{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0001","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[-10,-3,-2],\"x0\":-2,\"y0\":-11}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([-10, -3, -2], -2, -12, -11, 否)","explanation_zh":"應用擴展核對=([-10, -3, -2], -2, -12, -11, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214001,"parameters":{"coefficients":[-10,-3,-2],"x0":-2,"y0":-11,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-10,-3,-2],\"x0\":-2,\"y0\":-11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0002","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[-9,-2,0],\"x0\":-1,\"y0\":-7}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([-9, -2, 0], -1, -7, -7, 是)","explanation_zh":"應用擴展核對=([-9, -2, 0], -1, -7, -7, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214002,"parameters":{"coefficients":[-9,-2,0],"x0":-1,"y0":-7,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-9,-2,0],\"x0\":-1,\"y0\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0003","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[-8,-1,2],\"x0\":0,\"y0\":-8}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([-8, -1, 2], 0, -8, -8, 是)","explanation_zh":"應用擴展核對=([-8, -1, 2], 0, -8, -8, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214003,"parameters":{"coefficients":[-8,-1,2],"x0":0,"y0":-8,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-8,-1,2],\"x0\":0,\"y0\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0004","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[-7,0,-1],\"x0\":1,\"y0\":-8}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([-7, 0, -1], 1, -8, -8, 是)","explanation_zh":"應用擴展核對=([-7, 0, -1], 1, -8, -8, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214004,"parameters":{"coefficients":[-7,0,-1],"x0":1,"y0":-8,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-7,0,-1],\"x0\":1,\"y0\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0005","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[-6,1,1],\"x0\":2,\"y0\":1}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([-6, 1, 1], 2, 0, 1, 否)","explanation_zh":"應用擴展核對=([-6, 1, 1], 2, 0, 1, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214005,"parameters":{"coefficients":[-6,1,1],"x0":2,"y0":1,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-6,1,1],\"x0\":2,\"y0\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0006","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[-5,2,-2],\"x0\":-2,\"y0\":-17}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([-5, 2, -2], -2, -17, -17, 是)","explanation_zh":"應用擴展核對=([-5, 2, -2], -2, -17, -17, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214006,"parameters":{"coefficients":[-5,2,-2],"x0":-2,"y0":-17,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-5,2,-2],\"x0\":-2,\"y0\":-17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0007","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[-4,3,0],\"x0\":-1,\"y0\":-7}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([-4, 3, 0], -1, -7, -7, 是)","explanation_zh":"應用擴展核對=([-4, 3, 0], -1, -7, -7, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214007,"parameters":{"coefficients":[-4,3,0],"x0":-1,"y0":-7,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-4,3,0],\"x0\":-1,\"y0\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0008","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[-3,-3,2],\"x0\":0,\"y0\":-3}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([-3, -3, 2], 0, -3, -3, 是)","explanation_zh":"應用擴展核對=([-3, -3, 2], 0, -3, -3, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214008,"parameters":{"coefficients":[-3,-3,2],"x0":0,"y0":-3,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-3,-3,2],\"x0\":0,\"y0\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0009","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[-2,-2,-1],\"x0\":1,\"y0\":-4}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, -1], 1, -5, -4, 否)","explanation_zh":"應用擴展核對=([-2, -2, -1], 1, -5, -4, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214009,"parameters":{"coefficients":[-2,-2,-1],"x0":1,"y0":-4,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-2,-2,-1],\"x0\":1,\"y0\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0010","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[-1,-1,1],\"x0\":2,\"y0\":1}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1, 1], 2, 1, 1, 是)","explanation_zh":"應用擴展核對=([-1, -1, 1], 2, 1, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214010,"parameters":{"coefficients":[-1,-1,1],"x0":2,"y0":1,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-1,-1,1],\"x0\":2,\"y0\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0011","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[0,0,-2],\"x0\":-2,\"y0\":-8}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, -2], -2, -8, -8, 是)","explanation_zh":"應用擴展核對=([0, 0, -2], -2, -8, -8, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214011,"parameters":{"coefficients":[0,0,-2],"x0":-2,"y0":-8,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[0,0,-2],\"x0\":-2,\"y0\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0012","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[1,1,0],\"x0\":-1,\"y0\":0}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 0], -1, 0, 0, 是)","explanation_zh":"應用擴展核對=([1, 1, 0], -1, 0, 0, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214012,"parameters":{"coefficients":[1,1,0],"x0":-1,"y0":0,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[1,1,0],\"x0\":-1,\"y0\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0013","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[2,2,2],\"x0\":0,\"y0\":3}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 2], 0, 2, 3, 否)","explanation_zh":"應用擴展核對=([2, 2, 2], 0, 2, 3, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214013,"parameters":{"coefficients":[2,2,2],"x0":0,"y0":3,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[2,2,2],\"x0\":0,\"y0\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0014","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[3,3,-1],\"x0\":1,\"y0\":5}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([3, 3, -1], 1, 5, 5, 是)","explanation_zh":"應用擴展核對=([3, 3, -1], 1, 5, 5, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214014,"parameters":{"coefficients":[3,3,-1],"x0":1,"y0":5,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[3,3,-1],\"x0\":1,\"y0\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0015","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[4,-3,1],\"x0\":2,\"y0\":2}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([4, -3, 1], 2, 2, 2, 是)","explanation_zh":"應用擴展核對=([4, -3, 1], 2, 2, 2, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214015,"parameters":{"coefficients":[4,-3,1],"x0":2,"y0":2,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[4,-3,1],\"x0\":2,\"y0\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0016","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[5,-2,-2],\"x0\":-2,\"y0\":1}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([5, -2, -2], -2, 1, 1, 是)","explanation_zh":"應用擴展核對=([5, -2, -2], -2, 1, 1, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214016,"parameters":{"coefficients":[5,-2,-2],"x0":-2,"y0":1,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[5,-2,-2],\"x0\":-2,\"y0\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0017","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[6,-1,0],\"x0\":-1,\"y0\":8}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([6, -1, 0], -1, 7, 8, 否)","explanation_zh":"應用擴展核對=([6, -1, 0], -1, 7, 8, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214017,"parameters":{"coefficients":[6,-1,0],"x0":-1,"y0":8,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[6,-1,0],\"x0\":-1,\"y0\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0018","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[7,0,2],\"x0\":0,\"y0\":7}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([7, 