, 2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244049,"parameters":{"coefficients":[11,1,-1,1,1,-1],"x":1,"y":-1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[11,1,-1,1,1,-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-r04-partial-derivative-0050","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-partial-y；資料={\"coefficients\":[12,2,0,1,-1,-1],\"x\":2,\"y\":0}。請依 [coefficients,x,y,partial_y] 欄序回答。","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, -1], 2, 0, -2)","explanation_zh":"依賴結構核對=([12, 2, 0, 1, -1, -1], 2, 0, -2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-partial-y","seed":202610244050,"parameters":{"coefficients":[12,2,0,1,-1,-1],"x":2,"y":0,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-partial-y","semantic_template":"qf-mod24-partial-partial-y/{\"coefficients\":[12,2,0,1,-1,-1],\"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-r04-partial-derivative-0051","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[-12,-2,-3,1,-1,-1],\"x\":-2,\"y\":-3}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -1], -2, -3, [-3, 5])","explanation_zh":"依賴結構核對=([-12, -2, -3, 1, -1, -1], -2, -3, [-3, 5])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244051,"parameters":{"coefficients":[-12,-2,-3,1,-1,-1],"x":-2,"y":-3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-12,-2,-3,1,-1,-1],\"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-r04-partial-derivative-0052","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[-11,-1,-2,1,0,-1],\"x\":-1,\"y\":-2}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([-11, -1, -2, 1, 0, -1], -1, -2, [-3, 2])","explanation_zh":"依賴結構核對=([-11, -1, -2, 1, 0, -1], -1, -2, [-3, 2])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244052,"parameters":{"coefficients":[-11,-1,-2,1,0,-1],"x":-1,"y":-2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-11,-1,-2,1,0,-1],\"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-r04-partial-derivative-0053","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[-10,0,-1,1,1,-1],\"x\":0,\"y\":-1}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0, -1, 1, 1, -1], 0, -1, [-1, 1])","explanation_zh":"依賴結構核對=([-10, 0, -1, 1, 1, -1], 0, -1, [-1, 1])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244053,"parameters":{"coefficients":[-10,0,-1,1,1,-1],"x":0,"y":-1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-10,0,-1,1,1,-1],\"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-r04-partial-derivative-0054","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[-9,1,0,1,-1,-1],\"x\":1,\"y\":0}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1, 0, 1, -1, -1], 1, 0, [3, -1])","explanation_zh":"依賴結構核對=([-9, 1, 0, 1, -1, -1], 1, 0, [3, -1])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244054,"parameters":{"coefficients":[-9,1,0,1,-1,-1],"x":1,"y":0,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-9,1,0,1,-1,-1],\"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-r04-partial-derivative-0055","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[-8,2,1,1,0,-1],\"x\":2,\"y\":1}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, -1], 2, 1, [6, -1])","explanation_zh":"依賴結構核對=([-8, 2, 1, 1, 0, -1], 2, 1, [6, -1])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244055,"parameters":{"coefficients":[-8,2,1,1,0,-1],"x":2,"y":1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-8,2,1,1,0,-1],\"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-r04-partial-derivative-0056","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[-7,-2,2,1,1,-1],\"x\":-2,\"y\":2}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([-7, -2, 2, 1, 1, -1], -2, 2, [-4, -4])","explanation_zh":"依賴結構核對=([-7, -2, 2, 1, 1, -1], -2, 2, [-4, -4])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244056,"parameters":{"coefficients":[-7,-2,2,1,1,-1],"x":-2,"y":2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-7,-2,2,1,1,-1],\"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-r04-partial-derivative-0057","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[-6,-1,3,1,-1,-1],\"x\":-1,\"y\":3}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([-6, -1, 