{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261025-r07-partial-derivative-0001","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[-12,-2,-3,1,-1,-2]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -2], [-2, 2, -1], [1,x,y])","explanation_zh":"收束波核對=([-12, -2, -3, 1, -1, -2], [-2, 2, -1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257001,"parameters":{"coefficients":[-12,-2,-3,1,-1,-2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-12,-2,-3,1,-1,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0002","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[-11,-1,-2,2,0,-1]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-11, -1, -2, 2, 0, -1], [-1, 4, 0], [1,x,y])","explanation_zh":"收束波核對=([-11, -1, -2, 2, 0, -1], [-1, 4, 0], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257002,"parameters":{"coefficients":[-11,-1,-2,2,0,-1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-11,-1,-2,2,0,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0003","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[-10,0,-1,3,1,0]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0, -1, 3, 1, 0], [0, 6, 1], [1,x,y])","explanation_zh":"收束波核對=([-10, 0, -1, 3, 1, 0], [0, 6, 1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257003,"parameters":{"coefficients":[-10,0,-1,3,1,0],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-10,0,-1,3,1,0]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0004","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[-9,1,0,4,-1,1]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1, 0, 4, -1, 1], [1, 8, -1], [1,x,y])","explanation_zh":"收束波核對=([-9, 1, 0, 4, -1, 1], [1, 8, -1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257004,"parameters":{"coefficients":[-9,1,0,4,-1,1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-9,1,0,4,-1,1]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0005","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[-8,2,1,1,0,2]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, 2], [2, 2, 0], [1,x,y])","explanation_zh":"收束波核對=([-8, 2, 1, 1, 0, 2], [2, 2, 0], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257005,"parameters":{"coefficients":[-8,2,1,1,0,2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-8,2,1,1,0,2]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0006","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[-7,-2,2,2,1,-2]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-7, -2, 2, 2, 1, -2], [-2, 4, 1], [1,x,y])","explanation_zh":"收束波核對=([-7, -2, 2, 2, 1, -2], [-2, 4, 1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257006,"parameters":{"coefficients":[-7,-2,2,2,1,-2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-7,-2,2,2,1,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0007","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[-6,-1,3,3,-1,-1]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-6, -1, 3, 3, -1, -1], [-1, 6, -1], [1,x,y])","explanation_zh":"收束波核對=([-6, -1, 3, 3, -1, -1], [-1, 6, -1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257007,"parameters":{"coefficients":[-6,-1,3,3,-1,-1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-6,-1,3,3,-1,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0008","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[-5,0,-3,4,0,0]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-5, 0, -3, 4, 0, 0], [0, 8, 0], [1,x,y])","explanation_zh":"收束波核對=([-5, 0, -3, 4, 0, 0], [0, 8, 0], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257008,"parameters":{"coefficients":[-5,0,-3,4,0,0],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-5,0,-3,4,0,0]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0009","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[-4,1,-2,1,1,1]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, 1], [1, 2, 1], [1,x,y])","explanation_zh":"收束波核對=([-4, 1, -2, 1, 1, 1], [1, 2, 1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257009,"parameters":{"coefficients":[-4,1,-2,1,1,1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-4,1,-2,1,1,1]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0010","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[-3,2,-1,2,-1,2]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2, -1, 2, -1, 2], [2, 4, -1], [1,x,y])","explanation_zh":"收束波核對=([-3, 2, -1, 2, -1, 2], [2, 4, -1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257010,"parameters":{"coefficients":[-3,2,-1,2,-1,2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-3,2,-1,2,-1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0011","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[-2,-2,0,3,0,-2]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, 0, 3, 0, -2], [-2, 6, 0], [1,x,y])","explanation_zh":"收束波核對=([-2, -2, 0, 3, 0, -2], [-2, 6, 0], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257011,"parameters":{"coefficients":[-2,-2,0,3,0,-2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-2,-2,0,3,0,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0012","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[-1,-1,1,4,1,-1]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1, 1, 4, 1, -1], [-1, 8, 1], [1,x,y])","explanation_zh":"收束波核對=([-1, -1, 1, 4, 1, -1], [-1, 8, 1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257012,"parameters":{"coefficients":[-1,-1,1,4,1,-1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[-1,-1,1,4,1,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0013","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[0,0,2,1,-1,0]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, 0], [0, 2, -1], [1,x,y])","explanation_zh":"收束波核對=([0, 0, 2, 1, -1, 0], [0, 2, -1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257013,"parameters":{"coefficients":[0,0,2,1,-1,0],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[0,0,2,1,-1,0]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0014","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[1,1,3,2,0,1]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3, 2, 0, 1], [1, 4, 0], [1,x,y])","explanation_zh":"收束波核對=([1, 1, 3, 2, 0, 1], [1, 4, 0], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257014,"parameters":{"coefficients":[1,1,3,2,0,1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[1,1,3,2,0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0015","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[2,2,-3,3,1,2]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, -3, 3, 1, 2], [2, 6, 1], [1,x,y])","explanation_zh":"收束波核對=([2, 2, -3, 