0, 2], 0, 7, 7, 是)","explanation_zh":"應用擴展核對=([7, 0, 2], 0, 7, 7, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214018,"parameters":{"coefficients":[7,0,2],"x0":0,"y0":7,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[7,0,2],\"x0\":0,\"y0\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0019","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[8,1,-1],\"x0\":1,\"y0\":8}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([8, 1, -1], 1, 8, 8, 是)","explanation_zh":"應用擴展核對=([8, 1, -1], 1, 8, 8, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214019,"parameters":{"coefficients":[8,1,-1],"x0":1,"y0":8,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[8,1,-1],\"x0\":1,\"y0\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0020","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[9,2,1],\"x0\":2,\"y0\":17}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([9, 2, 1], 2, 17, 17, 是)","explanation_zh":"應用擴展核對=([9, 2, 1], 2, 17, 17, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214020,"parameters":{"coefficients":[9,2,1],"x0":2,"y0":17,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[9,2,1],\"x0\":2,\"y0\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0021","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[10,3,-2],\"x0\":-2,\"y0\":-3}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([10, 3, -2], -2, -4, -3, 否)","explanation_zh":"應用擴展核對=([10, 3, -2], -2, -4, -3, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214021,"parameters":{"coefficients":[10,3,-2],"x0":-2,"y0":-3,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[10,3,-2],\"x0\":-2,\"y0\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0022","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[11,-3,0],\"x0\":-1,\"y0\":14}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([11, -3, 0], -1, 14, 14, 是)","explanation_zh":"應用擴展核對=([11, -3, 0], -1, 14, 14, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214022,"parameters":{"coefficients":[11,-3,0],"x0":-1,"y0":14,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[11,-3,0],\"x0\":-1,\"y0\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0023","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[12,-2,2],\"x0\":0,\"y0\":12}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([12, -2, 2], 0, 12, 12, 是)","explanation_zh":"應用擴展核對=([12, -2, 2], 0, 12, 12, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214023,"parameters":{"coefficients":[12,-2,2],"x0":0,"y0":12,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[12,-2,2],\"x0\":0,\"y0\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0024","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[13,-1,-1],\"x0\":1,\"y0\":11}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([13, -1, -1], 1, 11, 11, 是)","explanation_zh":"應用擴展核對=([13, -1, -1], 1, 11, 11, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214024,"parameters":{"coefficients":[13,-1,-1],"x0":1,"y0":11,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[13,-1,-1],\"x0\":1,\"y0\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0025","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[14,0,1],\"x0\":2,\"y0\":19}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。","answer":{"kind":"expression","value":"([14, 0, 1], 2, 18, 19, 否)","explanation_zh":"應用擴展核對=([14, 0, 1], 2, 18, 19, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214025,"parameters":{"coefficients":[14,0,1],"x0":2,"y0":19,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-initial-audit","semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[14,0,1],\"x0\":2,\"y0\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0026","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[-10,-3,-2],\"x0\":-2,\"y0\":-11,\"dy0\":4}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([-10, -3, -2], -2, -12, 5, -11, 4, 否, 否, 否)","explanation_zh":"應用擴展核對=([-10, -3, -2], -2, -12, 5, -11, 4, 否, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214026,"parameters":{"coefficients":[-10,-3,-2],"x0":-2,"y0":-11,"dy0":4,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-10,-3,-2],\"dy0\":4,\"x0\":-2,\"y0\":-11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0027","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[-9,-2,0],\"x0\":-1,\"y0\":-7,\"dy0\":-2}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([-9, -2, 0], -1, -7, -2, -7, -2, 是, 是, 是)","explanation_zh":"應用擴展核對=([-9, -2, 0], -1, -7, -2, -7, -2, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214027,"parameters":{"coefficients":[-9,-2,0],"x0":-1,"y0":-7,"dy0":-2,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-9,-2,0],\"dy0\":-2,\"x0\":-1,\"y0\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0028","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[-8,-1,2],\"x0\":0,\"y0\":-8,\"dy0\":-1}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([-8, -1, 2], 0, -8, -1, -8, -1, 是, 是, 是)","explanation_zh":"應用擴展核對=([-8, -1, 2], 0, -8, -1, -8, -1, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214028,"parameters":{"coefficients":[-8,-1,2],"x0":0,"y0":-8,"dy0":-1,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-8,-1,2],\"dy0\":-1,\"x0\":0,\"y0\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0029","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[-7,0,-1],\"x0\":1,\"y0\":-8,\"dy0\":-3}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([-7, 0, -1], 1, -8, -2, -8, -3, 是, 否, 否)","explanation_zh":"應用擴展核對=([-7, 0, -1], 1, -8, -2, -8, -3, 是, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214029,"parameters":{"coefficients":[-7,0,-1],"x0":1,"y0":-8,"dy0":-3,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-7,0,-1],\"dy0\":-3,\"x0\":1,\"y0\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0030","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[-6,1,1],\"x0\":2,\"y0\":1,\"dy0\":5}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([-6, 1, 1], 2, 0, 5, 1, 5, 否, 是, 否)","explanation_zh":"應用擴展核對=([-6, 1, 1], 2, 0, 5, 1, 5, 否, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214030,"parameters":{"coefficients":[-6,1,1],"x0":2,"y0":1,"dy0":5,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-6,1,1],\"dy0\":5,\"x0\":2,\"y0\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0031","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[-5,2,-2],\"x0\":-2,\"y0\":-17,\"dy0\":10}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([-5, 2, -2], -2, -17, 10, -17, 10, 是, 是, 是)","explanation_zh":"應用擴展核對=([-5, 2, -2], -2, -17, 10, -17, 10, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214031,"parameters":{"coefficients":[-5,2,-2],"x0":-2,"y0":-17,"dy0":10,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-5,2,-2],\"dy0\":10,\"x0\":-2,\"y0\":-17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0032","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[-4,3,0],\"x0\":-1,\"y0\":-7,\"dy0\":2}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([-4, 3, 0], -1, -7, 3, -7, 2, 是, 否, 否)","explanation_zh":"應用擴展核對=([-4, 3, 0], -1, -7, 3, -7, 2, 是, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214032,"parameters":{"coefficients":[-4,3,0],"x0":-1,"y0":-7,"dy0":2,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-4,3,0],\"dy0\":2,\"x0\":-1,\"y0\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0033","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[-3,-3,2],\"x0\":0,\"y0\":-3,\"dy0\":-3}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([-3, -3, 2], 0, -3, -3, -3, -3, 是, 是, 是)","explanation_zh":"應用擴展核對=([-3, -3, 2], 0, -3, -3, -3, -3, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214033,"parameters":{"coefficients":[-3,-3,2],"x0":0,"y0":-3,"dy0":-3,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-3,-3,2],\"dy0\":-3,\"x0\":0,\"y0\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0034","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[-2,-2,-1],\"x0\":1,\"y0\":-4,\"dy0\":-4}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([-2, -2, -1], 