3, 1, -1, -1], -1, 3, [-6, -2])","explanation_zh":"依賴結構核對=([-6, -1, 3, 1, -1, -1], -1, 3, [-6, -2])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244057,"parameters":{"coefficients":[-6,-1,3,1,-1,-1],"x":-1,"y":3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-6,-1,3,1,-1,-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-r04-partial-derivative-0058","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[-5,0,-3,1,0,-1],\"x\":0,\"y\":-3}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([-5, 0, -3, 1, 0, -1], 0, -3, [0, 3])","explanation_zh":"依賴結構核對=([-5, 0, -3, 1, 0, -1], 0, -3, [0, 3])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244058,"parameters":{"coefficients":[-5,0,-3,1,0,-1],"x":0,"y":-3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-5,0,-3,1,0,-1],\"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-r04-partial-derivative-0059","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[-4,1,-2,1,1,-1],\"x\":1,\"y\":-2}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, -1], 1, -2, [1, 3])","explanation_zh":"依賴結構核對=([-4, 1, -2, 1, 1, -1], 1, -2, [1, 3])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244059,"parameters":{"coefficients":[-4,1,-2,1,1,-1],"x":1,"y":-2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-4,1,-2,1,1,-1],\"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-r04-partial-derivative-0060","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[-3,2,-1,1,-1,-1],\"x\":2,\"y\":-1}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2, -1, 1, -1, -1], 2, -1, [7, -1])","explanation_zh":"依賴結構核對=([-3, 2, -1, 1, -1, -1], 2, -1, [7, -1])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244060,"parameters":{"coefficients":[-3,2,-1,1,-1,-1],"x":2,"y":-1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-3,2,-1,1,-1,-1],\"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-r04-partial-derivative-0061","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[-2,-2,0,1,0,-1],\"x\":-2,\"y\":0}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, 0, 1, 0, -1], -2, 0, [-6, 0])","explanation_zh":"依賴結構核對=([-2, -2, 0, 1, 0, -1], -2, 0, [-6, 0])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244061,"parameters":{"coefficients":[-2,-2,0,1,0,-1],"x":-2,"y":0,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-2,-2,0,1,0,-1],\"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-r04-partial-derivative-0062","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[-1,-1,1,1,1,-1],\"x\":-1,\"y\":1}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1, 1, 1, 1, -1], -1, 1, [-2, -2])","explanation_zh":"依賴結構核對=([-1, -1, 1, 1, 1, -1], -1, 1, [-2, -2])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244062,"parameters":{"coefficients":[-1,-1,1,1,1,-1],"x":-1,"y":1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[-1,-1,1,1,1,-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-r04-partial-derivative-0063","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[0,0,2,1,-1,-1],\"x\":0,\"y\":2}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, -1], 0, 2, [-2, -2])","explanation_zh":"依賴結構核對=([0, 0, 2, 1, -1, -1], 0, 2, [-2, -2])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244063,"parameters":{"coefficients":[0,0,2,1,-1,-1],"x":0,"y":2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[0,0,2,1,-1,-1],\"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-r04-partial-derivative-0064","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[1,1,3,1,0,-1],\"x\":1,\"y\":3}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3, 1, 0, -1], 1, 3, [3, -3])","explanation_zh":"依賴結構核對=([1, 1, 3, 1, 0, -1], 1, 3, [3, -3])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244064,"parameters":{"coefficients":[1,1,3,1,0,-1],"x":1,"y":3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[1,1,3,1,0,-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-r04-partial-derivative-0065","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[2,2,-3,1,1,-1],\"x\":2,\"y\":-3}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, -3, 