3, 1, 2], [2, 6, 1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257015,"parameters":{"coefficients":[2,2,-3,3,1,2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[2,2,-3,3,1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0016","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[3,-2,-2,4,-1,-2]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([3, -2, -2, 4, -1, -2], [-2, 8, -1], [1,x,y])","explanation_zh":"收束波核對=([3, -2, -2, 4, -1, -2], [-2, 8, -1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257016,"parameters":{"coefficients":[3,-2,-2,4,-1,-2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[3,-2,-2,4,-1,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0017","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[4,-1,-1,1,0,-1]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], [-1, 2, 0], [1,x,y])","explanation_zh":"收束波核對=([4, -1, -1, 1, 0, -1], [-1, 2, 0], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257017,"parameters":{"coefficients":[4,-1,-1,1,0,-1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[4,-1,-1,1,0,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0018","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[5,0,0,2,1,0]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([5, 0, 0, 2, 1, 0], [0, 4, 1], [1,x,y])","explanation_zh":"收束波核對=([5, 0, 0, 2, 1, 0], [0, 4, 1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257018,"parameters":{"coefficients":[5,0,0,2,1,0],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[5,0,0,2,1,0]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0019","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[6,1,1,3,-1,1]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([6, 1, 1, 3, -1, 1], [1, 6, -1], [1,x,y])","explanation_zh":"收束波核對=([6, 1, 1, 3, -1, 1], [1, 6, -1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257019,"parameters":{"coefficients":[6,1,1,3,-1,1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[6,1,1,3,-1,1]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0020","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[7,2,2,4,0,2]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([7, 2, 2, 4, 0, 2], [2, 8, 0], [1,x,y])","explanation_zh":"收束波核對=([7, 2, 2, 4, 0, 2], [2, 8, 0], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257020,"parameters":{"coefficients":[7,2,2,4,0,2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[7,2,2,4,0,2]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0021","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[8,-2,3,1,1,-2]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -2], [-2, 2, 1], [1,x,y])","explanation_zh":"收束波核對=([8, -2, 3, 1, 1, -2], [-2, 2, 1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257021,"parameters":{"coefficients":[8,-2,3,1,1,-2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[8,-2,3,1,1,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0022","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[9,-1,-3,2,-1,-1]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([9, -1, -3, 2, -1, -1], [-1, 4, -1], [1,x,y])","explanation_zh":"收束波核對=([9, -1, -3, 2, -1, -1], [-1, 4, -1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257022,"parameters":{"coefficients":[9,-1,-3,2,-1,-1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[9,-1,-3,2,-1,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0023","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[10,0,-2,3,0,0]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([10, 0, -2, 3, 0, 0], [0, 6, 0], [1,x,y])","explanation_zh":"收束波核對=([10, 0, -2, 3, 0, 0], [0, 6, 0], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257023,"parameters":{"coefficients":[10,0,-2,3,0,0],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[10,0,-2,3,0,0]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0024","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[11,1,-1,4,1,1]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([11, 1, -1, 4, 1, 1], [1, 8, 1], [1,x,y])","explanation_zh":"收束波核對=([11, 1, -1, 4, 1, 1], [1, 8, 1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257024,"parameters":{"coefficients":[11,1,-1,4,1,1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[11,1,-1,4,1,1]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0025","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-x-polynomial；資料={\"coefficients\":[12,2,0,1,-1,2]}。請依 [coefficients,partial_x_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, 2], [2, 2, -1], [1,x,y])","explanation_zh":"收束波核對=([12, 2, 0, 1, -1, 2], [2, 2, -1], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-x-polynomial","seed":202610257025,"parameters":{"coefficients":[12,2,0,1,-1,2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-x-polynomial","semantic_template":"qf-mod25-partialdeep-partial-x-polynomial/{\"coefficients\":[12,2,0,1,-1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0026","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[-12,-2,-3,1,-1,-2]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -2], [-3, -1, -4], [1,x,y])","explanation_zh":"收束波核對=([-12, -2, -3, 1, -1, -2], [-3, -1, -4], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257026,"parameters":{"coefficients":[-12,-2,-3,1,-1,-2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-12,-2,-3,1,-1,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0027","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[-11,-1,-2,2,0,-1]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-11, -1, -2, 2, 0, -1], [-2, 0, -2], [1,x,y])","explanation_zh":"收束波核對=([-11, -1, -2, 2, 0, -1], [-2, 0, -2], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257027,"parameters":{"coefficients":[-11,-1,-2,2,0,-1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-11,-1,-2,2,0,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0028","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[-10,0,-1,3,1,0]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0, -1, 3, 1, 0], [-1, 1, 0], [1,x,y])","explanation_zh":"收束波核對=([-10, 0, -1, 3, 1, 0], [-1, 1, 0], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257028,"parameters":{"coefficients":[-10,0,-1,3,1,0],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-10,0,-1,3,1,0]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0029","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[-9,1,0,4,-1,1]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1, 0, 4, -1, 1], [0, -1, 2], [1,x,y])","explanation_zh":"收束波核對=([-9, 1, 0, 4, -1, 1], [0, -1, 2], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257029,"parameters":{"coefficients":[-9,1,0,4,-1,1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-9,1,0,4,-1,1]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0030","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[-8,2,1,1,0,2]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, 2], [1, 0, 4], [1,x,y])","explanation_zh":"收束波核對=([-8, 2, 1, 1, 0, 2], [1, 0, 4], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257030,"parameters":{"coefficients":[-8,2,1,1,0,2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-8,2,1,1,0,2]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0031","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[-7,-2,2,2,1,-2]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-7, -2, 2, 2, 1, -2], [2, 1, -4], [1,x,y])","explanation_zh":"收束波核對=([-7, -2, 2, 2, 1, -2], [2, 1, -4], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257031,"parameters":{"coefficients":[-7,-2,2,2,1,-2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-7,-2,2,2,1,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0032","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[-6,-1,3,3,-1,-1]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-6, -1, 3, 3, -1, -1], [3, -1, -2], [1,x,y])","explanation_zh":"收束波核對=([-6, -1, 3, 3, -1, -1], [3, -1, -2], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257032,"parameters":{"coefficients":[-6,-1,3,3,-1,-1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-6,-1,3,3,-1,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0033","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[-5,0,-3,4,0,0]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-5, 0, -3, 4, 0, 0], [-3, 0, 0], [1,x,y])","explanation_zh":"收束波核對=([-5, 0, -3, 4, 0, 0], [-3, 0, 0], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257033,"parameters":{"coefficients":[-5,0,-3,4,0,0],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-5,0,-3,4,0,0]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0034","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[-4,1,-2,1,1,1]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, 1], [-2, 1, 2], [1,x,y])","explanation_zh":"收束波核對=([-4, 1, -2, 1, 1, 1], [-2, 1, 2], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257034,"parameters":{"coefficients":[-4,1,-2,1,1,1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-4,1,-2,1,1,1]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0035","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[-3,2,-1,2,-1,2]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2, -1, 2, -1, 2], [-1, -1, 4], [1,x,y])","explanation_zh":"收束波核對=([-3, 2, -1, 2, -1, 2], [-1, -1, 4], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257035,"parameters":{"coefficients":[-3,2,-1,2,-1,2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-3,2,-1,2,-1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0036","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[-2,-2,0,3,0,-2]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, 0, 3, 0, -2], [0, 0, -4], [1,x,y])","explanation_zh":"收束波核對=([-2, -2, 0, 3, 0, -2], [0, 0, -4], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257036,"parameters":{"coefficients":[-2,-2,0,3,0,-2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-2,-2,0,3,0,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0037","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[-1,-1,1,4,1,-1]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1, 1, 4, 1, -1], [1, 1, -2], [1,x,y])","explanation_zh":"收束波核對=([-1, -1, 1, 4, 1, -1], [1, 1, -2], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257037,"parameters":{"coefficients":[-1,-1,1,4,1,-1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[-1,-1,1,4,1,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0038","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[0,0,2,1,-1,0]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, 0], [2, -1, 0], [1,x,y])","explanation_zh":"收束波核對=([0, 0, 2, 1, -1, 0], [2, -1, 0], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257038,"parameters":{"coefficients":[0,0,2,1,-1,0],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[0,0,2,1,-1,0]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0039","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[1,1,3,2,0,1]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3, 2, 0, 1], [3, 0, 2], [1,x,y])","explanation_zh":"收束波核對=([1, 1, 3, 2, 0, 1], [3, 0, 2], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257039,"parameters":{"coefficients":[1,1,3,2,0,1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[1,1,3,2,0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0040","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[2,2,-3,3,1,2]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, -3, 3, 1, 2], [-3, 1, 4], [1,x,y])","explanation_zh":"收束波核對=([2, 2, -3, 3, 1, 2], [-3, 1, 4], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257040,"parameters":{"coefficients":[2,2,-3,3,1,2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[2,2,-3,3,1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0041","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[3,-2,-2,4,-1,-2]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([3, -2, -2, 4, -1, -2], [-2, -1, -4], [1,x,y])","explanation_zh":"收束波核對=([3, -2, -2, 4, -1, -2], [-2, -1, -4], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257041,"parameters":{"coefficients":[3,-2,-2,4,-1,-2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[3,-2,-2,4,-1,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0042","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[4,-1,-1,1,0,-1]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], [-1, 0, -2], [1,x,y])","explanation_zh":"收束波核對=([4, -1, -1, 1, 0, -1], [-1, 0, -2], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257042,"parameters":{"coefficients":[4,-1,-1,1,0,-1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[4,-1,-1,1,0,-1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0043","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[5,0,0,2,1,0]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([5, 0, 0, 2, 1, 0], [0, 1, 0], [1,x,y])","explanation_zh":"收束波核對=([5, 0, 0, 2, 1, 0], [0, 1, 0], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257043,"parameters":{"coefficients":[5,0,0,2,1,0],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[5,0,0,2,1,0]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0044","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[6,1,1,3,-1,1]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([6, 