1, -5, -4, -4, -4, 否, 是, 否)","explanation_zh":"應用擴展核對=([-2, -2, -1], 1, -5, -4, -4, -4, 否, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214034,"parameters":{"coefficients":[-2,-2,-1],"x0":1,"y0":-4,"dy0":-4,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-2,-2,-1],\"dy0\":-4,\"x0\":1,\"y0\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0035","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[-1,-1,1],\"x0\":2,\"y0\":1,\"dy0\":2}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([-1, -1, 1], 2, 1, 3, 1, 2, 是, 否, 否)","explanation_zh":"應用擴展核對=([-1, -1, 1], 2, 1, 3, 1, 2, 是, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214035,"parameters":{"coefficients":[-1,-1,1],"x0":2,"y0":1,"dy0":2,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-1,-1,1],\"dy0\":2,\"x0\":2,\"y0\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0036","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[0,0,-2],\"x0\":-2,\"y0\":-8,\"dy0\":8}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, -2], -2, -8, 8, -8, 8, 是, 是, 是)","explanation_zh":"應用擴展核對=([0, 0, -2], -2, -8, 8, -8, 8, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214036,"parameters":{"coefficients":[0,0,-2],"x0":-2,"y0":-8,"dy0":8,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[0,0,-2],\"dy0\":8,\"x0\":-2,\"y0\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0037","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[1,1,0],\"x0\":-1,\"y0\":0,\"dy0\":1}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 0], -1, 0, 1, 0, 1, 是, 是, 是)","explanation_zh":"應用擴展核對=([1, 1, 0], -1, 0, 1, 0, 1, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214037,"parameters":{"coefficients":[1,1,0],"x0":-1,"y0":0,"dy0":1,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[1,1,0],\"dy0\":1,\"x0\":-1,\"y0\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0038","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[2,2,2],\"x0\":0,\"y0\":3,\"dy0\":1}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([2, 2, 2], 0, 2, 2, 3, 1, 否, 否, 否)","explanation_zh":"應用擴展核對=([2, 2, 2], 0, 2, 2, 3, 1, 否, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214038,"parameters":{"coefficients":[2,2,2],"x0":0,"y0":3,"dy0":1,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[2,2,2],\"dy0\":1,\"x0\":0,\"y0\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0039","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[3,3,-1],\"x0\":1,\"y0\":5,\"dy0\":1}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([3, 3, -1], 1, 5, 1, 5, 1, 是, 是, 是)","explanation_zh":"應用擴展核對=([3, 3, -1], 1, 5, 1, 5, 1, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214039,"parameters":{"coefficients":[3,3,-1],"x0":1,"y0":5,"dy0":1,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[3,3,-1],\"dy0\":1,\"x0\":1,\"y0\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0040","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[4,-3,1],\"x0\":2,\"y0\":2,\"dy0\":1}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([4, -3, 1], 2, 2, 1, 2, 1, 是, 是, 是)","explanation_zh":"應用擴展核對=([4, -3, 1], 2, 2, 1, 2, 1, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214040,"parameters":{"coefficients":[4,-3,1],"x0":2,"y0":2,"dy0":1,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[4,-3,1],\"dy0\":1,\"x0\":2,\"y0\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0041","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[5,-2,-2],\"x0\":-2,\"y0\":1,\"dy0\":5}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([5, -2, -2], -2, 1, 6, 1, 5, 是, 否, 否)","explanation_zh":"應用擴展核對=([5, -2, -2], -2, 1, 6, 1, 5, 是, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214041,"parameters":{"coefficients":[5,-2,-2],"x0":-2,"y0":1,"dy0":5,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[5,-2,-2],\"dy0\":5,\"x0\":-2,\"y0\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0042","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[6,-1,0],\"x0\":-1,\"y0\":8,\"dy0\":-1}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([6, -1, 0], -1, 7, -1, 8, -1, 否, 是, 否)","explanation_zh":"應用擴展核對=([6, -1, 0], -1, 7, -1, 8, -1, 否, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214042,"parameters":{"coefficients":[6,-1,0],"x0":-1,"y0":8,"dy0":-1,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[6,-1,0],\"dy0\":-1,\"x0\":-1,\"y0\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0043","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[7,0,2],\"x0\":0,\"y0\":7,\"dy0\":0}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([7, 0, 2], 0, 7, 0, 7, 0, 是, 是, 是)","explanation_zh":"應用擴展核對=([7, 0, 2], 0, 7, 0, 7, 0, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214043,"parameters":{"coefficients":[7,0,2],"x0":0,"y0":7,"dy0":0,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[7,0,2],\"dy0\":0,\"x0\":0,\"y0\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0044","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[8,1,-1],\"x0\":1,\"y0\":8,\"dy0\":-2}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([8, 1, -1], 1, 8, -1, 8, -2, 是, 否, 否)","explanation_zh":"應用擴展核對=([8, 1, -1], 1, 8, -1, 8, -2, 是, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214044,"parameters":{"coefficients":[8,1,-1],"x0":1,"y0":8,"dy0":-2,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[8,1,-1],\"dy0\":-2,\"x0\":1,\"y0\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0045","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[9,2,1],\"x0\":2,\"y0\":17,\"dy0\":6}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([9, 2, 1], 2, 17, 6, 17, 6, 是, 是, 是)","explanation_zh":"應用擴展核對=([9, 2, 1], 2, 17, 6, 17, 6, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214045,"parameters":{"coefficients":[9,2,1],"x0":2,"y0":17,"dy0":6,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[9,2,1],\"dy0\":6,\"x0\":2,\"y0\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0046","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[10,3,-2],\"x0\":-2,\"y0\":-3,\"dy0\":11}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([10, 3, -2], -2, -4, 11, -3, 11, 否, 是, 否)","explanation_zh":"應用擴展核對=([10, 3, -2], -2, -4, 11, -3, 11, 否, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214046,"parameters":{"coefficients":[10,3,-2],"x0":-2,"y0":-3,"dy0":11,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[10,3,-2],\"dy0\":11,\"x0\":-2,\"y0\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0047","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[11,-3,0],\"x0\":-1,\"y0\":14,\"dy0\":-4}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([11, -3, 0], -1, 14, -3, 14, -4, 是, 否, 否)","explanation_zh":"應用擴展核對=([11, -3, 0], -1, 14, -3, 14, -4, 是, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214047,"parameters":{"coefficients":[11,-3,0],"x0":-1,"y0":14,"dy0":-4,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[11,-3,0],\"dy0\":-4,\"x0\":-1,\"y0\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0048","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[12,-2,2],\"x0\":0,\"y0\":12,\"dy0\":-2}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([12, -2, 2], 0, 12, -2, 12, -2, 是, 是, 是)","explanation_zh":"應用擴展核對=([12, -2, 2], 0, 12, -2, 12, -2, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214048,"parameters":{"coefficients":[12,-2,2],"x0":0,"y0":12,"dy0":-2,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[12,-2,2],\"dy0\":-2,\"x0\":0,\"y0\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0049","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[13,-1,-1],\"x0\":1,\"y0\":11,\"dy0\":-3}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([13, -1, -1], 1, 11, -3, 11, -3, 是, 是, 是)","explanation_zh":"應用擴展核對=([13, -1, -1], 1, 11, -3, 11, -3, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214049,"parameters":{"coefficients":[13,-1,-1],"x0":1,"y0":11,"dy0":-3,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[13,-1,-1],\"dy0\":-3,\"x0\":1,\"y0\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0050","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-initial-audit；資料={\"coefficients\":[14,0,1],\"x0\":2,\"y0\":19,\"dy0\":3}。