1, 1, -1], 2, -3, [3, 5])","explanation_zh":"依賴結構核對=([2, 2, -3, 1, 1, -1], 2, -3, [3, 5])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244065,"parameters":{"coefficients":[2,2,-3,1,1,-1],"x":2,"y":-3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[2,2,-3,1,1,-1],\"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-r04-partial-derivative-0066","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[3,-2,-2,1,-1,-1],\"x\":-2,\"y\":-2}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([3, -2, -2, 1, -1, -1], -2, -2, [-4, 4])","explanation_zh":"依賴結構核對=([3, -2, -2, 1, -1, -1], -2, -2, [-4, 4])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244066,"parameters":{"coefficients":[3,-2,-2,1,-1,-1],"x":-2,"y":-2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[3,-2,-2,1,-1,-1],\"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-r04-partial-derivative-0067","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[4,-1,-1,1,0,-1],\"x\":-1,\"y\":-1}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], -1, -1, [-3, 1])","explanation_zh":"依賴結構核對=([4, -1, -1, 1, 0, -1], -1, -1, [-3, 1])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244067,"parameters":{"coefficients":[4,-1,-1,1,0,-1],"x":-1,"y":-1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[4,-1,-1,1,0,-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-r04-partial-derivative-0068","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[5,0,0,1,1,-1],\"x\":0,\"y\":0}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([5, 0, 0, 1, 1, -1], 0, 0, [0, 0])","explanation_zh":"依賴結構核對=([5, 0, 0, 1, 1, -1], 0, 0, [0, 0])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244068,"parameters":{"coefficients":[5,0,0,1,1,-1],"x":0,"y":0,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[5,0,0,1,1,-1],\"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-r04-partial-derivative-0069","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[6,1,1,1,-1,-1],\"x\":1,\"y\":1}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([6, 1, 1, 1, -1, -1], 1, 1, [2, -2])","explanation_zh":"依賴結構核對=([6, 1, 1, 1, -1, -1], 1, 1, [2, -2])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244069,"parameters":{"coefficients":[6,1,1,1,-1,-1],"x":1,"y":1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[6,1,1,1,-1,-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-r04-partial-derivative-0070","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[7,2,2,1,0,-1],\"x\":2,\"y\":2}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([7, 2, 2, 1, 0, -1], 2, 2, [6, -2])","explanation_zh":"依賴結構核對=([7, 2, 2, 1, 0, -1], 2, 2, [6, -2])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244070,"parameters":{"coefficients":[7,2,2,1,0,-1],"x":2,"y":2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[7,2,2,1,0,-1],\"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-r04-partial-derivative-0071","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[8,-2,3,1,1,-1],\"x\":-2,\"y\":3}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -1], -2, 3, [-3, -5])","explanation_zh":"依賴結構核對=([8, -2, 3, 1, 1, -1], -2, 3, [-3, -5])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244071,"parameters":{"coefficients":[8,-2,3,1,1,-1],"x":-2,"y":3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[8,-2,3,1,1,-1],\"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-r04-partial-derivative-0072","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[9,-1,-3,1,-1,-1],\"x\":-1,\"y\":-3}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([9, -1, -3, 1, -1, -1], -1, -3, [0, 4])","explanation_zh":"依賴結構核對=([9, -1, -3, 1, -1, -1], -1, -3, [0, 4])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244072,"parameters":{"coefficients":[9,-1,-3,1,-1,-1],"x":-1,"y":-3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[9,-1,-3,1,-1,-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-r04-partial-derivative-0073","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[10,0,-2,1,0,-1],\"x\":0,\"y\":-2}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([10, 