1, 1, 3, -1, 1], [1, -1, 2], [1,x,y])","explanation_zh":"收束波核對=([6, 1, 1, 3, -1, 1], [1, -1, 2], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257044,"parameters":{"coefficients":[6,1,1,3,-1,1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[6,1,1,3,-1,1]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0045","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[7,2,2,4,0,2]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([7, 2, 2, 4, 0, 2], [2, 0, 4], [1,x,y])","explanation_zh":"收束波核對=([7, 2, 2, 4, 0, 2], [2, 0, 4], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257045,"parameters":{"coefficients":[7,2,2,4,0,2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[7,2,2,4,0,2]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0046","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[8,-2,3,1,1,-2]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -2], [3, 1, -4], [1,x,y])","explanation_zh":"收束波核對=([8, -2, 3, 1, 1, -2], [3, 1, -4], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257046,"parameters":{"coefficients":[8,-2,3,1,1,-2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[8,-2,3,1,1,-2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0047","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[9,-1,-3,2,-1,-1]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([9, -1, -3, 2, -1, -1], [-3, -1, -2], [1,x,y])","explanation_zh":"收束波核對=([9, -1, -3, 2, -1, -1], [-3, -1, -2], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257047,"parameters":{"coefficients":[9,-1,-3,2,-1,-1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[9,-1,-3,2,-1,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0048","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[10,0,-2,3,0,0]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([10, 0, -2, 3, 0, 0], [-2, 0, 0], [1,x,y])","explanation_zh":"收束波核對=([10, 0, -2, 3, 0, 0], [-2, 0, 0], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257048,"parameters":{"coefficients":[10,0,-2,3,0,0],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[10,0,-2,3,0,0]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0049","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[11,1,-1,4,1,1]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([11, 1, -1, 4, 1, 1], [-1, 1, 2], [1,x,y])","explanation_zh":"收束波核對=([11, 1, -1, 4, 1, 1], [-1, 1, 2], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257049,"parameters":{"coefficients":[11,1,-1,4,1,1],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[11,1,-1,4,1,1]}"}},"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0050","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-polynomial；資料={\"coefficients\":[12,2,0,1,-1,2]}。請依 [coefficients,partial_y_coefficients,basis_order] 欄序回答。","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, 2], [0, -1, 4], [1,x,y])","explanation_zh":"收束波核對=([12, 2, 0, 1, -1, 2], [0, -1, 4], [1,x,y])；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-polynomial","seed":202610257050,"parameters":{"coefficients":[12,2,0,1,-1,2],"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-polynomial","semantic_template":"qf-mod25-partialdeep-partial-y-polynomial/{\"coefficients\":[12,2,0,1,-1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0051","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[-12,-2,-3,1,-1,-2],\"fixed_x\":-2,\"y\":-3}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -2], -2, [-4, -1, -2], [-1, -4], -3, 11)","explanation_zh":"收束波核對=([-12, -2, -3, 1, -1, -2], -2, [-4, -1, -2], [-1, -4], -3, 11)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257051,"parameters":{"coefficients":[-12,-2,-3,1,-1,-2],"fixed_x":-2,"y":-3,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-12,-2,-3,1,-1,-2],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0052","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[-11,-1,-2,2,0,-1],\"fixed_x\":-1,\"y\":-2}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([-11, -1, -2, 2, 0, -1], -1, [-8, -2, -1], [-2, -2], -2, 2)","explanation_zh":"收束波核對=([-11, -1, -2, 2, 0, -1], -1, [-8, -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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257052,"parameters":{"coefficients":[-11,-1,-2,2,0,-1],"fixed_x":-1,"y":-2,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-11,-1,-2,2,0,-1],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0053","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[-10,0,-1,3,1,0],\"fixed_x\":0,\"y\":-1}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0, -1, 3, 1, 0], 0, [-10, -1, 0], [-1, 0], -1, -1)","explanation_zh":"收束波核對=([-10, 0, -1, 3, 1, 0], 0, [-10, -1, 0], [-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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257053,"parameters":{"coefficients":[-10,0,-1,3,1,0],"fixed_x":0,"y":-1,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-10,0,-1,3,1,0],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0054","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[-9,1,0,4,-1,1],\"fixed_x\":1,\"y\":0}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1, 0, 4, -1, 1], 1, [-4, -1, 1], [-1, 2], 0, -1)","explanation_zh":"收束波核對=([-9, 1, 0, 4, -1, 1], 1, [-4, -1, 1], [-1, 2], 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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257054,"parameters":{"coefficients":[-9,1,0,4,-1,1],"fixed_x":1,"y":0,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-9,1,0,4,-1,1],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0055","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[-8,2,1,1,0,2],\"fixed_x\":2,\"y\":1}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, 2], 2, [0, 1, 2], [1, 4], 1, 5)","explanation_zh":"收束波核對=([-8, 2, 1, 1, 0, 2], 2, [0, 1, 2], [1, 4], 1, 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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257055,"parameters":{"coefficients":[-8,2,1,1,0,2],"fixed_x":2,"y":1,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-8,2,1,1,0,2],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0056","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[-7,-2,2,2,1,-2],\"fixed_x\":-2,\"y\":2}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([-7, -2, 2, 2, 1, -2], -2, [5, 0, -2], [0, -4], 2, -8)","explanation_zh":"收束波核對=([-7, -2, 2, 2, 1, -2], -2, [5, 0, -2], [0, -4], 