請依 [coefficients,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,both] 欄序回答。","answer":{"kind":"expression","value":"([14, 0, 1], 2, 18, 4, 19, 3, 否, 否, 否)","explanation_zh":"應用擴展核對=([14, 0, 1], 2, 18, 4, 19, 3, 否, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214050,"parameters":{"coefficients":[14,0,1],"x0":2,"y0":19,"dy0":3,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-initial-audit","semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[14,0,1],\"dy0\":3,\"x0\":2,\"y0\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0051","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[-12,-3],\"p\":0,\"q\":-2,\"xs\":[-1,0,1],\"x0\":-2,\"y0\":-5}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-12, -3], 0, -2, [-1, 0, 1], [[-1, -9, -3, -1, 否], [0, -12, -3, -1, 否], [1, -15, -3, -1, 否]], -2, -6, -5, 否, 否)","explanation_zh":"應用擴展核對=([-12, -3], 0, -2, [-1, 0, 1], [[-1, -9, -3, -1, 否], [0, -12, -3, -1, 否], [1, -15, -3, -1, 否]], -2, -6, -5, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214051,"parameters":{"coefficients":[-12,-3],"p":0,"q":-2,"xs":[-1,0,1],"x0":-2,"y0":-5,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-12,-3],\"p\":0,\"q\":-2,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0052","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[-11,-2],\"p\":0,\"q\":-2,\"xs\":[-1,0,1],\"x0\":-1,\"y0\":-9}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-11, -2], 0, -2, [-1, 0, 1], [[-1, -9, -2, 0, 是], [0, -11, -2, 0, 是], [1, -13, -2, 0, 是]], -1, -9, -9, 是, 是)","explanation_zh":"應用擴展核對=([-11, -2], 0, -2, [-1, 0, 1], [[-1, -9, -2, 0, 是], [0, -11, -2, 0, 是], [1, -13, -2, 0, 是]], -1, -9, -9, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214052,"parameters":{"coefficients":[-11,-2],"p":0,"q":-2,"xs":[-1,0,1],"x0":-1,"y0":-9,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-11,-2],\"p\":0,\"q\":-2,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":-9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0053","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[-10,-1],\"p\":0,\"q\":-1,\"xs\":[-1,0,1],\"x0\":0,\"y0\":-10}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-10, -1], 0, -1, [-1, 0, 1], [[-1, -9, -1, 0, 是], [0, -10, -1, 0, 是], [1, -11, -1, 0, 是]], 0, -10, -10, 是, 是)","explanation_zh":"應用擴展核對=([-10, -1], 0, -1, [-1, 0, 1], [[-1, -9, -1, 0, 是], [0, -10, -1, 0, 是], [1, -11, -1, 0, 是]], 0, -10, -10, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214053,"parameters":{"coefficients":[-10,-1],"p":0,"q":-1,"xs":[-1,0,1],"x0":0,"y0":-10,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-10,-1],\"p\":0,\"q\":-1,\"x0\":0,\"xs\":[-1,0,1],\"y0\":-10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0054","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[-9,0],\"p\":0,\"q\":0,\"xs\":[-1,0,1],\"x0\":1,\"y0\":-8}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-9, 0], 0, 0, [-1, 0, 1], [[-1, -9, 0, 0, 是], [0, -9, 0, 0, 是], [1, -9, 0, 0, 是]], 1, -9, -8, 否, 否)","explanation_zh":"應用擴展核對=([-9, 0], 0, 0, [-1, 0, 1], [[-1, -9, 0, 0, 是], [0, -9, 0, 0, 是], [1, -9, 0, 0, 是]], 1, -9, -8, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214054,"parameters":{"coefficients":[-9,0],"p":0,"q":0,"xs":[-1,0,1],"x0":1,"y0":-8,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-9,0],\"p\":0,\"q\":0,\"x0\":1,\"xs\":[-1,0,1],\"y0\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0055","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[-8,1],\"p\":0,\"q\":2,\"xs\":[-1,0,1],\"x0\":2,\"y0\":-6}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-8, 1], 0, 2, [-1, 0, 1], [[-1, -9, 1, -1, 否], [0, -8, 1, -1, 否], [1, -7, 1, -1, 否]], 2, -6, -6, 是, 否)","explanation_zh":"應用擴展核對=([-8, 1], 0, 2, [-1, 0, 1], [[-1, -9, 1, -1, 否], [0, -8, 1, -1, 否], [1, -7, 1, -1, 否]], 2, -6, -6, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214055,"parameters":{"coefficients":[-8,1],"p":0,"q":2,"xs":[-1,0,1],"x0":2,"y0":-6,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-8,1],\"p\":0,\"q\":2,\"x0\":2,\"xs\":[-1,0,1],\"y0\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0056","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[-7,2],\"p\":0,\"q\":2,\"xs\":[-1,0,1],\"x0\":-2,\"y0\":-11}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-7, 2], 0, 2, [-1, 0, 1], [[-1, -9, 2, 0, 是], [0, -7, 2, 0, 是], [1, -5, 2, 0, 是]], -2, -11, -11, 是, 是)","explanation_zh":"應用擴展核對=([-7, 2], 0, 2, [-1, 0, 1], [[-1, -9, 2, 0, 是], [0, -7, 2, 0, 是], [1, -5, 2, 0, 是]], -2, -11, -11, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214056,"parameters":{"coefficients":[-7,2],"p":0,"q":2,"xs":[-1,0,1],"x0":-2,"y0":-11,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-7,2],\"p\":0,\"q\":2,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":-11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0057","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[-6,3],\"p\":0,\"q\":3,\"xs\":[-1,0,1],\"x0\":-1,\"y0\":-8}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-6, 3], 0, 3, [-1, 0, 1], [[-1, -9, 3, 0, 是], [0, -6, 3, 0, 是], [1, -3, 3, 0, 是]], -1, -9, -8, 否, 否)","explanation_zh":"應用擴展核對=([-6, 3], 0, 3, [-1, 0, 1], [[-1, -9, 3, 0, 是], [0, -6, 3, 0, 是], [1, -3, 3, 0, 是]], -1, -9, -8, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214057,"parameters":{"coefficients":[-6,3],"p":0,"q":3,"xs":[-1,0,1],"x0":-1,"y0":-8,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-6,3],\"p\":0,\"q\":3,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0058","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[-5,-3],\"p\":0,\"q\":-3,\"xs\":[-1,0,1],\"x0\":0,\"y0\":-5}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-5, -3], 0, -3, [-1, 0, 1], [[-1, -2, -3, 0, 是], [0, -5, -3, 0, 是], [1, -8, -3, 0, 是]], 0, -5, -5, 是, 是)","explanation_zh":"應用擴展核對=([-5, -3], 0, -3, [-1, 0, 1], [[-1, -2, -3, 0, 是], [0, -5, -3, 0, 是], [1, -8, -3, 0, 是]], 0, -5, -5, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214058,"parameters":{"coefficients":[-5,-3],"p":0,"q":-3,"xs":[-1,0,1],"x0":0,"y0":-5,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-5,-3],\"p\":0,\"q\":-3,\"x0\":0,\"xs\":[-1,0,1],\"y0\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0059","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[-4,-2],\"p\":0,\"q\":-1,\"xs\":[-1,0,1],\"x0\":1,\"y0\":-6}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-4, -2], 0, -1, [-1, 0, 1], [[-1, -2, -2, -1, 否], [0, -4, -2, -1, 否], [1, -6, -2, -1, 否]], 1, -6, -6, 是, 否)","explanation_zh":"應用擴展核對=([-4, -2], 0, -1, [-1, 0, 1], [[-1, -2, -2, -1, 否], [0, -4, -2, -1, 否], [1, -6, -2, -1, 否]], 1, -6, -6, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214059,"parameters":{"coefficients":[-4,-2],"p":0,"q":-1,"xs":[-1,0,1],"x0":1,"y0":-6,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-4,-2],\"p\":0,\"q\":-1,\"x0\":1,\"xs\":[-1,0,1],\"y0\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0060","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[-3,-1],\"p\":0,\"q\":-1,\"xs\":[-1,0,1],\"x0\":2,\"y0\":-4}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-3, -1], 0, -1, [-1, 0, 1], [[-1, -2, -1, 0, 是], [0, -3, -1, 0, 是], [1, -4, -1, 0, 是]], 2, -5, -4, 否, 否)","explanation_zh":"應用擴展核對=([-3, -1], 0, -1, [-1, 0, 1], [[-1, -2, -1, 0, 是], [0, -3, -1, 0, 是], [1, -4, -1, 0, 是]], 2, -5, -4, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214060,"parameters":{"coefficients":[-3,-1],"p":0,"q":-1,"xs":[-1,0,1],"x0":2,"y0":-4,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-3,-1],\"p\":0,\"q\":-1,\"x0\":2,\"xs\":[-1,0,1],\"y0\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0061","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[-2,0],\"p\":0,\"q\":0,\"xs\":[-1,0,1],\"x0\":-2,\"y0\":-2}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-2, 