0, -2, 1, 0, -1], 0, -2, [0, 2])","explanation_zh":"依賴結構核對=([10, 0, -2, 1, 0, -1], 0, -2, [0, 2])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244073,"parameters":{"coefficients":[10,0,-2,1,0,-1],"x":0,"y":-2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[10,0,-2,1,0,-1],\"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-r04-partial-derivative-0074","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[11,1,-1,1,1,-1],\"x\":1,\"y\":-1}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([11, 1, -1, 1, 1, -1], 1, -1, [2, 2])","explanation_zh":"依賴結構核對=([11, 1, -1, 1, 1, -1], 1, -1, [2, 2])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244074,"parameters":{"coefficients":[11,1,-1,1,1,-1],"x":1,"y":-1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[11,1,-1,1,1,-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-r04-partial-derivative-0075","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-gradient；資料={\"coefficients\":[12,2,0,1,-1,-1],\"x\":2,\"y\":0}。請依 [coefficients,x,y,gradient] 欄序回答。","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, -1], 2, 0, [6, -2])","explanation_zh":"依賴結構核對=([12, 2, 0, 1, -1, -1], 2, 0, [6, -2])；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-gradient","seed":202610244075,"parameters":{"coefficients":[12,2,0,1,-1,-1],"x":2,"y":0,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-gradient","semantic_template":"qf-mod24-partial-gradient/{\"coefficients\":[12,2,0,1,-1,-1],\"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-r04-partial-derivative-0076","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[-12,-2,-3,1,-1,-1],\"fixed_y\":-2,\"x\":-3}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -1], -2, [-10, 0, 1], [0, 2], -3, -6)","explanation_zh":"依賴結構核對=([-12, -2, -3, 1, -1, -1], -2, [-10, 0, 1], [0, 2], -3, -6)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244076,"parameters":{"coefficients":[-12,-2,-3,1,-1,-1],"fixed_y":-2,"x":-3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-12,-2,-3,1,-1,-1],\"fixed_y\":-2,\"x\":-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-r04-partial-derivative-0077","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[-11,-1,-2,1,0,-1],\"fixed_y\":-1,\"x\":-2}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([-11, -1, -2, 1, 0, -1], -1, [-10, -1, 1], [-1, 2], -2, -5)","explanation_zh":"依賴結構核對=([-11, -1, -2, 1, 0, -1], -1, [-10, -1, 1], [-1, 2], -2, -5)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244077,"parameters":{"coefficients":[-11,-1,-2,1,0,-1],"fixed_y":-1,"x":-2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-11,-1,-2,1,0,-1],\"fixed_y\":-1,\"x\":-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-r04-partial-derivative-0078","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[-10,0,-1,1,1,-1],\"fixed_y\":0,\"x\":-1}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0, -1, 1, 1, -1], 0, [-10, 0, 1], [0, 2], -1, -2)","explanation_zh":"依賴結構核對=([-10, 0, -1, 1, 1, -1], 0, [-10, 0, 1], [0, 2], -1, -2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244078,"parameters":{"coefficients":[-10,0,-1,1,1,-1],"fixed_y":0,"x":-1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-10,0,-1,1,1,-1],\"fixed_y\":0,\"x\":-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-r04-partial-derivative-0079","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[-9,1,0,1,-1,-1],\"fixed_y\":1,\"x\":0}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1, 0, 1, -1, -1], 1, [-10, 0, 1], [0, 2], 0, 0)","explanation_zh":"依賴結構核對=([-9, 1, 0, 1, -1, -1], 1, [-10, 0, 1], [0, 2], 0, 0)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244079,"parameters":{"coefficients":[-9,1,0,1,-1,-1],"fixed_y":1,"x":0,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-9,1,0,1,-1,-1],\"fixed_y\":1,\"x\":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-r04-partial-derivative-0080","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[-8,2,1,1,0,-1],\"fixed_y\":2,\"x\":1}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, -1], 2, [-10, 2, 