2, -8)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257056,"parameters":{"coefficients":[-7,-2,2,2,1,-2],"fixed_x":-2,"y":2,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-7,-2,2,2,1,-2],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0057","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[-6,-1,3,3,-1,-1],\"fixed_x\":-1,\"y\":3}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([-6, -1, 3, 3, -1, -1], -1, [-2, 4, -1], [4, -2], 3, -2)","explanation_zh":"收束波核對=([-6, -1, 3, 3, -1, -1], -1, [-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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257057,"parameters":{"coefficients":[-6,-1,3,3,-1,-1],"fixed_x":-1,"y":3,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-6,-1,3,3,-1,-1],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0058","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[-5,0,-3,4,0,0],\"fixed_x\":0,\"y\":-3}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([-5, 0, -3, 4, 0, 0], 0, [-5, -3, 0], [-3, 0], -3, -3)","explanation_zh":"收束波核對=([-5, 0, -3, 4, 0, 0], 0, [-5, -3, 0], [-3, 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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257058,"parameters":{"coefficients":[-5,0,-3,4,0,0],"fixed_x":0,"y":-3,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-5,0,-3,4,0,0],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0059","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[-4,1,-2,1,1,1],\"fixed_x\":1,\"y\":-2}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, 1], 1, [-2, -1, 1], [-1, 2], -2, -5)","explanation_zh":"收束波核對=([-4, 1, -2, 1, 1, 1], 1, [-2, -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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257059,"parameters":{"coefficients":[-4,1,-2,1,1,1],"fixed_x":1,"y":-2,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-4,1,-2,1,1,1],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0060","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[-3,2,-1,2,-1,2],\"fixed_x\":2,\"y\":-1}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2, -1, 2, -1, 2], 2, [9, -3, 2], [-3, 4], -1, -7)","explanation_zh":"收束波核對=([-3, 2, -1, 2, -1, 2], 2, [9, -3, 2], [-3, 4], -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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257060,"parameters":{"coefficients":[-3,2,-1,2,-1,2],"fixed_x":2,"y":-1,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-3,2,-1,2,-1,2],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0061","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[-2,-2,0,3,0,-2],\"fixed_x\":-2,\"y\":0}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, 0, 3, 0, -2], -2, [14, 0, -2], [0, -4], 0, 0)","explanation_zh":"收束波核對=([-2, -2, 0, 3, 0, -2], -2, [14, 0, -2], [0, -4], 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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257061,"parameters":{"coefficients":[-2,-2,0,3,0,-2],"fixed_x":-2,"y":0,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-2,-2,0,3,0,-2],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0062","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[-1,-1,1,4,1,-1],\"fixed_x\":-1,\"y\":1}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1, 1, 4, 1, -1], -1, [4, 0, -1], [0, -2], 1, -2)","explanation_zh":"收束波核對=([-1, -1, 1, 4, 1, -1], -1, [4, 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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257062,"parameters":{"coefficients":[-1,-1,1,4,1,-1],"fixed_x":-1,"y":1,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[-1,-1,1,4,1,-1],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0063","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[0,0,2,1,-1,0],\"fixed_x\":0,\"y\":2}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, 0], 0, [0, 2, 0], [2, 0], 2, 2)","explanation_zh":"收束波核對=([0, 0, 2, 1, -1, 0], 0, [0, 2, 0], [2, 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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257063,"parameters":{"coefficients":[0,0,2,1,-1,0],"fixed_x":0,"y":2,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[0,0,2,1,-1,0],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0064","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[1,1,3,2,0,1],\"fixed_x\":1,\"y\":3}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3, 2, 0, 1], 1, [4, 3, 1], [3, 2], 3, 9)","explanation_zh":"收束波核對=([1, 1, 3, 2, 0, 1], 1, [4, 3, 1], [3, 2], 3, 9)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257064,"parameters":{"coefficients":[1,1,3,2,0,1],"fixed_x":1,"y":3,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[1,1,3,2,0,1],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0065","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[2,2,-3,3,1,2],\"fixed_x\":2,\"y\":-3}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, -3, 3, 1, 2], 2, [18, -1, 2], [-1, 4], -3, -13)","explanation_zh":"收束波核對=([2, 2, -3, 3, 1, 2], 2, [18, -1, 2], [-1, 4], -3, -13)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257065,"parameters":{"coefficients":[2,2,-3,3,1,2],"fixed_x":2,"y":-3,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[2,2,-3,3,1,2],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0066","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[3,-2,-2,4,-1,-2],\"fixed_x\":-2,\"y\":-2}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([3, -2, -2, 4, -1, -2], -2, [23, 0, -2], [0, -4], -2, 8)","explanation_zh":"收束波核對=([3, -2, -2, 4, -1, -2], -2, [23, 0, -2], [0, -4], -2, 8)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257066,"parameters":{"coefficients":[3,-2,-2,4,-1,-2],"fixed_x":-2,"y":-2,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[3,-2,-2,4,-1,-2],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0067","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[4,-1,-1,1,0,-1],\"fixed_x\":-1,\"y\":-1}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], -1, [6, -1, -1], [-1, -2], -1, 1)","explanation_zh":"收束波核對=([4, -1, -1, 1, 0, -1], -1, [6, -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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257067,"parameters":{"coefficients":[4,-1,-1,1,0,-1],"fixed_x":-1,"y":-1,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[4,-1,-1,1,0,-1],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0068","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[5,0,0,2,1,0],\"fixed_x\":0,\"y\":0}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([5, 0, 0, 2, 1, 0], 0, [5, 0, 0], [0, 0], 0, 0)","explanation_zh":"收束波核對=([5, 0, 0, 2, 1, 0], 0, [5, 0, 