0], 0, 0, [-1, 0, 1], [[-1, -2, 0, 0, 是], [0, -2, 0, 0, 是], [1, -2, 0, 0, 是]], -2, -2, -2, 是, 是)","explanation_zh":"應用擴展核對=([-2, 0], 0, 0, [-1, 0, 1], [[-1, -2, 0, 0, 是], [0, -2, 0, 0, 是], [1, -2, 0, 0, 是]], -2, -2, -2, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214061,"parameters":{"coefficients":[-2,0],"p":0,"q":0,"xs":[-1,0,1],"x0":-2,"y0":-2,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-2,0],\"p\":0,\"q\":0,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0062","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[-1,1],\"p\":0,\"q\":1,\"xs\":[-1,0,1],\"x0\":-1,\"y0\":-2}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-1, 1], 0, 1, [-1, 0, 1], [[-1, -2, 1, 0, 是], [0, -1, 1, 0, 是], [1, 0, 1, 0, 是]], -1, -2, -2, 是, 是)","explanation_zh":"應用擴展核對=([-1, 1], 0, 1, [-1, 0, 1], [[-1, -2, 1, 0, 是], [0, -1, 1, 0, 是], [1, 0, 1, 0, 是]], -1, -2, -2, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214062,"parameters":{"coefficients":[-1,1],"p":0,"q":1,"xs":[-1,0,1],"x0":-1,"y0":-2,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-1,1],\"p\":0,\"q\":1,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0063","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[0,2],\"p\":0,\"q\":3,\"xs\":[-1,0,1],\"x0\":0,\"y0\":1}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([0, 2], 0, 3, [-1, 0, 1], [[-1, -2, 2, -1, 否], [0, 0, 2, -1, 否], [1, 2, 2, -1, 否]], 0, 0, 1, 否, 否)","explanation_zh":"應用擴展核對=([0, 2], 0, 3, [-1, 0, 1], [[-1, -2, 2, -1, 否], [0, 0, 2, -1, 否], [1, 2, 2, -1, 否]], 0, 0, 1, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214063,"parameters":{"coefficients":[0,2],"p":0,"q":3,"xs":[-1,0,1],"x0":0,"y0":1,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[0,2],\"p\":0,\"q\":3,\"x0\":0,\"xs\":[-1,0,1],\"y0\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0064","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[1,3],\"p\":0,\"q\":3,\"xs\":[-1,0,1],\"x0\":1,\"y0\":4}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([1, 3], 0, 3, [-1, 0, 1], [[-1, -2, 3, 0, 是], [0, 1, 3, 0, 是], [1, 4, 3, 0, 是]], 1, 4, 4, 是, 是)","explanation_zh":"應用擴展核對=([1, 3], 0, 3, [-1, 0, 1], [[-1, -2, 3, 0, 是], [0, 1, 3, 0, 是], [1, 4, 3, 0, 是]], 1, 4, 4, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214064,"parameters":{"coefficients":[1,3],"p":0,"q":3,"xs":[-1,0,1],"x0":1,"y0":4,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[1,3],\"p\":0,\"q\":3,\"x0\":1,\"xs\":[-1,0,1],\"y0\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0065","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[2,-3],\"p\":0,\"q\":-3,\"xs\":[-1,0,1],\"x0\":2,\"y0\":-4}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([2, -3], 0, -3, [-1, 0, 1], [[-1, 5, -3, 0, 是], [0, 2, -3, 0, 是], [1, -1, -3, 0, 是]], 2, -4, -4, 是, 是)","explanation_zh":"應用擴展核對=([2, -3], 0, -3, [-1, 0, 1], [[-1, 5, -3, 0, 是], [0, 2, -3, 0, 是], [1, -1, -3, 0, 是]], 2, -4, -4, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214065,"parameters":{"coefficients":[2,-3],"p":0,"q":-3,"xs":[-1,0,1],"x0":2,"y0":-4,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[2,-3],\"p\":0,\"q\":-3,\"x0\":2,\"xs\":[-1,0,1],\"y0\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0066","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[3,-2],\"p\":0,\"q\":-2,\"xs\":[-1,0,1],\"x0\":-2,\"y0\":8}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([3, -2], 0, -2, [-1, 0, 1], [[-1, 5, -2, 0, 是], [0, 3, -2, 0, 是], [1, 1, -2, 0, 是]], -2, 7, 8, 否, 否)","explanation_zh":"應用擴展核對=([3, -2], 0, -2, [-1, 0, 1], [[-1, 5, -2, 0, 是], [0, 3, -2, 0, 是], [1, 1, -2, 0, 是]], -2, 7, 8, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214066,"parameters":{"coefficients":[3,-2],"p":0,"q":-2,"xs":[-1,0,1],"x0":-2,"y0":8,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[3,-2],\"p\":0,\"q\":-2,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0067","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[4,-1],\"p\":0,\"q\":0,\"xs\":[-1,0,1],\"x0\":-1,\"y0\":5}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([4, -1], 0, 0, [-1, 0, 1], [[-1, 5, -1, -1, 否], [0, 4, -1, -1, 否], [1, 3, -1, -1, 否]], -1, 5, 5, 是, 否)","explanation_zh":"應用擴展核對=([4, -1], 0, 0, [-1, 0, 1], [[-1, 5, -1, -1, 否], [0, 4, -1, -1, 否], [1, 3, -1, -1, 否]], -1, 5, 5, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214067,"parameters":{"coefficients":[4,-1],"p":0,"q":0,"xs":[-1,0,1],"x0":-1,"y0":5,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[4,-1],\"p\":0,\"q\":0,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0068","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[5,0],\"p\":0,\"q\":0,\"xs\":[-1,0,1],\"x0\":0,\"y0\":5}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([5, 0], 0, 0, [-1, 0, 1], [[-1, 5, 0, 0, 是], [0, 5, 0, 0, 是], [1, 5, 0, 0, 是]], 0, 5, 5, 是, 是)","explanation_zh":"應用擴展核對=([5, 0], 0, 0, [-1, 0, 1], [[-1, 5, 0, 0, 是], [0, 5, 0, 0, 是], [1, 5, 0, 0, 是]], 0, 5, 5, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214068,"parameters":{"coefficients":[5,0],"p":0,"q":0,"xs":[-1,0,1],"x0":0,"y0":5,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[5,0],\"p\":0,\"q\":0,\"x0\":0,\"xs\":[-1,0,1],\"y0\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0069","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[6,1],\"p\":0,\"q\":1,\"xs\":[-1,0,1],\"x0\":1,\"y0\":8}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([6, 1], 0, 1, [-1, 0, 1], [[-1, 5, 1, 0, 是], [0, 6, 1, 0, 是], [1, 7, 1, 0, 是]], 1, 7, 8, 否, 否)","explanation_zh":"應用擴展核對=([6, 1], 0, 1, [-1, 0, 1], [[-1, 5, 1, 0, 是], [0, 6, 1, 0, 是], [1, 7, 1, 0, 是]], 1, 7, 8, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214069,"parameters":{"coefficients":[6,1],"p":0,"q":1,"xs":[-1,0,1],"x0":1,"y0":8,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[6,1],\"p\":0,\"q\":1,\"x0\":1,\"xs\":[-1,0,1],\"y0\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0070","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[7,2],\"p\":0,\"q\":2,\"xs\":[-1,0,1],\"x0\":2,\"y0\":11}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([7, 2], 0, 2, [-1, 0, 1], [[-1, 5, 2, 0, 是], [0, 7, 2, 0, 是], [1, 9, 2, 0, 是]], 2, 11, 11, 是, 是)","explanation_zh":"應用擴展核對=([7, 2], 0, 2, [-1, 0, 1], [[-1, 5, 2, 0, 是], [0, 7, 2, 0, 是], [1, 9, 2, 0, 是]], 2, 11, 11, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214070,"parameters":{"coefficients":[7,2],"p":0,"q":2,"xs":[-1,0,1],"x0":2,"y0":11,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[7,2],\"p\":0,\"q\":2,\"x0\":2,\"xs\":[-1,0,1],\"y0\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0071","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[8,3],\"p\":0,\"q\":4,\"xs\":[-1,0,1],\"x0\":-2,\"y0\":2}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([8, 3], 0, 4, [-1, 0, 1], [[-1, 5, 3, -1, 否], [0, 8, 3, -1, 否], [1, 11, 3, -1, 否]], -2, 2, 2, 是, 否)","explanation_zh":"應用擴展核對=([8, 3], 0, 4, [-1, 0, 1], [[-1, 5, 3, -1, 否], [0, 8, 3, -1, 否], [1, 11, 3, -1, 否]], -2, 2, 2, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214071,"parameters":{"coefficients":[8,3],"p":0,"q":4,"xs":[-1,0,1],"x0":-2,"y0":2,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[8,3],\"p\":0,\"q\":4,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0072","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[9,-3],\"p\":0,\"q\":-3,\"xs\":[-1,0,1],\"x0\":-1,\"y0\":13}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([9, -3], 0, -3, [-1, 0, 1], [[-1, 12, -3, 0, 是], [0, 9, -3, 0, 是], [1, 6, -3, 0, 是]], -1, 12, 13, 否, 否)","explanation_zh":"應用擴展核對=([9, -3], 0, -3, [-1, 0, 1], [[-1, 12, -3, 0, 是], [0, 9, -3, 0, 是], [1, 6, -3, 0, 是]], -1, 12, 13, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214072,"parameters":{"coefficients":[9,-3],"p":0,"q":-3,"xs":[-1,0,1],"x0":-1,"y0":13,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[9,-3],\"p\":0,\"q\":-3,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0073","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[10,-2],\"p\":0,\"q\":-2,\"xs\":[-1,0,1],\"x0\":0,\"y0\":10}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([10, -2], 