1], [2, 2], 1, 4)","explanation_zh":"依賴結構核對=([-8, 2, 1, 1, 0, -1], 2, [-10, 2, 1], [2, 2], 1, 4)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244080,"parameters":{"coefficients":[-8,2,1,1,0,-1],"fixed_y":2,"x":1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-8,2,1,1,0,-1],\"fixed_y\":2,\"x\":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-r04-partial-derivative-0081","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[-7,-2,2,1,1,-1],\"fixed_y\":-2,\"x\":2}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([-7, -2, 2, 1, 1, -1], -2, [-15, -4, 1], [-4, 2], 2, 0)","explanation_zh":"依賴結構核對=([-7, -2, 2, 1, 1, -1], -2, [-15, -4, 1], [-4, 2], 2, 0)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244081,"parameters":{"coefficients":[-7,-2,2,1,1,-1],"fixed_y":-2,"x":2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-7,-2,2,1,1,-1],\"fixed_y\":-2,\"x\":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-r04-partial-derivative-0082","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[-6,-1,3,1,-1,-1],\"fixed_y\":-1,\"x\":3}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([-6, -1, 3, 1, -1, -1], -1, [-10, 0, 1], [0, 2], 3, 6)","explanation_zh":"依賴結構核對=([-6, -1, 3, 1, -1, -1], -1, [-10, 0, 1], [0, 2], 3, 6)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244082,"parameters":{"coefficients":[-6,-1,3,1,-1,-1],"fixed_y":-1,"x":3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-6,-1,3,1,-1,-1],\"fixed_y\":-1,\"x\":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-r04-partial-derivative-0083","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[-5,0,-3,1,0,-1],\"fixed_y\":0,\"x\":-3}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([-5, 0, -3, 1, 0, -1], 0, [-5, 0, 1], [0, 2], -3, -6)","explanation_zh":"依賴結構核對=([-5, 0, -3, 1, 0, -1], 0, [-5, 0, 1], [0, 2], -3, -6)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244083,"parameters":{"coefficients":[-5,0,-3,1,0,-1],"fixed_y":0,"x":-3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-5,0,-3,1,0,-1],\"fixed_y\":0,\"x\":-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-r04-partial-derivative-0084","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[-4,1,-2,1,1,-1],\"fixed_y\":1,\"x\":-2}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, -1], 1, [-7, 2, 1], [2, 2], -2, -2)","explanation_zh":"依賴結構核對=([-4, 1, -2, 1, 1, -1], 1, [-7, 2, 1], [2, 2], -2, -2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244084,"parameters":{"coefficients":[-4,1,-2,1,1,-1],"fixed_y":1,"x":-2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-4,1,-2,1,1,-1],\"fixed_y\":1,\"x\":-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-r04-partial-derivative-0085","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[-3,2,-1,1,-1,-1],\"fixed_y\":2,\"x\":-1}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2, -1, 1, -1, -1], 2, [-9, 0, 1], [0, 2], -1, -2)","explanation_zh":"依賴結構核對=([-3, 2, -1, 1, -1, -1], 2, [-9, 0, 1], [0, 2], -1, -2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244085,"parameters":{"coefficients":[-3,2,-1,1,-1,-1],"fixed_y":2,"x":-1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-3,2,-1,1,-1,-1],\"fixed_y\":2,\"x\":-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-r04-partial-derivative-0086","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[-2,-2,0,1,0,-1],\"fixed_y\":-2,\"x\":0}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, 0, 1, 0, -1], -2, [-6, -2, 1], [-2, 2], 0, -2)","explanation_zh":"依賴結構核對=([-2, -2, 0, 1, 0, -1], -2, [-6, -2, 1], [-2, 2], 0, -2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244086,"parameters":{"coefficients":[-2,-2,0,1,0,-1],"fixed_y":-2,"x":0,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-2,-2,0,1,0,-1],\"fixed_y\":-2,\"x\":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-r04-partial-derivative-0087","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[-1,-1,1,1,1,-1],\"fixed_y\":-1,\"x\":1}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1, 1, 1, 1, -1], -1, [-3, -2, 1], [-2, 2], 1, 0)","explanation_zh":"依賴結構核對=([-1, -1, 1, 1, 