0], [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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257068,"parameters":{"coefficients":[5,0,0,2,1,0],"fixed_x":0,"y":0,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[5,0,0,2,1,0],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0069","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[6,1,1,3,-1,1],\"fixed_x\":1,\"y\":1}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([6, 1, 1, 3, -1, 1], 1, [10, 0, 1], [0, 2], 1, 2)","explanation_zh":"收束波核對=([6, 1, 1, 3, -1, 1], 1, [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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257069,"parameters":{"coefficients":[6,1,1,3,-1,1],"fixed_x":1,"y":1,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[6,1,1,3,-1,1],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0070","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[7,2,2,4,0,2],\"fixed_x\":2,\"y\":2}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([7, 2, 2, 4, 0, 2], 2, [27, 2, 2], [2, 4], 2, 10)","explanation_zh":"收束波核對=([7, 2, 2, 4, 0, 2], 2, [27, 2, 2], [2, 4], 2, 10)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257070,"parameters":{"coefficients":[7,2,2,4,0,2],"fixed_x":2,"y":2,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[7,2,2,4,0,2],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0071","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[8,-2,3,1,1,-2],\"fixed_x\":-2,\"y\":3}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -2], -2, [16, 1, -2], [1, -4], 3, -11)","explanation_zh":"收束波核對=([8, -2, 3, 1, 1, -2], -2, [16, 1, -2], [1, -4], 3, -11)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257071,"parameters":{"coefficients":[8,-2,3,1,1,-2],"fixed_x":-2,"y":3,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[8,-2,3,1,1,-2],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0072","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[9,-1,-3,2,-1,-1],\"fixed_x\":-1,\"y\":-3}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([9, -1, -3, 2, -1, -1], -1, [12, -2, -1], [-2, -2], -3, 4)","explanation_zh":"收束波核對=([9, -1, -3, 2, -1, -1], -1, [12, -2, -1], [-2, -2], -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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257072,"parameters":{"coefficients":[9,-1,-3,2,-1,-1],"fixed_x":-1,"y":-3,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[9,-1,-3,2,-1,-1],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0073","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[10,0,-2,3,0,0],\"fixed_x\":0,\"y\":-2}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([10, 0, -2, 3, 0, 0], 0, [10, -2, 0], [-2, 0], -2, -2)","explanation_zh":"收束波核對=([10, 0, -2, 3, 0, 0], 0, [10, -2, 0], [-2, 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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257073,"parameters":{"coefficients":[10,0,-2,3,0,0],"fixed_x":0,"y":-2,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[10,0,-2,3,0,0],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0074","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[11,1,-1,4,1,1],\"fixed_x\":1,\"y\":-1}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([11, 1, -1, 4, 1, 1], 1, [16, 0, 1], [0, 2], -1, -2)","explanation_zh":"收束波核對=([11, 1, -1, 4, 1, 1], 1, [16, 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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257074,"parameters":{"coefficients":[11,1,-1,4,1,1],"fixed_x":1,"y":-1,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[11,1,-1,4,1,1],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0075","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-partial-y-slice-derivative；資料={\"coefficients\":[12,2,0,1,-1,2],\"fixed_x\":2,\"y\":0}。請依 [coefficients,fixed_x,slice_coefficients,derivative_coefficients,y,value] 欄序回答。","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, 2], 2, [20, -2, 2], [-2, 4], 0, -2)","explanation_zh":"收束波核對=([12, 2, 0, 1, -1, 2], 2, [20, -2, 2], [-2, 4], 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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-partial-y-slice-derivative","seed":202610257075,"parameters":{"coefficients":[12,2,0,1,-1,2],"fixed_x":2,"y":0,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-partial-y-slice-derivative","semantic_template":"qf-mod25-partialdeep-partial-y-slice-derivative/{\"coefficients\":[12,2,0,1,-1,2],\"fixed_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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0076","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[-12,-2,-3,1,-1,-2],\"x\":-2,\"y\":-3}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -2], -2, -3, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([-12, -2, -3, 1, -1, -2], -2, -3, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257076,"parameters":{"coefficients":[-12,-2,-3,1,-1,-2],"x":-2,"y":-3,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-12,-2,-3,1,-1,-2],\"x\":-2,\"y\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0077","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[-11,-1,-2,2,0,-1],\"x\":-1,\"y\":-2}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([-11, -1, -2, 2, 0, -1], -1, -2, [-5, 2], quotient=constant+h_coefficient*h for h≠0, -5, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([-11, -1, -2, 2, 0, -1], -1, -2, [-5, 2], quotient=constant+h_coefficient*h for h≠0, -5, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257077,"parameters":{"coefficients":[-11,-1,-2,2,0,-1],"x":-1,"y":-2,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-11,-1,-2,2,0,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0078","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[-10,0,-1,3,1,0],\"x\":0,\"y\":-1}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([-10, 0, -1, 3, 1, 0], 0, -1, [-1, 3], quotient=constant+h_coefficient*h for h≠0, -1, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([-10, 0, -1, 3, 1, 0], 0, -1, [-1, 3], quotient=constant+h_coefficient*h for h≠0, -1, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257078,"parameters":{"coefficients":[-10,0,-1,3,1,0],"x":0,"y":-1,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-10,0,-1,3,1,0],\"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0079","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[-9,1,0,4,-1,1],\"x\":1,\"y\":0}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([-9, 1, 0, 4, -1, 1], 1, 0, [9, 4], quotient=constant+h_coefficient*h for h≠0, 9, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([-9, 1, 0, 4, -1, 1], 1, 0, [9, 4], quotient=constant+h_coefficient*h