0, -2, [-1, 0, 1], [[-1, 12, -2, 0, 是], [0, 10, -2, 0, 是], [1, 8, -2, 0, 是]], 0, 10, 10, 是, 是)","explanation_zh":"應用擴展核對=([10, -2], 0, -2, [-1, 0, 1], [[-1, 12, -2, 0, 是], [0, 10, -2, 0, 是], [1, 8, -2, 0, 是]], 0, 10, 10, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214073,"parameters":{"coefficients":[10,-2],"p":0,"q":-2,"xs":[-1,0,1],"x0":0,"y0":10,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[10,-2],\"p\":0,\"q\":-2,\"x0\":0,\"xs\":[-1,0,1],\"y0\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0074","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[11,-1],\"p\":0,\"q\":-1,\"xs\":[-1,0,1],\"x0\":1,\"y0\":10}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([11, -1], 0, -1, [-1, 0, 1], [[-1, 12, -1, 0, 是], [0, 11, -1, 0, 是], [1, 10, -1, 0, 是]], 1, 10, 10, 是, 是)","explanation_zh":"應用擴展核對=([11, -1], 0, -1, [-1, 0, 1], [[-1, 12, -1, 0, 是], [0, 11, -1, 0, 是], [1, 10, -1, 0, 是]], 1, 10, 10, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214074,"parameters":{"coefficients":[11,-1],"p":0,"q":-1,"xs":[-1,0,1],"x0":1,"y0":10,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[11,-1],\"p\":0,\"q\":-1,\"x0\":1,\"xs\":[-1,0,1],\"y0\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0075","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-first-order-candidate-grid；資料={\"coefficients\":[12,0],\"p\":0,\"q\":1,\"xs\":[-1,0,1],\"x0\":2,\"y0\":13}。請依 [coefficients,p,q,xs,residual_rows,x0,y_x0,expected_y,initial_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([12, 0], 0, 1, [-1, 0, 1], [[-1, 12, 0, -1, 否], [0, 12, 0, -1, 否], [1, 12, 0, -1, 否]], 2, 12, 13, 否, 否)","explanation_zh":"應用擴展核對=([12, 0], 0, 1, [-1, 0, 1], [[-1, 12, 0, -1, 否], [0, 12, 0, -1, 否], [1, 12, 0, -1, 否]], 2, 12, 13, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214075,"parameters":{"coefficients":[12,0],"p":0,"q":1,"xs":[-1,0,1],"x0":2,"y0":13,"domain":"Polynomial_candidate_IVP","operation":"ivp-first-order-candidate-grid","semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[12,0],\"p\":0,\"q\":1,\"x0\":2,\"xs\":[-1,0,1],\"y0\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0076","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[-12,-3,-2],\"a\":1,\"b\":0,\"c\":0,\"rhs\":-3,\"xs\":[-1,0,1],\"x0\":-2,\"y0\":-13,\"dy0\":4}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-12, -3, -2], 1, 0, 0, -3, [-1, 0, 1], [[-1, -11, 1, -4, -1, 否], [0, -12, -3, -4, -1, 否], [1, -17, -7, -4, -1, 否]], -2, -14, 5, -13, 4, 否, 否, 否)","explanation_zh":"應用擴展核對=([-12, -3, -2], 1, 0, 0, -3, [-1, 0, 1], [[-1, -11, 1, -4, -1, 否], [0, -12, -3, -4, -1, 否], [1, -17, -7, -4, -1, 否]], -2, -14, 5, -13, 4, 否, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214076,"parameters":{"coefficients":[-12,-3,-2],"a":1,"b":0,"c":0,"rhs":-3,"xs":[-1,0,1],"x0":-2,"y0":-13,"dy0":4,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-12,-3,-2],\"dy0\":4,\"rhs\":-3,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":-13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0077","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[-11,-2,-1],\"a\":1,\"b\":0,\"c\":0,\"rhs\":-2,\"xs\":[-1,0,1],\"x0\":-1,\"y0\":-10,\"dy0\":0}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-11, -2, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -10, 0, -2, 0, 是], [0, -11, -2, -2, 0, 是], [1, -14, -4, -2, 0, 是]], -1, -10, 0, -10, 0, 是, 是, 是)","explanation_zh":"應用擴展核對=([-11, -2, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -10, 0, -2, 0, 是], [0, -11, -2, -2, 0, 是], [1, -14, -4, -2, 0, 是]], -1, -10, 0, -10, 0, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214077,"parameters":{"coefficients":[-11,-2,-1],"a":1,"b":0,"c":0,"rhs":-2,"xs":[-1,0,1],"x0":-1,"y0":-10,"dy0":0,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-11,-2,-1],\"dy0\":0,\"rhs\":-2,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":-10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0078","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[-10,-1,0],\"a\":1,\"b\":0,\"c\":0,\"rhs\":0,\"xs\":[-1,0,1],\"x0\":0,\"y0\":-10,\"dy0\":-1}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-10, -1, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, -9, -1, 0, 0, 是], [0, -10, -1, 0, 0, 是], [1, -11, -1, 0, 0, 是]], 0, -10, -1, -10, -1, 是, 是, 是)","explanation_zh":"應用擴展核對=([-10, -1, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, -9, -1, 0, 0, 是], [0, -10, -1, 0, 0, 是], [1, -11, -1, 0, 0, 是]], 0, -10, -1, -10, -1, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214078,"parameters":{"coefficients":[-10,-1,0],"a":1,"b":0,"c":0,"rhs":0,"xs":[-1,0,1],"x0":0,"y0":-10,"dy0":-1,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-10,-1,0],\"dy0\":-1,\"rhs\":0,\"x0\":0,\"xs\":[-1,0,1],\"y0\":-10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0079","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[-9,0,1],\"a\":1,\"b\":0,\"c\":0,\"rhs\":2,\"xs\":[-1,0,1],\"x0\":1,\"y0\":-7,\"dy0\":2}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-9, 0, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, -8, -2, 2, 0, 是], [0, -9, 0, 2, 0, 是], [1, -8, 2, 2, 0, 是]], 1, -8, 2, -7, 2, 否, 是, 否)","explanation_zh":"應用擴展核對=([-9, 0, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, -8, -2, 2, 0, 是], [0, -9, 0, 2, 0, 是], [1, -8, 2, 2, 0, 是]], 1, -8, 2, -7, 2, 否, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214079,"parameters":{"coefficients":[-9,0,1],"a":1,"b":0,"c":0,"rhs":2,"xs":[-1,0,1],"x0":1,"y0":-7,"dy0":2,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-9,0,1],\"dy0\":2,\"rhs\":2,\"x0\":1,\"xs\":[-1,0,1],\"y0\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0080","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[-8,1,2],\"a\":1,\"b\":0,\"c\":0,\"rhs\":5,\"xs\":[-1,0,1],\"x0\":2,\"y0\":2,\"dy0\":9}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-8, 1, 2], 1, 0, 0, 5, [-1, 0, 1], [[-1, -7, -3, 4, -1, 否], [0, -8, 1, 4, -1, 否], [1, -5, 5, 4, -1, 否]], 2, 2, 9, 2, 9, 是, 是, 否)","explanation_zh":"應用擴展核對=([-8, 1, 2], 1, 0, 0, 5, [-1, 0, 1], [[-1, -7, -3, 4, -1, 否], [0, -8, 1, 4, -1, 否], [1, -5, 5, 4, -1, 否]], 2, 2, 9, 2, 9, 是, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214080,"parameters":{"coefficients":[-8,1,2],"a":1,"b":0,"c":0,"rhs":5,"xs":[-1,0,1],"x0":2,"y0":2,"dy0":9,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-8,1,2],\"dy0\":9,\"rhs\":5,\"x0\":2,\"xs\":[-1,0,1],\"y0\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0081","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[-7,2,-2],\"a\":1,\"b\":0,\"c\":0,\"rhs\":-4,\"xs\":[-1,0,1],\"x0\":-2,\"y0\":-19,\"dy0\":9}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-7, 2, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, -11, 6, -4, 0, 是], [0, -7, 2, -4, 0, 是], [1, -7, -2, -4, 0, 是]], -2, -19, 10, -19, 9, 是, 否, 否)","explanation_zh":"應用擴展核對=([-7, 2, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, -11, 6, -4, 0, 是], [0, -7, 2, -4, 0, 是], [1, -7, -2, -4, 0, 是]], -2, -19, 10, -19, 9, 是, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214081,"parameters":{"coefficients":[-7,2,-2],"a":1,"b":0,"c":0,"rhs":-4,"xs":[-1,0,1],"x0":-2,"y0":-19,"dy0":9,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-7,2,-2],\"dy0\":9,\"rhs\":-4,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":-19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0082","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[-6,3,-1],\"a\":1,\"b\":0,\"c\":0,\"rhs\":-2,\"xs\":[-1,0,1],\"x0\":-1,\"y0\":-9,\"dy0\":5}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-6, 3, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -10, 5, -2, 0, 是], [0, -6, 3, -2, 0, 是], [1, -4, 1, -2, 0, 是]], -1, -10, 5, -9, 5, 否, 是, 否)","explanation_zh":"應用擴展核對=([-6, 3, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -10, 5, -2, 0, 是], [0, -6, 3, -2, 0, 是], [1, -4, 1, -2, 0, 是]], -1, -10, 