1, -1], -1, [-3, -2, 1], [-2, 2], 1, 0)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244087,"parameters":{"coefficients":[-1,-1,1,1,1,-1],"fixed_y":-1,"x":1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[-1,-1,1,1,1,-1],\"fixed_y\":-1,\"x\":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-r04-partial-derivative-0088","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[0,0,2,1,-1,-1],\"fixed_y\":0,\"x\":2}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, -1], 0, [0, 0, 1], [0, 2], 2, 4)","explanation_zh":"依賴結構核對=([0, 0, 2, 1, -1, -1], 0, [0, 0, 1], [0, 2], 2, 4)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244088,"parameters":{"coefficients":[0,0,2,1,-1,-1],"fixed_y":0,"x":2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[0,0,2,1,-1,-1],\"fixed_y\":0,\"x\":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-r04-partial-derivative-0089","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[1,1,3,1,0,-1],\"fixed_y\":1,\"x\":3}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3, 1, 0, -1], 1, [3, 1, 1], [1, 2], 3, 7)","explanation_zh":"依賴結構核對=([1, 1, 3, 1, 0, -1], 1, [3, 1, 1], [1, 2], 3, 7)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244089,"parameters":{"coefficients":[1,1,3,1,0,-1],"fixed_y":1,"x":3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[1,1,3,1,0,-1],\"fixed_y\":1,\"x\":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-r04-partial-derivative-0090","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[2,2,-3,1,1,-1],\"fixed_y\":2,\"x\":-3}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, -3, 1, 1, -1], 2, [-8, 4, 1], [4, 2], -3, -2)","explanation_zh":"依賴結構核對=([2, 2, -3, 1, 1, -1], 2, [-8, 4, 1], [4, 2], -3, -2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244090,"parameters":{"coefficients":[2,2,-3,1,1,-1],"fixed_y":2,"x":-3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[2,2,-3,1,1,-1],\"fixed_y\":2,\"x\":-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-r04-partial-derivative-0091","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[3,-2,-2,1,-1,-1],\"fixed_y\":-2,\"x\":-2}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([3, -2, -2, 1, -1, -1], -2, [3, 0, 1], [0, 2], -2, -4)","explanation_zh":"依賴結構核對=([3, -2, -2, 1, -1, -1], -2, [3, 0, 1], [0, 2], -2, -4)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244091,"parameters":{"coefficients":[3,-2,-2,1,-1,-1],"fixed_y":-2,"x":-2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[3,-2,-2,1,-1,-1],\"fixed_y\":-2,\"x\":-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-r04-partial-derivative-0092","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[4,-1,-1,1,0,-1],\"fixed_y\":-1,\"x\":-1}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], -1, [4, -1, 1], [-1, 2], -1, -3)","explanation_zh":"依賴結構核對=([4, -1, -1, 1, 0, -1], -1, [4, -1, 1], [-1, 2], -1, -3)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244092,"parameters":{"coefficients":[4,-1,-1,1,0,-1],"fixed_y":-1,"x":-1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[4,-1,-1,1,0,-1],\"fixed_y\":-1,\"x\":-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-r04-partial-derivative-0093","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[5,0,0,1,1,-1],\"fixed_y\":0,\"x\":0}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([5, 0, 0, 1, 1, -1], 0, [5, 0, 1], [0, 2], 0, 0)","explanation_zh":"依賴結構核對=([5, 0, 0, 1, 1, -1], 0, [5, 0, 1], [0, 2], 0, 0)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244093,"parameters":{"coefficients":[5,0,0,1,1,-1],"fixed_y":0,"x":0,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[5,0,0,1,1,-1],\"fixed_y\":0,\"x\":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-r04-partial-derivative-0094","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[6,1,1,1,-1,-1],\"fixed_y\":1,\"x\":1}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([6, 1, 1, 1, -1, -1], 1, [6, 0, 1], [0, 2], 1, 2)","explanation_zh":"依賴結構核對=([6, 1, 1, 1, -1, -1], 1, [6, 0, 1], [0, 2], 1, 