for h≠0, 9, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257079,"parameters":{"coefficients":[-9,1,0,4,-1,1],"x":1,"y":0,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-9,1,0,4,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0080","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[-8,2,1,1,0,2],\"x\":2,\"y\":1}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, 2], 2, 1, [6, 1], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([-8, 2, 1, 1, 0, 2], 2, 1, [6, 1], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257080,"parameters":{"coefficients":[-8,2,1,1,0,2],"x":2,"y":1,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-8,2,1,1,0,2],\"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0081","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[-7,-2,2,2,1,-2],\"x\":-2,\"y\":2}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([-7, -2, 2, 2, 1, -2], -2, 2, [-8, 2], quotient=constant+h_coefficient*h for h≠0, -8, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([-7, -2, 2, 2, 1, -2], -2, 2, [-8, 2], quotient=constant+h_coefficient*h for h≠0, -8, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257081,"parameters":{"coefficients":[-7,-2,2,2,1,-2],"x":-2,"y":2,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-7,-2,2,2,1,-2],\"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0082","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[-6,-1,3,3,-1,-1],\"x\":-1,\"y\":3}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([-6, -1, 3, 3, -1, -1], -1, 3, [-10, 3], quotient=constant+h_coefficient*h for h≠0, -10, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([-6, -1, 3, 3, -1, -1], -1, 3, [-10, 3], quotient=constant+h_coefficient*h for h≠0, -10, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257082,"parameters":{"coefficients":[-6,-1,3,3,-1,-1],"x":-1,"y":3,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-6,-1,3,3,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0083","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[-5,0,-3,4,0,0],\"x\":0,\"y\":-3}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([-5, 0, -3, 4, 0, 0], 0, -3, [0, 4], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([-5, 0, -3, 4, 0, 0], 0, -3, [0, 4], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257083,"parameters":{"coefficients":[-5,0,-3,4,0,0],"x":0,"y":-3,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-5,0,-3,4,0,0],\"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0084","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[-4,1,-2,1,1,1],\"x\":1,\"y\":-2}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, 1], 1, -2, [1, 1], quotient=constant+h_coefficient*h for h≠0, 1, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([-4, 1, -2, 1, 1, 1], 1, -2, [1, 1], quotient=constant+h_coefficient*h for h≠0, 1, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257084,"parameters":{"coefficients":[-4,1,-2,1,1,1],"x":1,"y":-2,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0085","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[-3,2,-1,2,-1,2],\"x\":2,\"y\":-1}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([-3, 2, -1, 2, -1, 2], 2, -1, [11, 2], quotient=constant+h_coefficient*h for h≠0, 11, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([-3, 2, -1, 2, -1, 2], 2, -1, [11, 2], quotient=constant+h_coefficient*h for h≠0, 11, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257085,"parameters":{"coefficients":[-3,2,-1,2,-1,2],"x":2,"y":-1,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-3,2,-1,2,-1,2],\"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0086","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[-2,-2,0,3,0,-2],\"x\":-2,\"y\":0}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, 0, 3, 0, -2], -2, 0, [-14, 3], quotient=constant+h_coefficient*h for h≠0, -14, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([-2, -2, 0, 3, 0, -2], -2, 0, [-14, 3], quotient=constant+h_coefficient*h for h≠0, -14, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257086,"parameters":{"coefficients":[-2,-2,0,3,0,-2],"x":-2,"y":0,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-2,-2,0,3,0,-2],\"x\":-2,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0087","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[-1,-1,1,4,1,-1],\"x\":-1,\"y\":1}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1, 1, 4, 1, -1], -1, 1, [-8, 4], quotient=constant+h_coefficient*h for h≠0, -8, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([-1, -1, 1, 4, 1, -1], -1, 1, [-8, 4], quotient=constant+h_coefficient*h for h≠0, -8, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257087,"parameters":{"coefficients":[-1,-1,1,4,1,-1],"x":-1,"y":1,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[-1,-1,1,4,1,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0088","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[0,0,2,1,-1,0],\"x\":0,\"y\":2}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, 0], 0, 2, [-2, 1], quotient=constant+h_coefficient*h for h≠0, -2, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([0, 0, 2, 1, -1, 0], 0, 2, [-2, 1], quotient=constant+h_coefficient*h for h≠0, -2, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257088,"parameters":{"coefficients":[0,0,2,1,-1,0],"x":0,"y":2,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[0,0,2,1,-1,0],\"x\":0,\"y\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0089","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[1,1,3,2,0,1],\"x\":1,\"y\":3}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 3, 2, 0, 1], 1, 3, [5, 2], quotient=constant+h_coefficient*h for h≠0, 5, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([1, 1, 3, 2, 0, 1], 1, 3, [5, 2], quotient=constant+h_coefficient*h for h≠0, 5, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257089,"parameters":{"coefficients":[1,1,3,2,0,1],"x":1,"y":3,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[1,1,3,2,0,1],\"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0090","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[2,2,-3,3,1,2],\"x\":2,\"y\":-3}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, -3, 3, 1, 2], 2, -3, [11, 3], quotient=constant+h_coefficient*h for h≠0, 11, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([2, 2, -3, 3, 1, 2], 2, -3, [11, 3], quotient=constant+h_coefficient*h for h≠0, 11, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257090,"parameters":{"coefficients":[2,2,-3,3,1,2],"x":2,"y":-3,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[2,2,-3,3,1,2],\"x\":2,\"y\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0091","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[3,-2,-2,4,-1,-2],\"x\":-2,\"y\":-2}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([3, -2, -2, 4, -1, -2], -2, -2, [-16, 4], quotient=constant+h_coefficient*h for h≠0, -16, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([3, -2, -2, 4, -1, -2], -2, -2, [-16, 4], quotient=constant+h_coefficient*h for h≠0, -16, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257091,"parameters":{"coefficients":[3,-2,-2,4,-1,-2],"x":-2,"y":-2,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[3,-2,-2,4,-1,-2],\"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0092","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[4,-1,-1,1,0,-1],\"x\":-1,\"y\":-1}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], -1, -1, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([4, -1, -1, 1, 0, -1], -1, -1, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257092,"parameters":{"coefficients":[4,-1,-1,1,0,-1],"x":-1,"y":-1,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0093","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[5,0,0,2,1,0],\"x\":0,\"y\":0}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([5, 0, 0, 2, 1, 0], 0, 0, [0, 2], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([5, 0, 0, 2, 1, 0], 0, 0, [0, 2], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257093,"parameters":{"coefficients":[5,0,0,2,1,0],"x":0,"y":0,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[5,0,0,2,1,0],\"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0094","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[6,1,1,3,-1,1],\"x\":1,\"y\":1}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([6, 1, 1, 3, -1, 1], 1, 1, [6, 3], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([6, 1, 1, 3, -1, 1], 1, 1, [6, 3], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257094,"parameters":{"coefficients":[6,1,1,3,-1,1],"x":1,"y":1,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[6,1,1,3,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0095","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[7,2,2,4,0,2],\"x\":2,\"y\":2}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([7, 2, 2, 4, 0, 2], 2, 2, [18, 4], quotient=constant+h_coefficient*h for h≠0, 18, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([7, 2, 2, 4, 0, 2], 2, 2, [18, 4], quotient=constant+h_coefficient*h for h≠0, 18, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257095,"parameters":{"coefficients":[7,2,2,4,0,2],"x":2,"y":2,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[7,2,2,4,0,2],\"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0096","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[8,-2,3,1,1,-2],\"x\":-2,\"y\":3}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -2], -2, 3, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([8, -2, 3, 1, 1, -2], -2, 3, [-3, 1], quotient=constant+h_coefficient*h for h≠0, -3, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257096,"parameters":{"coefficients":[8,-2,3,1,1,-2],"x":-2,"y":3,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[8,-2,3,1,1,-2],\"x\":-2,\"y\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0097","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[9,-1,-3,2,-1,-1],\"x\":-1,\"y\":-3}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([9, -1, -3, 2, -1, -1], -1, -3, [-2, 2], quotient=constant+h_coefficient*h for h≠0, -2, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([9, -1, -3, 2, -1, -1], -1, -3, [-2, 2], quotient=constant+h_coefficient*h for h≠0, -2, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257097,"parameters":{"coefficients":[9,-1,-3,2,-1,-1],"x":-1,"y":-3,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[9,-1,-3,2,-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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0098","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[10,0,-2,3,0,0],\"x\":0,\"y\":-2}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([10, 0, -2, 3, 0, 0], 0, -2, [0, 3], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([10, 0, -2, 3, 0, 0], 0, -2, [0, 3], quotient=constant+h_coefficient*h for h≠0, 0, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257098,"parameters":{"coefficients":[10,0,-2,3,0,0],"x":0,"y":-2,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[10,0,-2,3,0,0],\"x\":0,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0099","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[11,1,-1,4,1,1],\"x\":1,\"y\":-1}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([11, 1, -1, 4, 1, 1], 1, -1, [8, 4], quotient=constant+h_coefficient*h for h≠0, 8, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([11, 1, -1, 4, 1, 1], 1, -1, [8, 4], quotient=constant+h_coefficient*h for h≠0, 8, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257099,"parameters":{"coefficients":[11,1,-1,4,1,1],"x":1,"y":-1,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[11,1,-1,4,1,1],\"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、BVP適定性、全域最佳解、Markov長期性、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-20261025-r07-partial-derivative-0100","batch_id":"batch-20261013-advance-20261025-r07-partial-derivative-foundations","language":"zh-Hant","source_mko_ids":["mko-partial-derivative"],"title_zh":"偏導密度","stem_zh":"固定另一座標的多項式切片。操作=partialdeep-slice-difference-quotient-limit；資料={\"coefficients\":[12,2,0,1,-1,2],\"x\":2,\"y\":0}。請依 [coefficients,x,y,quotient_coefficients,quotient_formula,limit,scope] 欄序回答。","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, 2], 2, 0, [6, 1], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference)","explanation_zh":"收束波核對=([12, 2, 0, 1, -1, 2], 2, 0, [6, 1], quotient=constant+h_coefficient*h for h≠0, 6, symbolic_limit_not_finite_difference)；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；符號消去h後才取極限；不以有限差分證偏導或推出全微分。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏導密度"]},"variant":{"family_id":"qf-mod25-partialdeep-slice-difference-quotient-limit","seed":202610257100,"parameters":{"coefficients":[12,2,0,1,-1,2],"x":2,"y":0,"domain":"Polynomial_partial_derivative_density","operation":"partialdeep-slice-difference-quotient-limit","semantic_template":"qf-mod25-partialdeep-slice-difference-quotient-limit/{\"coefficients\":[12,2,0,1,-1,2],\"x\":2,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、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"}}