5, -9, 5, 否, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214082,"parameters":{"coefficients":[-6,3,-1],"a":1,"b":0,"c":0,"rhs":-2,"xs":[-1,0,1],"x0":-1,"y0":-9,"dy0":5,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-6,3,-1],\"dy0\":5,\"rhs\":-2,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":-9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0083","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[-5,-3,0],\"a\":1,\"b\":0,\"c\":0,\"rhs\":0,\"xs\":[-1,0,1],\"x0\":0,\"y0\":-5,\"dy0\":-3}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-5, -3, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, -2, -3, 0, 0, 是], [0, -5, -3, 0, 0, 是], [1, -8, -3, 0, 0, 是]], 0, -5, -3, -5, -3, 是, 是, 是)","explanation_zh":"應用擴展核對=([-5, -3, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, -2, -3, 0, 0, 是], [0, -5, -3, 0, 0, 是], [1, -8, -3, 0, 0, 是]], 0, -5, -3, -5, -3, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214083,"parameters":{"coefficients":[-5,-3,0],"a":1,"b":0,"c":0,"rhs":0,"xs":[-1,0,1],"x0":0,"y0":-5,"dy0":-3,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-5,-3,0],\"dy0\":-3,\"rhs\":0,\"x0\":0,\"xs\":[-1,0,1],\"y0\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0084","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[-4,-2,1],\"a\":1,\"b\":0,\"c\":0,\"rhs\":3,\"xs\":[-1,0,1],\"x0\":1,\"y0\":-5,\"dy0\":0}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-4, -2, 1], 1, 0, 0, 3, [-1, 0, 1], [[-1, -1, -4, 2, -1, 否], [0, -4, -2, 2, -1, 否], [1, -5, 0, 2, -1, 否]], 1, -5, 0, -5, 0, 是, 是, 否)","explanation_zh":"應用擴展核對=([-4, -2, 1], 1, 0, 0, 3, [-1, 0, 1], [[-1, -1, -4, 2, -1, 否], [0, -4, -2, 2, -1, 否], [1, -5, 0, 2, -1, 否]], 1, -5, 0, -5, 0, 是, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214084,"parameters":{"coefficients":[-4,-2,1],"a":1,"b":0,"c":0,"rhs":3,"xs":[-1,0,1],"x0":1,"y0":-5,"dy0":0,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-4,-2,1],\"dy0\":0,\"rhs\":3,\"x0\":1,\"xs\":[-1,0,1],\"y0\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0085","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[-3,-1,2],\"a\":1,\"b\":0,\"c\":0,\"rhs\":4,\"xs\":[-1,0,1],\"x0\":2,\"y0\":4,\"dy0\":7}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-3, -1, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 0, -5, 4, 0, 是], [0, -3, -1, 4, 0, 是], [1, -2, 3, 4, 0, 是]], 2, 3, 7, 4, 7, 否, 是, 否)","explanation_zh":"應用擴展核對=([-3, -1, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 0, -5, 4, 0, 是], [0, -3, -1, 4, 0, 是], [1, -2, 3, 4, 0, 是]], 2, 3, 7, 4, 7, 否, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214085,"parameters":{"coefficients":[-3,-1,2],"a":1,"b":0,"c":0,"rhs":4,"xs":[-1,0,1],"x0":2,"y0":4,"dy0":7,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-3,-1,2],\"dy0\":7,\"rhs\":4,\"x0\":2,\"xs\":[-1,0,1],\"y0\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0086","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[-2,0,-2],\"a\":1,\"b\":0,\"c\":0,\"rhs\":-4,\"xs\":[-1,0,1],\"x0\":-2,\"y0\":-10,\"dy0\":7}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-2, 0, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, -4, 4, -4, 0, 是], [0, -2, 0, -4, 0, 是], [1, -4, -4, -4, 0, 是]], -2, -10, 8, -10, 7, 是, 否, 否)","explanation_zh":"應用擴展核對=([-2, 0, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, -4, 4, -4, 0, 是], [0, -2, 0, -4, 0, 是], [1, -4, -4, -4, 0, 是]], -2, -10, 8, -10, 7, 是, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214086,"parameters":{"coefficients":[-2,0,-2],"a":1,"b":0,"c":0,"rhs":-4,"xs":[-1,0,1],"x0":-2,"y0":-10,"dy0":7,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-2,0,-2],\"dy0\":7,\"rhs\":-4,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":-10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0087","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[-1,1,-1],\"a\":1,\"b\":0,\"c\":0,\"rhs\":-2,\"xs\":[-1,0,1],\"x0\":-1,\"y0\":-3,\"dy0\":3}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([-1, 1, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -3, 3, -2, 0, 是], [0, -1, 1, -2, 0, 是], [1, -1, -1, -2, 0, 是]], -1, -3, 3, -3, 3, 是, 是, 是)","explanation_zh":"應用擴展核對=([-1, 1, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -3, 3, -2, 0, 是], [0, -1, 1, -2, 0, 是], [1, -1, -1, -2, 0, 是]], -1, -3, 3, -3, 3, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214087,"parameters":{"coefficients":[-1,1,-1],"a":1,"b":0,"c":0,"rhs":-2,"xs":[-1,0,1],"x0":-1,"y0":-3,"dy0":3,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-1,1,-1],\"dy0\":3,\"rhs\":-2,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0088","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[0,2,0],\"a\":1,\"b\":0,\"c\":0,\"rhs\":1,\"xs\":[-1,0,1],\"x0\":0,\"y0\":1,\"dy0\":2}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([0, 2, 0], 1, 0, 0, 1, [-1, 0, 1], [[-1, -2, 2, 0, -1, 否], [0, 0, 2, 0, -1, 否], [1, 2, 2, 0, -1, 否]], 0, 0, 2, 1, 2, 否, 是, 否)","explanation_zh":"應用擴展核對=([0, 2, 0], 1, 0, 0, 1, [-1, 0, 1], [[-1, -2, 2, 0, -1, 否], [0, 0, 2, 0, -1, 否], [1, 2, 2, 0, -1, 否]], 0, 0, 2, 1, 2, 否, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214088,"parameters":{"coefficients":[0,2,0],"a":1,"b":0,"c":0,"rhs":1,"xs":[-1,0,1],"x0":0,"y0":1,"dy0":2,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[0,2,0],\"dy0\":2,\"rhs\":1,\"x0\":0,\"xs\":[-1,0,1],\"y0\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0089","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[1,3,1],\"a\":1,\"b\":0,\"c\":0,\"rhs\":2,\"xs\":[-1,0,1],\"x0\":1,\"y0\":5,\"dy0\":5}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([1, 3, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, -1, 1, 2, 0, 是], [0, 1, 3, 2, 0, 是], [1, 5, 5, 2, 0, 是]], 1, 5, 5, 5, 5, 是, 是, 是)","explanation_zh":"應用擴展核對=([1, 3, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, -1, 1, 2, 0, 是], [0, 1, 3, 2, 0, 是], [1, 5, 5, 2, 0, 是]], 1, 5, 5, 5, 5, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214089,"parameters":{"coefficients":[1,3,1],"a":1,"b":0,"c":0,"rhs":2,"xs":[-1,0,1],"x0":1,"y0":5,"dy0":5,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[1,3,1],\"dy0\":5,\"rhs\":2,\"x0\":1,\"xs\":[-1,0,1],\"y0\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0090","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[2,-3,2],\"a\":1,\"b\":0,\"c\":0,\"rhs\":4,\"xs\":[-1,0,1],\"x0\":2,\"y0\":4,\"dy0\":5}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([2, -3, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 7, -7, 4, 0, 是], [0, 2, -3, 4, 0, 是], [1, 1, 1, 4, 0, 是]], 2, 4, 5, 4, 5, 是, 是, 是)","explanation_zh":"應用擴展核對=([2, -3, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 7, -7, 4, 0, 是], [0, 2, -3, 4, 0, 是], [1, 1, 1, 4, 0, 是]], 2, 4, 5, 4, 5, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214090,"parameters":{"coefficients":[2,-3,2],"a":1,"b":0,"c":0,"rhs":4,"xs":[-1,0,1],"x0":2,"y0":4,"dy0":5,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[2,-3,2],\"dy0\":5,\"rhs\":4,\"x0\":2,\"xs\":[-1,0,1],\"y0\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0091","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[3,-2,-2],\"a\":1,\"b\":0,\"c\":0,\"rhs\":-4,\"xs\":[-1,0,1],\"x0\":-2,\"y0\":0,\"dy0\":5}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([3, -2, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, 3, 2, -4, 0, 是], [0, 3, -2, -4, 0, 是], [1, -1, -6, -4, 0, 是]], -2, -1, 6, 0, 5, 否, 否, 否)","explanation_zh":"應用擴展核對=([3, -2, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, 3, 2, -4, 0, 是], [0, 3, -2, -4, 0, 是], [1, -1, -6, -4, 0, 是]], -2, -1, 6, 0, 5, 否, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214091,"parameters":{"coefficients":[3,-2,-2],"a":1,"b":0,"c":0,"rhs":-4,"xs":[-1,0,1],"x0":-2,"y0":0,"dy0":5,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[3,-2,-2],\"dy0\":5,\"rhs\":-4,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0092","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[4,-1,-1],\"a\":1,\"b\":0,\"c\":0,\"rhs\":-1,\"xs\":[-1,0,1],\"x0\":-1,\"y0\":4,\"dy0\":1}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([4, -1, -1], 