2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244094,"parameters":{"coefficients":[6,1,1,1,-1,-1],"fixed_y":1,"x":1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[6,1,1,1,-1,-1],\"fixed_y\":1,\"x\":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-r04-partial-derivative-0095","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[7,2,2,1,0,-1],\"fixed_y\":2,\"x\":2}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([7, 2, 2, 1, 0, -1], 2, [7, 2, 1], [2, 2], 2, 6)","explanation_zh":"依賴結構核對=([7, 2, 2, 1, 0, -1], 2, [7, 2, 1], [2, 2], 2, 6)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244095,"parameters":{"coefficients":[7,2,2,1,0,-1],"fixed_y":2,"x":2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[7,2,2,1,0,-1],\"fixed_y\":2,\"x\":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-r04-partial-derivative-0096","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[8,-2,3,1,1,-1],\"fixed_y\":-2,\"x\":3}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -1], -2, [-2, -4, 1], [-4, 2], 3, 2)","explanation_zh":"依賴結構核對=([8, -2, 3, 1, 1, -1], -2, [-2, -4, 1], [-4, 2], 3, 2)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244096,"parameters":{"coefficients":[8,-2,3,1,1,-1],"fixed_y":-2,"x":3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[8,-2,3,1,1,-1],\"fixed_y\":-2,\"x\":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-r04-partial-derivative-0097","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[9,-1,-3,1,-1,-1],\"fixed_y\":-1,\"x\":-3}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([9, -1, -3, 1, -1, -1], -1, [11, 0, 1], [0, 2], -3, -6)","explanation_zh":"依賴結構核對=([9, -1, -3, 1, -1, -1], -1, [11, 0, 1], [0, 2], -3, -6)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244097,"parameters":{"coefficients":[9,-1,-3,1,-1,-1],"fixed_y":-1,"x":-3,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[9,-1,-3,1,-1,-1],\"fixed_y\":-1,\"x\":-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-r04-partial-derivative-0098","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[10,0,-2,1,0,-1],\"fixed_y\":0,\"x\":-2}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([10, 0, -2, 1, 0, -1], 0, [10, 0, 1], [0, 2], -2, -4)","explanation_zh":"依賴結構核對=([10, 0, -2, 1, 0, -1], 0, [10, 0, 1], [0, 2], -2, -4)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244098,"parameters":{"coefficients":[10,0,-2,1,0,-1],"fixed_y":0,"x":-2,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[10,0,-2,1,0,-1],\"fixed_y\":0,\"x\":-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-r04-partial-derivative-0099","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[11,1,-1,1,1,-1],\"fixed_y\":1,\"x\":-1}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([11, 1, -1, 1, 1, -1], 1, [9, 2, 1], [2, 2], -1, 0)","explanation_zh":"依賴結構核對=([11, 1, -1, 1, 1, -1], 1, [9, 2, 1], [2, 2], -1, 0)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244099,"parameters":{"coefficients":[11,1,-1,1,1,-1],"fixed_y":1,"x":-1,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[11,1,-1,1,1,-1],\"fixed_y\":1,\"x\":-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-r04-partial-derivative-0100","batch_id":"batch-20261013-advance-20261024-r04-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導數","stem_zh":"固定另一座標的多項式切片。操作=partial-x-slice-derivative；資料={\"coefficients\":[12,2,0,1,-1,-1],\"fixed_y\":2,\"x\":0}。請依 [coefficients,fixed_y,slice_coefficients,derivative_coefficients,x,value] 欄序回答。","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, -1], 2, [8, 0, 1], [0, 2], 0, 0)","explanation_zh":"依賴結構核對=([12, 2, 0, 1, -1, -1], 2, [8, 0, 1], [0, 2], 0, 0)；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；偏導依h趨近0的切片極限定義；偏導存在不推出連續或全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導數"]},"variant":{"family_id":"qf-mod24-partial-x-slice-derivative","seed":202610244100,"parameters":{"coefficients":[12,2,0,1,-1,-1],"fixed_y":2,"x":0,"domain":"Polynomial_R2_partial_derivatives","operation":"partial-x-slice-derivative","semantic_template":"qf-mod24-partial-x-slice-derivative/{\"coefficients\":[12,2,0,1,-1,-1],\"fixed_y\":2,\"x\":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"}}