1, 0, 0, -1, [-1, 0, 1], [[-1, 4, 1, -2, -1, 否], [0, 4, -1, -2, -1, 否], [1, 2, -3, -2, -1, 否]], -1, 4, 1, 4, 1, 是, 是, 否)","explanation_zh":"應用擴展核對=([4, -1, -1], 1, 0, 0, -1, [-1, 0, 1], [[-1, 4, 1, -2, -1, 否], [0, 4, -1, -2, -1, 否], [1, 2, -3, -2, -1, 否]], -1, 4, 1, 4, 1, 是, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214092,"parameters":{"coefficients":[4,-1,-1],"a":1,"b":0,"c":0,"rhs":-1,"xs":[-1,0,1],"x0":-1,"y0":4,"dy0":1,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[4,-1,-1],\"dy0\":1,\"rhs\":-1,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0093","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[5,0,0],\"a\":1,\"b\":0,\"c\":0,\"rhs\":0,\"xs\":[-1,0,1],\"x0\":0,\"y0\":5,\"dy0\":0}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([5, 0, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, 5, 0, 0, 0, 是], [0, 5, 0, 0, 0, 是], [1, 5, 0, 0, 0, 是]], 0, 5, 0, 5, 0, 是, 是, 是)","explanation_zh":"應用擴展核對=([5, 0, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, 5, 0, 0, 0, 是], [0, 5, 0, 0, 0, 是], [1, 5, 0, 0, 0, 是]], 0, 5, 0, 5, 0, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214093,"parameters":{"coefficients":[5,0,0],"a":1,"b":0,"c":0,"rhs":0,"xs":[-1,0,1],"x0":0,"y0":5,"dy0":0,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[5,0,0],\"dy0\":0,\"rhs\":0,\"x0\":0,\"xs\":[-1,0,1],\"y0\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0094","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[6,1,1],\"a\":1,\"b\":0,\"c\":0,\"rhs\":2,\"xs\":[-1,0,1],\"x0\":1,\"y0\":9,\"dy0\":3}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([6, 1, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, 6, -1, 2, 0, 是], [0, 6, 1, 2, 0, 是], [1, 8, 3, 2, 0, 是]], 1, 8, 3, 9, 3, 否, 是, 否)","explanation_zh":"應用擴展核對=([6, 1, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, 6, -1, 2, 0, 是], [0, 6, 1, 2, 0, 是], [1, 8, 3, 2, 0, 是]], 1, 8, 3, 9, 3, 否, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214094,"parameters":{"coefficients":[6,1,1],"a":1,"b":0,"c":0,"rhs":2,"xs":[-1,0,1],"x0":1,"y0":9,"dy0":3,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[6,1,1],\"dy0\":3,\"rhs\":2,\"x0\":1,\"xs\":[-1,0,1],\"y0\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0095","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[7,2,2],\"a\":1,\"b\":0,\"c\":0,\"rhs\":4,\"xs\":[-1,0,1],\"x0\":2,\"y0\":19,\"dy0\":10}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([7, 2, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 7, -2, 4, 0, 是], [0, 7, 2, 4, 0, 是], [1, 11, 6, 4, 0, 是]], 2, 19, 10, 19, 10, 是, 是, 是)","explanation_zh":"應用擴展核對=([7, 2, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 7, -2, 4, 0, 是], [0, 7, 2, 4, 0, 是], [1, 11, 6, 4, 0, 是]], 2, 19, 10, 19, 10, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214095,"parameters":{"coefficients":[7,2,2],"a":1,"b":0,"c":0,"rhs":4,"xs":[-1,0,1],"x0":2,"y0":19,"dy0":10,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[7,2,2],\"dy0\":10,\"rhs\":4,\"x0\":2,\"xs\":[-1,0,1],\"y0\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0096","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[8,3,-2],\"a\":1,\"b\":0,\"c\":0,\"rhs\":-3,\"xs\":[-1,0,1],\"x0\":-2,\"y0\":-6,\"dy0\":10}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([8, 3, -2], 1, 0, 0, -3, [-1, 0, 1], [[-1, 3, 7, -4, -1, 否], [0, 8, 3, -4, -1, 否], [1, 9, -1, -4, -1, 否]], -2, -6, 11, -6, 10, 是, 否, 否)","explanation_zh":"應用擴展核對=([8, 3, -2], 1, 0, 0, -3, [-1, 0, 1], [[-1, 3, 7, -4, -1, 否], [0, 8, 3, -4, -1, 否], [1, 9, -1, -4, -1, 否]], -2, -6, 11, -6, 10, 是, 否, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214096,"parameters":{"coefficients":[8,3,-2],"a":1,"b":0,"c":0,"rhs":-3,"xs":[-1,0,1],"x0":-2,"y0":-6,"dy0":10,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[8,3,-2],\"dy0\":10,\"rhs\":-3,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0097","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[9,-3,-1],\"a\":1,\"b\":0,\"c\":0,\"rhs\":-2,\"xs\":[-1,0,1],\"x0\":-1,\"y0\":12,\"dy0\":-1}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([9, -3, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, 11, -1, -2, 0, 是], [0, 9, -3, -2, 0, 是], [1, 5, -5, -2, 0, 是]], -1, 11, -1, 12, -1, 否, 是, 否)","explanation_zh":"應用擴展核對=([9, -3, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, 11, -1, -2, 0, 是], [0, 9, -3, -2, 0, 是], [1, 5, -5, -2, 0, 是]], -1, 11, -1, 12, -1, 否, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214097,"parameters":{"coefficients":[9,-3,-1],"a":1,"b":0,"c":0,"rhs":-2,"xs":[-1,0,1],"x0":-1,"y0":12,"dy0":-1,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[9,-3,-1],\"dy0\":-1,\"rhs\":-2,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0098","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[10,-2,0],\"a\":1,\"b\":0,\"c\":0,\"rhs\":0,\"xs\":[-1,0,1],\"x0\":0,\"y0\":10,\"dy0\":-2}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([10, -2, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, 12, -2, 0, 0, 是], [0, 10, -2, 0, 0, 是], [1, 8, -2, 0, 0, 是]], 0, 10, -2, 10, -2, 是, 是, 是)","explanation_zh":"應用擴展核對=([10, -2, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, 12, -2, 0, 0, 是], [0, 10, -2, 0, 0, 是], [1, 8, -2, 0, 0, 是]], 0, 10, -2, 10, -2, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214098,"parameters":{"coefficients":[10,-2,0],"a":1,"b":0,"c":0,"rhs":0,"xs":[-1,0,1],"x0":0,"y0":10,"dy0":-2,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[10,-2,0],\"dy0\":-2,\"rhs\":0,\"x0\":0,\"xs\":[-1,0,1],\"y0\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0099","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[11,-1,1],\"a\":1,\"b\":0,\"c\":0,\"rhs\":2,\"xs\":[-1,0,1],\"x0\":1,\"y0\":11,\"dy0\":1}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([11, -1, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, 13, -3, 2, 0, 是], [0, 11, -1, 2, 0, 是], [1, 11, 1, 2, 0, 是]], 1, 11, 1, 11, 1, 是, 是, 是)","explanation_zh":"應用擴展核對=([11, -1, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, 13, -3, 2, 0, 是], [0, 11, -1, 2, 0, 是], [1, 11, 1, 2, 0, 是]], 1, 11, 1, 11, 1, 是, 是, 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214099,"parameters":{"coefficients":[11,-1,1],"a":1,"b":0,"c":0,"rhs":2,"xs":[-1,0,1],"x0":1,"y0":11,"dy0":1,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[11,-1,1],\"dy0\":1,\"rhs\":2,\"x0\":1,\"xs\":[-1,0,1],\"y0\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261021-r04-initial-value-problem-0100","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","language":"zh-Hant","source_mko_ids":["mko-initial-value-problem"],"title_zh":"多項式初值候選","stem_zh":"整係數多項式候選與明示有限網格。操作=ivp-second-order-candidate-grid；資料={\"coefficients\":[12,0,2],\"a\":1,\"b\":0,\"c\":0,\"rhs\":5,\"xs\":[-1,0,1],\"x0\":2,\"y0\":21,\"dy0\":8}。請依 [coefficients,a,b,c,rhs,xs,residual_rows,x0,y_x0,y_prime_x0,expected_y,expected_y_prime,y_match,y_prime_match,finite_grid_and_initial_match] 欄序回答。","answer":{"kind":"expression","value":"([12, 0, 2], 1, 0, 0, 5, [-1, 0, 1], [[-1, 14, -4, 4, -1, 否], [0, 12, 0, 4, -1, 否], [1, 14, 4, 4, -1, 否]], 2, 20, 8, 21, 8, 否, 是, 否)","explanation_zh":"應用擴展核對=([12, 0, 2], 1, 0, 0, 5, [-1, 0, 1], [[-1, 14, -4, 4, -1, 否], [0, 12, 0, 4, -1, 否], [1, 14, 4, 4, -1, 否]], 2, 20, 8, 21, 8, 否, 是, 否)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不宣稱解存在或唯一；有限網格殘差不是全區間證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式初值候選"]},"variant":{"family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214100,"parameters":{"coefficients":[12,0,2],"a":1,"b":0,"c":0,"rhs":5,"xs":[-1,0,1],"x0":2,"y0":21,"dy0":8,"domain":"Polynomial_candidate_IVP","operation":"ivp-second-order-candidate-grid","semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[12,0,2],\"dy0\":8,\"rhs\":5,\"x0\":2,\"xs\":[-1,0,1],\"y0\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}